Giuseppe Briatico
email: giuseppe.bria@gmail.com

Hervé Bocherens

Denis Geraads

Rita T Melis

Margherita Mussi

A view from the highlands: the Pleistocene environments between 2.02 and 0.6 Ma at Melka Kunture (Upper Awash, Ethiopia) based upon stable isotope analysis

Pleistocene environments are among the most studied issues in paleoecology and human evolution research in eastern Africa. Many data have been recorded from archaeological sites located at low and medium elevations (≤ 1500 m), whereas few contexts are known at 2000 m and above. Here, we present a substantial isotopic study from Melka Kunture, a complex of prehistoric sites located at 2000 -2200 m above sea level in the Ethiopian highlands. We analyzed the stable carbon and oxygen composition of 308 faunal tooth enamel samples from sites dated between 2.02 and 0.6 Ma to investigate the animal diets and habitats. The carbon isotopic results indicate that the analyzed taxa had C4-dominated and mixed C3-C4 diets with no significant diachronic changes in terms of feeding behavior. This is consistent with faunal and phytolith analyses, which suggested environments characterized by open grasslands (with both C3 and C4 grasses), patches of bushes and thickets, and aquatic vegetation. However, palynological data documented mountain forests, woodlands, and high-elevation grasslands. The carbon isotopic comparison with eastern African localities shows that differences in elevation did not influence animal feeding strategies and habitat partitioning, even though plant species vary according to altitudinal gradients. In contrast, the oxygen isotopic comparison suggests significant differences consistent with the altitude effect. Our approach allowed us to detect diverse aspects of animal behavior, habitat, and vegetation that should be considered when reconstructing past environments.

Introduction

Over the past decades, intensive paleoanthropological research has shown that fossil records in eastern Africa preserve key information on the early stages of the evolutionary history of hominins. Diverse hypotheses have been developed to explain why and how the evolution of mammals is linked to climatic and environmental changes throughout the Pleistocene [START_REF] Potts | Evolution and climatic variability[END_REF][START_REF] Reed | Early hominid evolution and ecological change through the African Plio-Pleistocene[END_REF][START_REF] Demenocal | African climate change and faunal evolution during the Pliocene-Pleistocene[END_REF][START_REF] Trauth | High-and low-latitude controls and East African forcing of Plio-Pleistocene East African climate and early human evolution[END_REF][START_REF] Maslin | Plio-Pleistocene East African pulsed climate variability and its influence on early human evolution[END_REF][6][START_REF] Faith | Rethinking the ecological drivers of hominin evolution[END_REF] . These hypotheses have been based on the current understanding of the global paleoclimate records, as well as on the development of the East Africa Rift System, where tectonic events have generated significant regional variations in climate, hydrological landscape, and distribution of vegetation 8- 10 . The paleoenvironmental archives are mostly from Pleistocene archaeological and paleontological sites in eastern Africa at low and medium elevations (≤ 1500 m a.s.l.) (e.g., Olduvai Gorge, Lake Turkana Basin, Busidima Formation) [START_REF] Bedaso | Paleoenvironmental reconstruction of the Asbole fauna (Busidima Formation, Afar, Ethiopia)[END_REF][START_REF] Van Der Merwe | Isotopic ecology of fossil fauna from Olduvai Gorge at ca 1.8 Ma, compared with modern fauna[END_REF][START_REF] Ascari | Fossil herbivores and crocodiles as paleoclimatic indicators of environmental shifts from Bed I and Bed II times of the Olduvai Gorge, Tanzania. Palaeogeogr[END_REF][START_REF] Rivals | Dietary traits of the ungulates from the HWK EE site at Olduvai Gorge (Tanzania): Diachronic changes and seasonality[END_REF][START_REF] Uno | Large mammal diets and paleoecology across the Oldowan-Acheulean transition at Olduvai Gorge, Tanzania from stable isotope and tooth wear analyses[END_REF][START_REF] Negash | Dietary trends in herbivores from Shungura Formation, southwestern Ethiopia[END_REF][START_REF] Semaw | Co-occurrence of Acheulian and Oldowan artifacts with Homo erectus cranial fossils from Gona, Afar, Ethiopia[END_REF][START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] , whereas few archaeological sites are known at higher elevations (≥ 2000 m a.s.l.), such as Melka Kunture (hereafter MK) [START_REF] Chavaillon | Studies on the Early Paleolithic site of Melka Kunture, Ethiopia. Origines[END_REF][START_REF] Mussi | Une vue d ensemble sur Melka Kunture, grand complexe de sites pléistocènes dans la vallée supérieure de l Awash (Ethiopie)[END_REF][START_REF] Mussi | 2.000.000 y 5.000[END_REF] , Melka Wakena [START_REF] Hovers | The expansion of the Acheulian to the Southeastern Ethiopian Highlands: Insights from the new early Pleistocene site-complex of Melka Wakena[END_REF] , Gadeb [START_REF] Clark | Transitions: Homo erectus and the acheulian: the Ethiopian sites of Gadeb and the middle Awash[END_REF] , Chilga Kernet [START_REF] Todd | Chilga Kernet: an Acheulean landscape on Ethiopia's western plateau[END_REF] , Fanta [START_REF] Lanzarone | Examining fluvial stratigraphic architecture using ground-penetrating radar at the Fanta stream fossil and archaeological site, Central Ethiopia[END_REF] , and Mount Dendi [START_REF] Vogelsang | When Hominins Conquered Highlands -an Acheulean Site at 3000 m a.s.l. on Mount Dendi/Ethiopia[END_REF] in Ethiopia; Kilombe [START_REF] Gowlett | At the heart of the African Acheulean: the physical, social and cognitive landscapes of Kilombe[END_REF] in Kenya; and Isimila [START_REF] Willoughby | The Middle and Later Stone Age in the Iringa Region of southern Tanzania[END_REF] in Tanzania. Among the Ethiopian sites at high elevations, MK yielded a long and well-assessed geoarchaeological sequence spanning from 2.02 Ma to the Holocene [START_REF] Morgan | A chronological framework for a long and persistent archaeological record: Melka Kunture, Ethiopia[END_REF][START_REF] Tamrat | Magnetostratigraphic study of the Melka Kunture archaeological site (Ethiopia) and its chronological implications[END_REF][START_REF] Perini | Magnetochronology and age models of deposition of the Melka Kunture stratigraphic sequence (Upper Awash, Ethiopia) and age assessments of the main archaeological levels therein contained[END_REF][START_REF] Galán | Age and formation processes of an Acheulean site with extensive accumulation of large cutting tools: Garba I (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Mutri | review. Living on the Awash. Everyday life and activities of a Later Stone Age community[END_REF] , where the archaeological layers that recorded faunal and human remains, pollen, phytoliths, and ichnological evidence are coupled with accurate stratigraphic positioning. Here, we present and discuss a comprehensive isotopic study of stable carbon and oxygen in 308 fauna tooth enamel samples from MK at sites dated between 2.02 and 0.6 Ma. We utilize the isotopic data to provide information on the diet and habitat of the Pleistocene fauna and evaluate possible dietary variability over time. We integrate our interpretation with other proxies (e.g., faunal taxonomy, pollen, and phytolith data) to better characterize the environmental conditions of the site area. Additionally, we compare our isotopic results with other published data from eastern African sites at low and medium elevations to evaluate whether the differences in altitude and vegetation composition influenced animal feeding strategies and behavior. This is the first substantial isotopic study along the MK archaeological sequence. Archeological background MK is a cluster of prehistoric sites located approximately 50 km southwest of Addis Ababa, on the western shoulder of the Main Ethiopian Rift, between 2000 and 2200 m a.s.l. [START_REF] Chavaillon | Studies on the Early Paleolithic site of Melka Kunture, Ethiopia. Origines[END_REF][START_REF] Mussi | Une vue d ensemble sur Melka Kunture, grand complexe de sites pléistocènes dans la vallée supérieure de l Awash (Ethiopie)[END_REF][START_REF] Mussi | 2.000.000 y 5.000[END_REF] , extending over 100 km 2 on the banks of the Upper Awash River (Fig. 1A -B). The stratigraphic sequence was controlled by the interplay between variable tectonic, fluvial, and volcanic activities placed in a low-energy floodplain [START_REF] Kieffer | Volcanic markers in coarse alluvium at Melka Kunture (Upper Awash, Ethiopia)[END_REF][START_REF] Raynal | Garba IV and the Melka Kunture Formation. A preliminary lithostratigraphic approach[END_REF][START_REF] Salvini | Topographic and geological mapping in the prehistoric area of Melka Kunture (Ethiopia)[END_REF][START_REF] Pioli | Volcaniclastic sedimentation in a closed, marginal rift basin: the case of the Melka Kunture area (upper Awash, Ethiopia)[END_REF] . The chronology has been assessed by combining magnetochronological studies [START_REF] Tamrat | Magnetostratigraphic study of the Melka Kunture archaeological site (Ethiopia) and its chronological implications[END_REF][START_REF] Perini | Magnetochronology and age models of deposition of the Melka Kunture stratigraphic sequence (Upper Awash, Ethiopia) and age assessments of the main archaeological levels therein contained[END_REF] , radiometric dating (K-Ar and Ar/Ar) [START_REF] Morgan | A chronological framework for a long and persistent archaeological record: Melka Kunture, Ethiopia[END_REF] , Electron Spin Resonance spectrometry on quartz grains [START_REF] Galán | Age and formation processes of an Acheulean site with extensive accumulation of large cutting tools: Garba I (Melka Kunture, Upper Awash, Ethiopia)[END_REF] , and radiocarbon dating [START_REF] Mutri | review. Living on the Awash. Everyday life and activities of a Later Stone Age community[END_REF] . The archaeological sequence begins with the Early Stone Age, with Oldowan lithic tools (2.02 Ma) and Early, middle, and final Acheulean artifacts (1.95 -0.6 Ma) [START_REF] Perini | Magnetochronology and age models of deposition of the Melka Kunture stratigraphic sequence (Upper Awash, Ethiopia) and age assessments of the main archaeological levels therein contained[END_REF][START_REF] Galán | Age and formation processes of an Acheulean site with extensive accumulation of large cutting tools: Garba I (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Gallotti | The Unknown Oldowan: ~1.7-Million-Year-Old Standardized Obsidian Small Tools from Garba IV, Melka Kunture, Ethiopia[END_REF][START_REF] Gallotti | Two Acheuleans, two humankinds: From 1.5 to 0.85 Ma at Melka Kunture (Upper Awash, Ethiopian highlands)[END_REF][START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Mussi | review. Early Homo erectus lived at high altitudes and produced both Oldowan and Acheulean tools[END_REF] . The Early Middle Stone Age is documented at ~200 ka, whereas the Late Stone Age, mainly found in surface dispersions, is now investigated in a stratified deposit dated to the Holocene [START_REF] Mussi | 2.000.000 y 5.000[END_REF][START_REF] Mutri | review. Living on the Awash. Everyday life and activities of a Later Stone Age community[END_REF] . Fossil remains of Homo erectus, of a hominin related to H. heidelbergensis, and of an archaic H. sapiens were discovered within clear stratigraphic contexts and are directly associated with lithic technocomplexes and faunal remains [START_REF] Mussi | review. Early Homo erectus lived at high altitudes and produced both Oldowan and Acheulean tools[END_REF][START_REF] Cabec | Insights into the palaeobiology of an early Homo infant: multidisciplinary investigation of the GAR IVE hemi-mandible, Melka Kunture[END_REF][START_REF] Vincenzo | The massive fossil humerus from the Oldowan horizon of Gombore I, Melka Kunture (Ethiopia,> 1.39 Ma)[END_REF][START_REF] Profico | Filling the gap. Human cranial remains from Gombore II (Melka Kunture, Ethiopia; ca. 850 ka) and the origin of Homo heidelbergensis[END_REF][START_REF] Mussi | Ethiopia): a MSA site with archaic Homo sapiens remains revisited[END_REF] .

The faunal assemblage is dominated by Hippopotamus cf. amphibius, a large-sized hippo similar to the modern one. In contrast, the dwarf hippo (Hippopotamus cf. aethiopicus) is less represented. Alcelaphini is the most common bovid tribe recorded so far, whereas a few specimens refer to buffaloes (Bovini), kobs (Reduncini), and gazelles (Antilopini). The development of endemic mammal sub-species (e.g., Connochaetes gentry leptoceras and Damaliscus strepsiceras) points to a relative degree of isolation of the highlands due to the topographic relief. Equids are represented by the bones and teeth of Equus and Hipparion s.l., but the taxonomic identification was difficult. Suids are quite rare, with remains of Kolpochoerus, Metridiochoerus, and Phacochoerus sp. For giraffids, the record includes Giraffa sp. and the short-limbed Sivatherium maurusium. A few remains of elephants, rhinos, and crocodiles are recorded. The baboon Theropithecus cf. oswaldi, a rare genus at MK, is the only non-human primate. Carnivores are extremely rare, consisting of a few remains of Pseudocivetta ingens, Megantereon, Lupulella, and Crocuta sp. In addition, two rodent species have been identified as Oenomys kunturensis and Tachyoryctes konjitae [START_REF] Geraads | Studies on the Early Paleolithic site of Melka Kunture, Ethiopia[END_REF][START_REF] Geraads | Ethiopia: Early Pleistocene Faunas of the Ethiopian Highlands[END_REF] (Tab. S1). Mammals, including hominins, are also attested by footprints in many Pleistocene horizons within the Gombore gully [START_REF] Altamura | A Middle Pleistocene hippo tracksite at Gombore II-2 (Melka Kunture, Upper Awash, Ethiopia). Palaeogeogr., Palaeoclimatol[END_REF][START_REF] Altamura | Archaeology and ichnology at Gombore II-2, Melka Kunture, Ethiopia: everyday life of a mixed-age hominin group 700,000 years ago[END_REF][START_REF] Altamura | Ichnological and archaeological evidence from Gombore II OAM, Melka Kunture, Ethiopia: an integrated approach to reconstruct local environments and biological presences between 1.2-0.85[END_REF] . All over the MK sequence, the vegetation belonged to the Dry evergreen Afromontane Forest and grassland complex (DAF). This vegetation type ranged from forest to grassland and bushland, with variations in the distribution of trees, grasses, and herbs over time (Tab. S2; Fig. S1). The DAF complex currently characterizes the higher mountain ranges of eastern Africa with a cool and rainy climate. The plant species greatly differs from the warmer and drier plant and tree species of the African savanna at lower elevations [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF] . In Ethiopia, the DAF vegetation is found between 1800 and 3000 m a.s.l. In the MK area, where it would still develop, it has disappeared in modern times due to anthropic impact [START_REF] Friis | Atlas of the Potential Vegetation of Ethiopia[END_REF][START_REF] Mairal | Geographic barriers and Pleistocene climate change shaped patterns of genetic variation in the Eastern Afromontane biodiversity hotspot[END_REF] . According to the Ethiopian Meteorological Institute [START_REF]Ethiopian Meteorological Institute -Data and climatology directoret (Federal Democratic Republic of Ethiopia, Ministry of water and energy)[END_REF] , in the Upper Awash Valley at ~2000 m and more, 1000 mm of total rainfall was recorded in 2022, with mean maximum and minimum temperatures of 26°C and 11°C, respectively (Fig. S2). Modern rainfall distribution is bimodal and influenced by the Indian Monsoon and seasonal movement of the Inter Tropical Convergence Zone (ITCZ), with a short rainy season (Belg) from February to May, long summer rains (Kiremt) from June to September, and a dry season (Bega) from October to January [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF][START_REF] Seleshi | Recent changes in rainfall and rainy days in Ethiopia[END_REF] . Figure 1. (A) Location of MK in the Upper Awash Valley of Ethiopia; (B) map of the archaeological sites involved in this study; (C) MK stratigraphic sequences: the sedimentary interpretation of Garba, Gombore, and Simbiro is from one of us (R.T.M.), whereas the succession from Karre is according to Chavaillon and Piperno [START_REF] Chavaillon | Studies on the Early Paleolithic site of Melka Kunture, Ethiopia. Origines[END_REF] . Chronology is from Morgan et al. [START_REF] Morgan | A chronological framework for a long and persistent archaeological record: Melka Kunture, Ethiopia[END_REF] , and Perini et al. [START_REF] Perini | Magnetochronology and age models of deposition of the Melka Kunture stratigraphic sequence (Upper Awash, Ethiopia) and age assessments of the main archaeological levels therein contained[END_REF] .

Results

We report on δ 13 C and δ 18 O values of 6 mammalian families (Hippopotamidae, Bovidae, Equidae, Suidae, Hyaenidae, and Giraffidae) in order of their specimen abundance and combined with each archaeological locality, stratigraphic level, and chronology (Fig. 1C; Tab. S3). Overall, 308 isotopic results are reported with median and average values and ranges for each mammalian family. The sampling strategy reflects relative faunal abundance. Indeed, Hippopotamidae and Bovidae are the most abundant taxa in faunal assemblage and isotopic datasets.

Hippopotamidae. The median δ 13 C value of Hippopotamus cf. amphibius (n = 138) was -0.07 ‰, ranging from -6.6 ‰ to + 2.8 ‰. The δ 18 O values ranged from + 18.2 ‰ to + 30.9 ‰ with a median δ 18 O value of + 24.3 ‰ (Fig. 2A). The statistics indicate a non-normal distribution of δ 13 C values (Shapiro-Wilk Test: p = 0.0001), whereas δ 18 O values follow a normal distribution (Shapiro-Wilk Test: p = 0.9782), which were also confirmed by visual Kernel density and Normal Quantile Plot observations. Statistically significant differences in hippo δ 13 C (Wilcoxon/Kruskal-Wallis Rank-Sum Test: p = 0.0036) and δ 18 O values (Anova Test: p = 0.0064) were observed among the archaeological localities. Bovidae. The analyzed bovids (n = 105) comprise five tribes (Alcelaphini, Antilopini, Bovini, Hippotragini, and Reduncini) and others not assigned to the tribe and species level.

Alcelaphini. Samples from alcelaphin bovids (n = 55) had a median δ 13 C value of + 2.9 ‰ with values ranging from -2.2 ‰ to + 5.2 ‰, whereas the median δ 18 O value was + 27.5 ‰ with values ranging from + 21 ‰ to + 33.1 ‰ (Fig. S3A). Bovidae sensu lato. This group comprises bovid specimens (n = 37) not identified at the tribe and species levels. The median δ 13 C value was + 1.3 ‰ with values ranging from -4.6 ‰ to + 5 ‰, whereas the median δ 18 O value was + 25.3 ‰, ranging from + 20.2 ‰ to + 31.7 ‰ (Fig. S3F).

Overall, bovid δ 13 C values ranged from -9.7 ‰ to + 5.2 ‰ (median = + 1.9 ‰) (Fig. 2B). The δ 13 C values do not follow a normal distribution (Shapiro-Wilk Test: p = 0.0001), and the Wilcoxon/Kruskal-Wallis Rank-Sum Test (p = 0.0124) showed differences in mean δ 13 C values over time. Alcelaphini and Bovidae sensu lato showed higher δ 13 C values than those of Antilopini, Bovini, Hippotragini, and Reduncini. However, we noted that the small sample size of Antilopini, Bovini, Hippotragini, and Reduncini limited the effectiveness of statistical analysis. Normal Quantile Plot and Shapiro-Wilk Test (p = 0.8177) indicate a normal distribution of δ 18 O values, with statistically significant differences (Anova Test: p = 0.0001) among the archaeological sites and levels. Equidae. This group comprises specimens mostly not assigned to mammalian tribes and species, whereas a few samples belong to Equus (n = 4) and Hipparion (n = 2) sp. The overall median δ 13 C value of equid samples (n = 45) was + 0.1 ‰, with values ranging from -7.6 ‰ to + 6.4 ‰. The median δ 18 O value was + 26.7 ‰, ranging from + 23.1 ‰ to + 31.3 ‰ (Fig. 2C). Both δ 13 C and δ [START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] O values are normally distributed (Shapiro-Wilk Tests: p = 0.0798; p = 0.1170, respectively).

Across the sites, statistically significant variability in δ 13 C and δ 18 O values was observed (Anova Tests: p = 0.0427; p = 0.0179, respectively).

Suidae. This family (n = 12) comprises a few samples of Metridochoerus (n = 4), Kolpochoerus (n = 2), and other suids (n = 6). Overall, average δ 13 C value was + 0.05 ‰, ranging from -2.3 ‰ to + 3.3 ‰, whereas median δ 18 O value was + 26.8 ‰, ranging from + 22.6 ‰ to + 29.6 ‰ (Fig. 2D).

Hyaenidae. Five samples of hyena had an average δ 13 C value of + 0.1 ‰ (range between -3.3 ‰ and + 2.5 ‰), whereas average δ 18 O value was + 25.5 ‰, ranging from + 24.3 ‰ to + 27 ‰ (Fig. 2E).

Giraffidae. Three samples of giraffids show an average δ 13 C value of + 2.2 ‰ (range between + 1.9 ‰ and + 2.7 ‰). The average δ 18 O value was + 27.4 ‰, ranging from + 24.5 ‰ to + 29 ‰ (Fig. 2F). Due to the limited number of Suidae, Giraffidae, and Hyaenidae samples, a robust statistical evaluation was impossible. Therefore, our interpretation is only a preliminary one.

org). Discussion

The δ 13 C values suggest that the hippos had a diet dominated by C4 forage, although some individuals showed lower δ 13 C values, which reflects the opportunistic consumption of C3 grasses, fruits, aquatic vegetation, or a combination of these resources [START_REF] Ansell | Feeding habits of Hippopotamus amphibius Linn[END_REF][START_REF] Boisserie | Diets of modern and late Miocene hippopotamids: evidence from carbon isotope composition and microwear of tooth enamel[END_REF][START_REF] Cerling | Stable isotope ecology of the common hippopotamus[END_REF][START_REF] Souron | Intra-tooth isotopic profiles of canines from extant Hippopotamus amphibius and late Pliocene hippopotamids (Shungura Formation, Ethiopia): insights into the seasonality of diet and climate[END_REF][START_REF] Briatico | Isotopic insights into the Early Acheulean (1.95 Ma -1.66 Ma) high-elevation paleoenvironments at Melka Kunture (Upper Awash Valley, Ethiopia)[END_REF][START_REF] Briatico | Combining stable isotopes and ichnological evidence: the case study of Hippopotamus cf. amphibius from Melka Kunture (Upper Awash, Ethiopia)[END_REF] . Hippos are particularly suitable for tracking local paleovegetation [START_REF] Harris | Stable isotope ecology of fossil hippopotamids from the Lake Turkana Basin of East Africa[END_REF] . Indeed, modern hippos are known to spend most of their daily time close to a body of water, even though they regularly travel some kilometers to forage at night for preferred grass [START_REF] Eltringham | The Hippos[END_REF] . At MK, there is direct evidence of a 700-ka hippo trail produced by H. cf. amphibius, which is consistent with modern hippo behavior [START_REF] Altamura | A Middle Pleistocene hippo tracksite at Gombore II-2 (Melka Kunture, Upper Awash, Ethiopia). Palaeogeogr., Palaeoclimatol[END_REF][START_REF] Altamura | Archaeology and ichnology at Gombore II-2, Melka Kunture, Ethiopia: everyday life of a mixed-age hominin group 700,000 years ago[END_REF][START_REF] Briatico | Combining stable isotopes and ichnological evidence: the case study of Hippopotamus cf. amphibius from Melka Kunture (Upper Awash, Ethiopia)[END_REF] . Bovid and equid δ [START_REF] Ascari | Fossil herbivores and crocodiles as paleoclimatic indicators of environmental shifts from Bed I and Bed II times of the Olduvai Gorge, Tanzania. Palaeogeogr[END_REF] C values indicate that they had C4 diets, with lower δ 13 C values interpreted as the outcome of mixed

C D E F A B C3-C4 diet.
In contrast, suids and giraffids had full C4 diets, whereas hyena δ 13 C values reflect the various isotopic signatures of their prey [START_REF] Bocherens | Trophic level isotopic enrichment of carbon and nitrogen in bone collagen: Case studies from recent and ancient terrestrial ecosystems[END_REF] , which had both C4 and mixed C3-C4 diets (Fig. 2). The consumption of grazers and mixed feeders by hyenas is confirmed by carnivore tooth marks found on a hippo scapula and tibia at Gombore II-2, currently dated to 750 ka [START_REF] Perini | Magnetochronology and age models of deposition of the Melka Kunture stratigraphic sequence (Upper Awash, Ethiopia) and age assessments of the main archaeological levels therein contained[END_REF][START_REF] Altamura | A Middle Pleistocene hippo tracksite at Gombore II-2 (Melka Kunture, Upper Awash, Ethiopia). Palaeogeogr., Palaeoclimatol[END_REF][START_REF] Altamura | Archaeology and ichnology at Gombore II-2, Melka Kunture, Ethiopia: everyday life of a mixed-age hominin group 700,000 years ago[END_REF] .

The δ [START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] O values of all the taxa from different localities and stratigraphic levels point to variations in the oxygen isotopic composition of animal drinking water sources, food water, and possibly climatic and environmental changes over time. In addition, hippos showed lower δ 18 O values than other taxa (e.g., bovid, equid, suid, and giraffid), which is consistent with the expected differences between semiaquatic and terrestrial habitats (Fig. 3), as already evidenced at MK [START_REF] Briatico | Isotopic insights into the Early Acheulean (1.95 Ma -1.66 Ma) high-elevation paleoenvironments at Melka Kunture (Upper Awash Valley, Ethiopia)[END_REF][START_REF] Briatico | Combining stable isotopes and ichnological evidence: the case study of Hippopotamus cf. amphibius from Melka Kunture (Upper Awash, Ethiopia)[END_REF][START_REF] Bocherens | Isotopic biogeochemistry ( 13 C, 18 O) of mammalian enamel from African Pleistocene hominid sites[END_REF] and elsewhere [START_REF] Clementz | Differentiating aquatic mammal habitat and foraging ecology with stable isotopes in tooth enamel[END_REF][START_REF] Clementz | Identifying aquatic habits of herbivorous mammals through stable isotope analysis[END_REF][START_REF] Harris | Stable isotope ecology of fossil hippopotamids from the Lake Turkana Basin of East Africa[END_REF] . The isotopic data agree with the analysis of faunal assemblage, suggesting environments characterized by open grasslands, patches of bushes and thickets, and aquatic vegetation. No evidence of pure browsers is attested by either faunal or isotopic analyses [START_REF] Geraads | Ethiopia: Early Pleistocene Faunas of the Ethiopian Highlands[END_REF][START_REF] Briatico | Isotopic insights into the Early Acheulean (1.95 Ma -1.66 Ma) high-elevation paleoenvironments at Melka Kunture (Upper Awash Valley, Ethiopia)[END_REF][START_REF] Briatico | Combining stable isotopes and ichnological evidence: the case study of Hippopotamus cf. amphibius from Melka Kunture (Upper Awash, Ethiopia)[END_REF] . This is probably the result of taphonomic issues and the consequent sampling bias. In contrast, hippos are a significant component of the MK paleontological record because of their semi-aquatic habitat, which allows for a greater abundance within fluvial deposits. Caution is required when paleoenvironmental reconstruction is based on limited faunal remains. Pollen and phytolith analyses provide further environmental insights [START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF] . Within the Dry evergreen Afromontane Forest and grassland complex, which developed throughout the MK archaeological sequence, the proportion of mountain forests, woodlands, and grasslands was subjected to significant fluctuations. The two periods when mountain grasslands were more extensive than forests are those evidenced at Garba IVD (1.95 Ma) and Garba IB (0.6 Ma) (Fig. S1), which are far from each other in time. Among various sites, there is ample evidence of phases with much more wooded and humid vegetation, with no trend toward increased aridity along the MK sequence. The changes should rather be seen as related to the global glacial and interglacial cycles [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF] . The elevation also allowed for the development of diverse environments with C4 grass but also C3 types of grass, which at MK have been recorded by phytolith analysis [START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF] , and currently become dominant above 2000 m a.s.l. [START_REF] Tieszen | The Distribution of C3 and C4 Grasses and Carbon Isotope Discrimination along an Altitudinal and Moisture Gradient in Kenya[END_REF] . This is well-tracked by hippo, bovid, and equid dietary preferences in several archaeological localities with distinct chronologies, such as at Gombore IA (~1.6 Ma), Gombore Iγ (1.51 Ma), Simbiro III MS B (~1.2 Ma), Gombore II-1 (1.0 Ma), and Gombore II-2 (0. 75 Ma). Changes in vegetation composition did not have a major impact on the feeding behavior of mammals, which selected the preferred plant species in the varied mosaic environments. Furthermore, we cannot rule out the possibility that C3 isotopic signals are associated with seasonal variations in the proportion of C3 and C4 grasses reflected in the animal diet. This was demonstrated elsewhere by Souron et al. [START_REF] Souron | Intra-tooth isotopic profiles of canines from extant Hippopotamus amphibius and late Pliocene hippopotamids (Shungura Formation, Ethiopia): insights into the seasonality of diet and climate[END_REF] using intra-tooth isotopic profiles. In order to test whether the differences in elevation and vegetation influenced the animal feeding strategies and habitats, we compared δ 13 C and δ 18 O values (n = 1.064) of fauna tooth enamel from published data of eastern African sites at low and medium altitudes (≤ 1500 m a.s.l.), such as Olduvai Gorge [START_REF] Van Der Merwe | Isotopic ecology of fossil fauna from Olduvai Gorge at ca 1.8 Ma, compared with modern fauna[END_REF][START_REF] Ascari | Fossil herbivores and crocodiles as paleoclimatic indicators of environmental shifts from Bed I and Bed II times of the Olduvai Gorge, Tanzania. Palaeogeogr[END_REF][START_REF] Rivals | Dietary traits of the ungulates from the HWK EE site at Olduvai Gorge (Tanzania): Diachronic changes and seasonality[END_REF][START_REF] Uno | Large mammal diets and paleoecology across the Oldowan-Acheulean transition at Olduvai Gorge, Tanzania from stable isotope and tooth wear analyses[END_REF] , Lake Turkana Basin [START_REF] Negash | Dietary trends in herbivores from Shungura Formation, southwestern Ethiopia[END_REF][START_REF] Harris | Stable isotope ecology of fossil hippopotamids from the Lake Turkana Basin of East Africa[END_REF][START_REF] Braun | Early hominin diet included diverse terrestrial and aquatic animals 1.95 Ma in East Turkana, Kenya[END_REF] , and Busidima Formation [START_REF] Bedaso | Paleoenvironmental reconstruction of the Asbole fauna (Busidima Formation, Afar, Ethiopia)[END_REF] , with isotopic data from the higher elevation site of MK presented in this study. This isotopic dataset includes 6 faunal families (Hippopotamidae, Bovidae, Equidae, Suidae, Giraffidae, and Hyaenidae) dated between 2.1 and 0.6 Ma (Tab. S4) (Supplementary C). The comparison of δ 13 C values indicates a high degree of overlap in terms of dietary habits, with average diets dominated by C4 and mixed C3-C4 vegetation (Fig. 4). The only exception is represented by some bovids (Tragelaphini and Antilopini) and giraffids (Giraffa s.l., G. cf. stillei, G. cf. jumae) from the Lake Turkana Basin, Olduvai Gorge, and Busidima Formation, which only consumed C3 vegetation (Fig. S5). The median δ 13 C value of MK fauna is higher than those of other sites, except for the hippos from MK and Olduvai Gorge that show δ 13 C values close to each other, and the equids from MK and Lake Turkana that have equal median δ 13 C values (Tab. S5, Fig. S5). Based on carbon isotopic analysis, we argue that differences in elevation did not influence animal feeding strategies and habitat partitioning, even though plant species vary according to altitudinal gradients [START_REF] Tieszen | The Distribution of C3 and C4 Grasses and Carbon Isotope Discrimination along an Altitudinal and Moisture Gradient in Kenya[END_REF] . Furthermore, we emphasize that carbon isotopic data alone is not sufficient to directly reconstruct the paleoenvironments. Indeed, at MK, δ 13 C values suggest the presence of extended grasslands in the landscape. However, there is evidence from pollen analysis that the local mountain environment was characterized by a variable proportion through time of forests, woodlands, and grasslands (with both C3 and C4 grasses) [START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF] .

Comparing δ [START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] O values, we observe that taxa from MK had lower δ 18 O values than those from the other localities (Tab. S6, Fig. S6), which is consistent with the altitude effect. This is attributed to the progressive condensation of atmospheric vapor and rainout along the mountain slopes and its cool-off, with a consequent loss of H2 [START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] O in the form of rain from cloud moisture. As a result, high-altitude precipitation shows lower δ 18 O values than low-altitude precipitation. The altitude effect is reinforced by the difference in temperatures between low-and high-elevation localities 72,73,74 . Nevertheless, when interpreting δ 18 O values from fossil tooth enamel as a paleoclimatic proxy, we should consider that many factors (continentality, altitude, temperature, seasonality, rainfall amount, and relative humidity) may influence δ [START_REF] Mercader | Earliest Olduvai hominins exploited unstable environments ~2 million years ago[END_REF] O variability, making it difficult to establish a clear and direct link with past temperatures [START_REF] Pederzani | Oxygen Isotopes in Bioarchaeology: Principles and Applications, Challenges and Opportunities[END_REF] .

Conclusion

Isotopic analyses of fauna tooth enamel from MK provide information on feeding behavior and their adaptation. Overall, between 2.02 and 0.6 Ma, the sampled taxa show a low variability in foraging strategies, which are all within the range of C4 and mixed C3-C4 diets. The DAF vegetation changed over the Lower and Middle Pleistocene times, with variable proportions of mountain plant species [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF] , including both C3 and C4 grasses [START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF] , but this did not influence animal feeding habits. The analyzed mammals were able to select the preferred plants and successfully adapt to the mountain ecosystems. Comparisons with δ 13 C values from eastern African sites at low, medium (≤ 1500 m), and high elevations (≥ 2000 m) point to variability in the isotopic signal but not in the animal diets, which is constantly within the range of C4 and mixed C3-C4 diet. The plant species of the African lowlands are different from those at high-elevation [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF][START_REF] Tieszen | The Distribution of C3 and C4 Grasses and Carbon Isotope Discrimination along an Altitudinal and Moisture Gradient in Kenya[END_REF] , but the animal habitat and behavior in selecting food were not influenced along the altitudinal gradient by the changing vegetation.

The δ 18 O values recorded significant differences, which were explained as resulting from the altitude effect. The climate of modern-day Ethiopia is influenced by variable altitudes, with mountains up to 4000 m and more, high plateaus between 1300 and 3000 m, and lowlands stretching to sea level. These differences are also reflected in diverse local climates [START_REF] Gonfiantini | The altitude effect on the isotopic composition of tropical rains[END_REF] . Between 2.02 -0.6 Ma, the human presence at MK is directly recorded by paleoanthropological remains [START_REF] Mussi | review. Early Homo erectus lived at high altitudes and produced both Oldowan and Acheulean tools[END_REF][START_REF] Cabec | Insights into the palaeobiology of an early Homo infant: multidisciplinary investigation of the GAR IVE hemi-mandible, Melka Kunture[END_REF][START_REF] Vincenzo | The massive fossil humerus from the Oldowan horizon of Gombore I, Melka Kunture (Ethiopia,> 1.39 Ma)[END_REF][START_REF] Profico | Filling the gap. Human cranial remains from Gombore II (Melka Kunture, Ethiopia; ca. 850 ka) and the origin of Homo heidelbergensis[END_REF][START_REF] Mussi | Ethiopia): a MSA site with archaic Homo sapiens remains revisited[END_REF] , as well as by lithic artifacts [START_REF] Galán | Age and formation processes of an Acheulean site with extensive accumulation of large cutting tools: Garba I (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Mutri | review. Living on the Awash. Everyday life and activities of a Later Stone Age community[END_REF][START_REF] Gallotti | The Unknown Oldowan: ~1.7-Million-Year-Old Standardized Obsidian Small Tools from Garba IV, Melka Kunture, Ethiopia[END_REF][START_REF] Gallotti | Two Acheuleans, two humankinds: From 1.5 to 0.85 Ma at Melka Kunture (Upper Awash, Ethiopian highlands)[END_REF][START_REF] Mussi | A surge in obsidian exploitation more than 1.2 million years ago at Simbiro III (Melka Kunture, Upper Awash, Ethiopia)[END_REF][START_REF] Mussi | review. Early Homo erectus lived at high altitudes and produced both Oldowan and Acheulean tools[END_REF] and fossil footprints [START_REF] Altamura | A Middle Pleistocene hippo tracksite at Gombore II-2 (Melka Kunture, Upper Awash, Ethiopia). Palaeogeogr., Palaeoclimatol[END_REF][START_REF] Altamura | Archaeology and ichnology at Gombore II-2, Melka Kunture, Ethiopia: everyday life of a mixed-age hominin group 700,000 years ago[END_REF][START_REF] Altamura | Ichnological and archaeological evidence from Gombore II OAM, Melka Kunture, Ethiopia: an integrated approach to reconstruct local environments and biological presences between 1.2-0.85[END_REF] . Together with the isotopic results, this allows speculating on human behavior and adaptation. The hominins were clearly able to live and thrive in mountain mosaic ecosystems [START_REF] Bonnefille | Variability in the Mountain Environment at Melka Kunture Archaeological Site, Ethiopia, During the Early Pleistocene (~1.7 Ma) and the Mid-Pleistocene Transition[END_REF][START_REF] Mussi | The environment of the Ethiopian highlands at the Mid Transition: Fauna, flora, and hominins in the 850-700 ka sequence at Gombore II (Melka Kunture)[END_REF] , likely exploiting the diverse food resources provided by the DAF vegetation spectrum. The detailed paleoecological reconstruction is the outcome of the implementation of multiple methodologies, such as stable isotopes, pollen, phytoliths, and faunal taxonomy. Based on our experience with an integrative and complementary approach, caution should be exercised when paleoenvironmental interpretations are determined instead by a single proxy or analyses that are limited by taphonomic bias. Material and methods Stable isotope analyses. We collected fauna tooth enamel samples (n = 308) from several archaeological localities within the MK complex of sites to analyze the stable carbon and oxygen isotopic compositions. The specimens include Artiodactyla (Hippopotamidae, Bovidae, Suidae, Giraffidae), Perissodactyla (Equidae), and Carnivora (Hyaenidae) in order to represent the full dietary diversity. Samples collection, preparation, and isotopic analyses were performed in accordance with the internal lab protocol of the Biogeology Research Group of the University of Tübingen (Germany). Enamel was sampled using a drilling device equipped with a diamondtipped bit to obtain 12 -15 mg of powder. Powdered samples were soaked in 2 -3 % NaOCl for 24 h at 20 °C to oxidize organic residues and rinsed thrice with Millipore water (Milli-Q H 2 O) to remove all NaOCl. The remaining samples were treated with 0.1 M acetic acid-calcium acetate buffer (pH = 4.66) for 24 h at 20 °C to remove exogenous carbonate. Samples were rinsed thrice with Milli-Q H 2 O and placed in an oven to dry at 40 °C for 72 hours. Only 2.5 -3 mg of structural carbonate was subjected to Isotopic Ratio Mass Spectrometry (IRMS). Further details are provided in Supplementary A and B. Statistical analysis. Statistical tests were performed using JMP 17 Pro (licensed by Sapienza University of Rome), with the significance level set at p = 0.05. The Shapiro-Wilk test was used to verify whether the data followed a normal distribution. When the hypothesis of the normal distribution was accepted, parametric tests such as the t-test and the analysis of variance (ANOVA) test were used to test whether the δ 13 C and δ 18 O values significantly differed over time for each archaeological site. For the same purpose, non-parametric tests such as the Mann-Whitney U-test (also known as Wilcoxon Rank Sum Test) or the Kruskal-Wallis test were used when the hypothesis of the normal distribution was rejected. The t-test and Mann-Whitney Utest were applied with two groups or levels, whereas ANOVA and Kruskal-Wallis tests were applied with three or more groups.

  Antilopini. Two samples of Antilopini had δ[START_REF] Ascari | Fossil herbivores and crocodiles as paleoclimatic indicators of environmental shifts from Bed I and Bed II times of the Olduvai Gorge, Tanzania. Palaeogeogr[END_REF] C values of -9.7 ‰ and -2.8 ‰ (average = -6.2 ‰). The δ 18 O values were + 26.5 ‰ and + 30 ‰ (average = + 28.2 ‰) (Fig.S3B). Bovini. Bovin samples (n = 5) had δ 13 C values ranging from -2.3 ‰ to + 3.8 ‰ (average = + 0.3 ‰). The δ 18 O values ranged from + 26 ‰ to + 31.1 ‰ (average = + 27.9 ‰) (Fig. S3C). Hippotragini. Samples of Hippotragini (n = 3) had δ 13 C values ranging from -2 ‰ to + 2.1 ‰ (average = -0.3 ‰). The δ 18 O values ranged from + 24.4 ‰ to + 29.1 ‰ (average = + 27 ‰) (Fig. S3D). Reduncini. Samples of Reduncini (n = 3) had δ 13 C values ranging from -5.5 ‰ to + 1.4 ‰ (average = -1.1 ‰). The δ 18 O values ranged from + 24.3 ‰ to + 30.7 ‰ (average = + 27.1 ‰) (Fig. S3E).

Figure 2 .

 2 Figure 2. Box and whisker plots of δ 13 C and δ 18 O values of hippos (A), bovids (B), equids (C), suids (D), hyenas (E), and giraffids (F) from the MK archaeological sites (2.02 -0.6 Ma). Isotopic data are plotted at the family level. Further details on Bovidae are provided in Fig.S3(Supplementary Information). The vertical line in the boxes marks the median values, the box ends are the lower and upper quartiles, and the lines define the range of data except for outliers. Yellow and orange shades indicate mixed C3-C4 and C4-dominated diets, respectively[START_REF] Uno | Large mammal diets and paleoecology across the Oldowan-Acheulean transition at Olduvai Gorge, Tanzania from stable isotope and tooth wear analyses[END_REF][START_REF] Cerling | Carbon Isotope Fractionation between Diet and Bioapatite in Ungulate Mammals and Implications for Ecological and Paleoecological Studies[END_REF] . Animal silhouettes are from Phylopic (https://www.phylopic.org).

Figure 3 .

 3 Figure 3. Box and whisker plots of δ 18 O values of hippos (semi-aquatic habitat) and bovids, equids, suids, and giraffids (terrestrial habitat) from MK. The horizontal line in the boxes marks the median values, the box ends are the lower and upper quartiles, and the lines define the range of data except for outliers.

Figure 4 .

 4 Figure 4. Scatter plots of δ 13 C and δ 18 O values of hippos (A), bovids (B), equids (C), suids (D), hyenas (E), and giraffids (F) from Lake Turkana Basin, MK, Olduvai Gorge, and Busidima Formation. Animal silhouettes are from Phylopic (https://www.phylopic.org).

Acknowledgements

We thank the Authority for Research and Conservation of Cultural Heritage (ARCCH) and the National Museum of Ethiopia (Addis Ababa), which granted permits for sampling the paleontological collection and for exporting the samples. We are thankful to all the staff of the Museum for having facilitated the laboratory work in many ways. Thanks to Peter Tung for technical support in the isotope laboratory at the University of Tübingen, and Sandro Tomaro for checking the statistical analyses. G.B. is personally grateful to Dr. Raymonde Bonnefille for her valuable comments on the palynology.

Funding

This article is the outcome of G.B. Ph.D. project supported by the Sapienza Università di Roma (Italy), in co-tutorship with the Eberhard Karls University of Tübingen (Germany). The project was also funded by research grants provided by the Deutscher Akademischer Austauschdienst (DAAD) (Forschungsstipendien -Kurzstipendien 57552336), and the Sapienza Università di Roma ("Borse di mobilità all'estero" -Decreto N.16052020, Prot. N.0044296) awarded to G.B. The Hiwi contract provided by the Biogeology Research Group (University of Tübingen) and awarded to G.B. in November 2022 contributed to the preparation of additional samples. Stable isotope analyses were funded by the Biogeology Research Group, "Grandi Scavi di Ateneo" -grant SA11715C7C936C01 of Sapienza Università di Roma, and grant ARC-001666 of Ministero degli Affari Esteri e della Cooperazione Internazionale, awarded to M.M.

Data Availability

The data generated and analyzed during the study are available in this published article and its Supplementary Information files.

Author contributions G.B. designed the research and wrote the original manuscript with contributions from all authors (H.B., D.G., R.T.M., M.M.). G.B. collected the samples, performed the pretreatment, and elaborated on the isotopic results. All authors discussed, revised, and approved the manuscript and agreed to its submission to the journal. Competing interests The authors declare no competing interests. Additional information Supplementary data 1 (PDF document). Supplementary data 2 (Excel document).


domain: spi scco

text:
Ossen El Sawaf
email: elsawaf@lma.cnrs-mrs.fr

Sabine Meunier

Jacques Chatron

Emmanuel Friot

Marc Pachebat

Renaud Côte

Philippe Herzog

Guy Rabau

Temporal Integration at thresholds at very low and infrasonic frequencies

Keywords: Hearing thresholds, Infrasound, critical duration

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The human ear can perceive sound well below 20 Hz, i.e. infrasound, provided that the levels involved are high enough to be heard. It is also known that below a critical duration, the shorter the sounds are, the higher their intensity must be to be detected. Furthermore, this duration seems to increase with decreasing frequency. While most studies about hearing at infrasonic frequencies mention the necessity to use stimuli of sufficient length to be above the critical duration, to our knowledge only one study has actually measured the critical duration for a couple frequencies below 20 Hz and showed that at 4 Hz, the critical duration would be above 4000 ms. The presented work aims at bringing more precision to the measurement of the critical duration. In this paper, we will present the experimental setup built in order to reproduce infrasound at audible levels and the test procedure that we will use to measure the critical duration at different durations and frequencies below 63 Hz.

INTRODUCTION

While infrasonic frequencies are not a part of everyday communication in human, the human ear is still actually capable of hearing them and as such will still be a carrier of information. When speaking of audible sounds, infrasonic ---------frequencies are often not accounted for because it is not a type of sound that we are often exposed to. However, nowadays sources of infrasounds are getting more common, among other things because of the increasing size of machines, one of the most notable examples being wind turbines. As such, several studies have aimed to study infrasound perception by measuring features such as hearing thresholds, loudness function or spectral integration ([1]- [START_REF] Jurado | Frequency selectivity for frequencies below 100 Hz: Comparisons with mid-frequencies[END_REF]). A common aspect to all those studies is the attention brought to the length of the stimuli. Indeed, it is known that the absolute threshold tends to decrease with increasing stimulus duration. The limit above which the thresholds does not decrease anymore is called the critical duration. The critical duration tends to increase with decreasing frequency as shown by Watson & Gengel [START_REF] Watson | Signal Duration and Signal Frequency in Relation to Auditory Sensitivity[END_REF]. However, to our knowledge, only one paper has specifically studied this aspect in infrasonic frequencies. Jurado et al [START_REF] Jurado | Dependency of threshold and loudness on sound duration at low and infrasonic frequencies[END_REF] measured critical duration at 32, 16 and 4 Hz and has shown that for 32 and 16 Hz the critical duration is about 1000 ms while at 4 Hz the absolute thresholds was still decreasing for a stimulus duration of 4000 ms (which was their longest duration evaluated). Here, we will use a room designed specifically to broadcast infrasonic frequency to study temporal integration below 32 Hz. This project is part of the French National Research Agency (ANR) funded project "RIBEolH" which aims to assess impact of wind turbine noise on humans.

EXPERIMENTAL SETUP

Testing facility

Infrasonic frequencies have absolute thresholds so high that it is a difficult task to just broadcast them at sufficient levels. To do so, a specific room has been designed at the Laboratory of Mechanics and Acoustics (LMA) to reach the needed levels with minimal harmonic distortion as this would alter measured threshold. Indeed, as shown by Møller & Pedersen [START_REF] Møller | Hearing at low and infrasonic frequencies[END_REF], hearing thresholds increase very quickly with decreasing frequency below 20 Hz, as such, harmonic distortion occurring at higher frequency that the one tested will have a much lover hearing threshold and thus might be heard instead of the frequency of interest. To broadcast infrasonic frequencies the room is equipped with 32 subwoofer speakers on opposite wall (see Fig. 1) that allows to reach the needed levels to make these frequencies audible. The room is able to reach significant level with minimal distortions (i.e. distortions levels must be lower than their associated hearing thresholds) as shown in Fig. 2. A more thorough description of the room is proposed by Pachebat et al [START_REF] Pachebat | Une cabine acoustique pour la restitution contrôlée des basses et très basses fréquences (3Hz-300Hz) : qualification, identification acoustique et premier exemple de contrôle[END_REF] (in French).

Listeners' hearing thresholds

To ensure that harmonic distortions will indeed not be heard by the listeners, their hearing thresholds at higher frequencies, corresponding to the harmonics, will be measured before undergoing the infrasonic thresholds measurements.

Test procedure

A 2-interval forced choice procedure with a 2 down-1up rule will be used to measure the absolute thresholds of frequencies at and below 63 Hz and for different duration. Based on the results of [START_REF] Jurado | Dependency of threshold and loudness on sound duration at low and infrasonic frequencies[END_REF], a maximum duration of 1500 ms will be used for frequencies down to 16 Hz. Below that, longer durations will be used. Starting level will be around 10 dB above the hearing thresholds for infrasounds [START_REF] Møller | Hearing at low and infrasonic frequencies[END_REF]. This depend on the tested frequency as loudness has been shown to increase faster at very low frequencies [START_REF] Møller | Hearing at low and infrasonic frequencies[END_REF] As such, special care will be taken to ensure starting levels not too loud for the listeners.

RESULTS

Results will be presented at the 2023 Forum Acusticum convention.

ACKNOWLEDGEMENT

This work was supported by a grant from the French National Research Agency (ANR).

[1] B. Friedrich, H. Joost, T. Fedtke, and J. L. Verhey, « Spectral integration of infrasound at threshold »,

Temporal

  Integration at thresholds at very low and infrasonic frequencies O. El Sawaf 1* S. Meunier 1 J. Chatron 1 E. Friot 1 M. Pachebat 1 R. Côte 1 P. Herzog 2 G. Rabeau 1 1 Aix Marseille Univ, CNRS, Centrale Méditerranée, LMA UMR 7031, Marseille, France 2 ARTEAC-LAB SAS, 8 allées Léon Gambetta, Place de l'Innovation (CIC), 13001 Marseille, France ABSTRACT *

Figure 1 .

 1 Figure 1. Picture of the room for broadcast of infrasonic frequencies. The room is equipped with 32 subwoofers speakers (f ≤ 40 Hz), 40 low frequency speakers (40 < f ≤ 3000 Hz) and 6 medium frequency speakers(3000 < f ≤ 17000 Hz).

Figure 2 .

 2 Figure 2. Example of 8 Hz signal measured in the room. A level of 111 dB (8 dB above expected threshold from [2]) is reached while all harmonics distortions' levels are below the standardized thresholds from ISO 226:2003.


Sabine Meunier
email: meunier@lma.cnrs-mrs.fr

Dominique Habault

Hervé Moulinec

Philippe Lasaygues

Denis Guthleben

Pierre-Olivier Mattei

Elena Rosu

P.-O Mattei

HISTORY OF THE LABORATORY OF MECHANICS AND ACOUSTICS IN MARSEILLES (FRANCE): FROM THE SECOND WORLD WAR TO THE PRESENT DAY

Keywords: Acoustics, Mechanics, history ---------

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The Laboratory of Mechanics and Acoustics is, at its name indicates, specialized in mechanics and acoustics. It is composed of about 130 people (researchers and teacherresearchers, engineers, technicians, administrative staff, doctoral students). It is a research unit attached to Aix-Marseille University (AMU), the National Center for Scientific Research (Centre National de la Recherche Scientifique, CNRS) and the École Centrale Méditerranée. Its origin dates back to the beginning of the 20th century, but it has been a CNRS laboratory for "only" 80 years. This text is strongly inspired by the writings of Claude Gazanhes ( [START_REF] Gazanhes | Du laboratoire de la guerre sous-marine de Toulon au laboratoire de mécanique et d'acoustique de Marseille[END_REF], [START_REF] Gazanhes | Historique du laboratoire de Mécanique et d'Acoustique de Marseille 1941-1991[END_REF]).

THE VERY BEGINNING

The origins of the laboratory go back to the Laboratoire de la Guerre Sous-Marine (Submarine Warfare Laboratory), created in Toulon in 1917 by the French Navy. It was in this laboratory, among others, that Paul Langevin developed the first high-powered ultrasonic transmitters for the detection of enemy submarines by echoes. At the end of the war, in 1920, the missions of this laboratory grew; it became the Laboratoire du Centre d'Etudes de Toulon (Laboratory of the Toulon Study Center, LCET) and its scientific direction was entrusted to François Canac who played a major role in the history of the laboratory of which he was the founder and then the director until 1958. For F. Canac, inspired by the National Physical Laboratory, Teddington (UK), the objective was that the LCET, well beyond its missions for the French Navy, should be an organization of upstream research and "technology watch", capable of developing, for a government department or industrial sector, applications resulting from the latest discoveries, of building prototypes, what is called today "technology transfer".

THE SECOND WORLD WAR

At the beginning of the Second World War, the laboratory was renamed Centre de Recherches de la Marine (CRM). At the end of June 1940, the CRM, staff and equipment, was hurriedly embarked on a ship to Oran and then a train to Algiers. It seemed that the CRM would disappear in the turmoil. But it was headed by an energetic man with a strong personality. In August 1940, F. Canac proposed that the laboratory be placed under the supervision of the Ministry of Public Instruction: "The working methods (of the laboratory) are precisely those which it is necessary to introduce (very thorough research, complete development in view of pragmatic problems). It would bring its specialists, its material, its documentation". F. Canac's appeal was heard. The CNRS had just been created; its administrative and financial situation was still under study but everything happened very quickly. On January 1, 1941, the CRM became the first "Laboratoire Propre" (i.e. with a unique supervision) of the CNRS out of Paris and took the name of Centre de Recherches Scientifiques Industrielles et Maritimes (Center for Scientific, Industrial and Maritime Research, CRSIM). The research at CRSIM was divided into five major themes: Acoustics, Optics, Chemistry-corrosion, X-rays, Psychotechnics. The Acoustics part was strongly developed, notably under the influence of F. Canac who was passionately interested in the Acoustics of the Ancient Theaters [START_REF] Canac | L'Acoustique des théâtres antiques, ses enseignements[END_REF]. It included electroacoustics, vibrations, architectural acoustics, ultra-sounds in air and water. The CRSIM already had many facilities: a large anechoic chamber (13 x 5 x 5 m 3 ), a small anechoic chamber (4 x 2 x 2 m 3 ), a reverberation room (10 x 2.6 x 2.6 m 3 ), a ripple tank for the study of room models, a 18 m channel with an absorbing basin. Figure 1 presents a view of the large anechoic chamber and reminds us of the difficulties of the time: the absorbing walls were made of pyramidal trunks covered with blankets supplied by the Health Service (Figure 1). The laboratory moved to the Joseph-Aiguier site, still in Marseille, where it joined other CNRS laboratories. Thanks to this operation, the laboratory grew and benefited from the installation of new important and powerful equipment for acoustics: among others, a large anechoic chamber for the wave propagation, a smaller anechoic chamber for psychoacoustic tests, a station for architectural acoustics built according to international standards and including two adjacent reverberant rooms for studying the acoustics transparency walls, a music studio with a piano, several tanks dedicated to underwater acoustics and ultra-sound propagation.

THE LABORATORY OF MECHANICS AND ACOUSTICS

In 1975, the laboratory refocused on Solid Mechanics and Acoustics and became the LMA. Mechanics of continuous media, computer science, and real time techniques has developed from this time as well as experimental facilities for mechanics. The acoustics department was composed of 5 teams: acoustic theories, applied acoustics, aerodynamic acoustics, musical acoustics and propagation in liquid media.

Later, new themes would appear like ultrasounds, protection of environment, active noise control and psychoacoustics (which was part of the research themes of the CRP but has been abandoned for more than 10 years), submarine acoustics.

In 2012, the Laboratoire de Contrôle Non Destructif (Non-Destructive Control Laboratory, LCND), based in Aix-en-Provence (at Aix-Marseille University), joined the LMA. Since then, the LMA has a branch and premises at the IUT in Aix-en-Provence (Mechanical Engineering Department). At that time, the laboratory was restructured into 3 teams (described below) which continue today.

In 2015, the LMA moved to the Technopôle de Château-Gombert where the fluid mechanics and physics laboratories, as well as the engineering schools École Centrale Marseille (now École Centrale Méditerranée) and Polytech-Marseille, were already present.

The new laboratory offers an exceptional experimental environment, that was designed by its staff during the construction project. In particular, a set of 3 anechoic rooms represents a rare equipment in the community: an anechoic room equipped with the necessary cabling for the study of active anechoicity at low frequency (Figure 1); a semianechoic room coupled with an excitation room underneath it (where complex sounds are made), designed to accommodate large-scale projects directly related to industrial applications; an anechoic room that can receive participants for psychoacoustic experiments.

ACKNOWLEDGMENTS

We would like to thank Alain Rimeymeille for the pictures.

Figure 1 :

 1 Figure 1: the anechoic chamber of the CRSIM

  Today the LMA is composed of 3 teams: Materials and Structures, Waves and Imaging, Sounds. The work of the Materials and Structures team concerns solid mechanics, with experimental, theoretical and numerical aspects. The objectives of its activities range from the characterization of materials or structures (fatigue behavior for example) to the establishment of models or innovative modeling tools to simulate solid media (materials, structures or interfaces). The research conducted in the Waves and Imaging team aims at developing acoustic methods for modeling, characterization and/or imaging of heterogeneous media with fundamental and applied approaches. The research carried out in the Sound team concerns audible sounds: their production, propagation, control, synthesis and listening. Since the studied sounds are audible, Human remains an important element of the scientific context, either he/she participates in the production of the sound (sung voice, musical instruments), or he/she undergoes its nuisance (transport noises), or he/she uses it on purpose (communication). And the human being is also the object of the research (perception, hearing).

Figure 2 :

 2 Figure 2: the large anechoic room of the LMA at Château-Gombert

It was created in 1941, under the name of Centre de Navy, under the direction of François Canac, to work on submarine detection during World War I. It was the first laboratory of the National Center for Scientific Research out of Paris. In this laboratory, Paul Langevin developed the first high-powered ultrasonic transmitters for the detection of submarines. F. Canac headed the CRSIM until 1958. He was one of the founders of the journal Acustica. The acoustics service developed considerably under his leadership. The CRSIM became the Centre de Recherches Physiques (Physical Research Center, CRP) in 1962 and was directed by Théodore Vogel. In 1973, the laboratory had a Department of Mechanics and a Department of Acoustics. On July 10, 1973, the laboratory took the name of Laboratoire de Mécanique


domain: math spi

text:
E Azeraf

W Pieczynski

Equivalence between conditional random fields and partially Markov models

Keywords: Conditional random fields, Filtering, Hidden Markov models, Pairwise Markov chains, Partially Markov models

Hidden Markov models (HMM) have been used successfully in various applications for around sixty years. Moreover, conditional random fields (CRFs) are considered an alternative to HMMs and appear in the literature as different and somewhat competing models. It has recently been shown that basic linear chain CRFs (LC-CRFs), considered different from HMMs, are in fact equivalent to them in the sense that for each LC-CRF there exists an HMM, which can be specified, whose a posteriori distribution is identical to the given LC-CRF. In this article, we extend this result to general CRFs. We consider partially Markovian models (PMM), which are sequentially defined models extending HMMs, and we show that they are equivalent to CRFs. Equivalence means that the posterior distributions of the PMMs are CRFs and, conversely, for each given CRF there exists a PMM -which we specify -whose posterior distribution is the given CRF. The first assertion is well known and the proof of the second constitutes the main contribution of this article. The advantage of presenting a CRF in its PMM form is that the latter is sequential, whereas its original former is not. This property is essential in certain sequential processing, such as filtering or forecasting.

INTRODUCTION

ET 𝑍 1:𝑁 = (𝑍 1 , … , 𝑍 𝑁 ) be a stochastic sequence, with 𝑍 𝑛 = (𝑋 𝑛 , 𝑌 𝑛 ). Random variables 𝑋 1 , … , 𝑋 𝑁 take their values in a finite set Λ, while 𝑌 1 , … , 𝑌 𝑁 take their values either in a discrete or continuous set Ω. Probabilistic model is a distribution -or a family of distributionswhich will be denoted with 𝑝(𝑧 1:𝑁 ) , or 𝑝(𝑥 1:𝑁 , 𝑦 1:𝑁 ) . More generally, for 1 ≤ 𝑚 < 𝑛 ≤ 𝑁, we will note 𝑧 𝑚:𝑛 = (𝑧 𝑚 , … , 𝑧 𝑛 ) , and similarly for 𝑥 𝑚:𝑛 , 𝑦 𝑚:𝑛 . Conditional distributions will be denoted with 𝑝(. |. ). For example, we have 𝑝(𝑥 𝑛 |𝑦 1:𝑁 ) = 𝑃[𝑋 𝑛 = 𝑥 𝑛 |𝑌 1:𝑁 = 𝑦 1:𝑁 ] , 𝑝(𝑥 𝑛+1 |𝑥 1:𝑛 ) = 𝑃[𝑋 𝑛+1 = 𝑥 𝑛+1 |𝑋 1:𝑛 = 𝑥 1:𝑛 ], and so on.

The aim of the paper is to show the equivalence between sequential models called "Partially Markov models" (PMMs, [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF]) and non-sequential models called "Conditional random fields" (CRFs, [START_REF] Lafferty | Conditional random fields: probabilistic models for segmenting and labeling sequence data[END_REF], [START_REF] Sutton | An Introduction to Conditional Random Fields[END_REF]. The interest is that sequential PMMs are easier to manipulate than non-sequential CRFs; in particular, they allow new observations to be taken into account more easily. This contribution shows that apparently less general PMMs have actually the same modeling power as CRFs. PMMs are defined by the distribution 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) verifying (1.1), while CRFs considered in this paper are defined by the distribution 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) verifying (1.2). , (1.5) with 𝑐 constant and 𝜑, 𝜙 real valued functions. Actually, such distributions are particular cases of (1.2); indeed, they are (1.2) with the conditions 𝑝(𝑥 1 |𝑦 1:𝑁 ) > 0 for 𝑝(𝑥 1 |𝑦 1:𝑁 ), and 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) > 0 for every 𝑛 = 1, …, 𝑁 -1.

𝑞(𝑥

It is easy to see that the posterior distribution 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ) of a PMM (1.1) is a CRF (1.2); the converse proposition is trickier. The problem we deal with is to answer the following questions. Having a CRF 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) , does there exist a PMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) such CRFs found huge deal of applications, and continue to be widely used as attested by hundreds journal papers appearing each year. Examples include image segmentation [START_REF] Chen | DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs[END_REF], [START_REF] Batatia | A deep learning method with CRF for instance segmentation of metal-organic frameworks in scanning electron microscopy images[END_REF], [START_REF] Lapuyade-Lahorgue | Segmentation of multicorrelated images with copula models and conditionally random fields[END_REF], named entity recognition [32], or still part-of-speech tagging [START_REF] Gahbiche-Braham | Joint segmentation and POS tagging for Arabic using a CRF-based classifier[END_REF]. Let us also cite [START_REF] Liliana | A Review on conditional random Fields as a sequential classifier in machine learning[END_REF] as review article and [START_REF] Klinger | Conditional Random Fields for Named Entity Recognition[END_REF], [START_REF] Sutton | An Introduction to Conditional Random Fields[END_REF] as books, among others. PMMs have been little studied so far; we can cite [START_REF] Lanchantin | Unsupervised segmentation of triplet Markov chains hidden with long-memory noise[END_REF] where they were used to estimate non-stationary data hidden with long memory noise. This opens up new possibilities, as it means that sequential PMMs can be used in place of non-sequential CRFs without losing CRFs modelling power.

----------------  E.
The equivalence between CRFs and PMMs makes it possible to show the equivalence between "hidden" CRFs (HCRFs, [START_REF] Quattoni | Conditional random fields for object recognition[END_REF], [START_REF] Quattoni | Hidden conditional random fields[END_REF]) and "triplet" PMMs (called triplet partially Markov chains in [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF]). HCRFs consist of adding a latent variable 𝑈 1:𝑁 , and considering the triplet (𝑋 1:𝑁 , 𝑈 1:𝑁 , 𝑌 1:𝑁 ) . By setting 𝑉 1:𝑁 = (𝑋 1:𝑁 , 𝑈 1:𝑁 ) , HCRF is given assuming that (𝑉 1:𝑁 , 𝑌 1:𝑁 ) is a CRF. HCRFs are more general than CRFs because in HCRF (𝑋 1:𝑁 , 𝑈 1:𝑁 , 𝑌 1:𝑁 ) the couple (𝑋 1:𝑁 , 𝑈 1:𝑁 ) is Markov (conditionally on 𝑌 1:𝑁 ), and 𝑋 1:𝑁 may be Markovian or may not be Markovian. HCRFs are applicable in all situations CRFs are, and there is an abundant literature showing their interest. In particular, they have been applied to telephone classification [START_REF] Gunawardana | Hidden conditional random fields for phone classification[END_REF], [START_REF] Sung | Hidden conditional random fields for phone recognition[END_REF], detection of cell division [START_REF] Ong | Detecting cell division of Pseudomonas aeruginosa bacteria from bright-field microscopy images with hidden conditional random fields[END_REF], human action recognition [START_REF] Vrigkas | Robust incremental hidden conditional random fields for human action recognition[END_REF], [START_REF] Wang | Hidden part models for human action recognition: probabilistic versus max margin[END_REF], human facial expression recognition [START_REF] Siddiqi | Human facial expression recognition using stepwise linear discriminant analysis and hidden conditional random fields[END_REF], gesture recognition [START_REF] Sy | Hidden conditional random fields for gesture recognition[END_REF], fault diagnosis [START_REF] Tang | Wireless sensor-networks conditions monitoring and fault diagnosis using neighborhood hidden conditional random field[END_REF], CET image segmentation [START_REF] Trullo | Segmentation of organs at risk in thoracic CT images using a SharpMask architecture and conditional random fields[END_REF], and many others. A similar idea has been proposed for PMMs in [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF], leading to "triplet" PMMs (TPMMs). TPMMs have been applied to non-stationary Markov chains hidden with long memory noise in [START_REF] Lanchantin | Unsupervised segmentation of triplet Markov chains hidden with long-memory noise[END_REF]. According to the theorem, we can state that HCRFs and TPPMs are equivalent. This opens up enormous possibilities of applications of TPPMs in situations where HCRFs have performed well and where the sequential nature of TPMMs is of interest Once the equivalence between PMMs and CRFs has been established, we propose in section III the following contribution, which is related to particular PMMs called "pairwise Markov chains" (PMCs, [START_REF] Pieczynski | Pairwise Markov chains[END_REF]). The distribution of PMCs is given by 𝑞(𝑥 1:𝑁 , 𝑦 ) . This shows that PMCs are clearly more general than HMMs and, indeed, the interest of the former over the latter is highlighted in various articles. In particular, they have been applied to image segmentation [START_REF] Ameur | Color images segmentation using pairwise Markov chain[END_REF], [START_REF] Atiampo | Unsupervised image segmentation with pairwise Markov chains based on nonparametric estimation of copula using orthogonal polynomials[END_REF], [START_REF] Gangloff | Deep parameterizations of pairwise and triplet Markov models for unsupervised classification of sequential data[END_REF], [START_REF] Pieczynski | Pairwise Markov chains[END_REF], sound classification [START_REF] Le Cam | Fuzzy pairwise Markov chain to segment correlated noisy data[END_REF], fuzzy segmentation [START_REF] Le Cam | Fuzzy pairwise Markov chain to segment correlated noisy data[END_REF], or multiple target tracking [START_REF] Liu | Particle probability hypothesis density filter based on pairwise Markov chains[END_REF], [START_REF] Mahler | The pairwise-Markov Bernoulli filter[END_REF], [START_REF] Petetin | Bayesian multi-object filtering for pairwise Markov chains[END_REF]. An extension to apply Bayesian variational Bayesian methods is proposed in [START_REF] Morales | Variational bayesian inference for pairwise Markov models[END_REF], and some theoretical studies can be found in [START_REF] Kuljus | Pairwise Markov models and hybrid segmentation approach[END_REF], [START_REF] Lember | Exponential forgetting of smoothing distributions for pairwise Markov models[END_REF], [START_REF] Lember | Regenerativity of Viterbi process for pairwise Markov models[END_REF], [START_REF] Lember | Local Viterbi property in decoding[END_REF]. Since PMCs are applicable under the same conditions as HMMs while being more general, it is interesting to ask which family of CRFs is equivalent to PMCs. Our contribution consists in characterizing the family of CRFs equivalent to PMCs. More precisely, we show that a CRF 𝑝(𝑥

𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) = (1.9) 𝜅(𝑦 1:𝑁 )𝑒𝑥𝑝[∑ 𝑉 𝑛 (𝑥 𝑛 , 𝑥 𝑛+1 ) 𝑁-1 𝑛=1 + ∑ 𝑈 𝑛 (𝑥 𝑛 , 𝑦 𝑛 ) 𝑁 𝑛=1
],

where 𝜅(𝑦 1:𝑁 ) is the normalizing constant, we find again the fact that it is equivalent to classic HMM (1.8), which has been recently established in a direct way in [START_REF] Azeraf | Equivalence between LC-CRF and HMM, and discriminative computing of HMM-based MPM and MAP[END_REF]. Subsequently, another case studied shows that the classic Minimum Entropy Markov Models (MEMMs [START_REF] Mccallum | Maximum Entropy Markov Models for information extraction and segmentation[END_REF]) are equivalent to particular PMCs. Dependence graphs of HMMs, PMCs, and PMMs are presented in Figure.

This article is organized as follows. In the next section, we show the equivalence, in the sense of definition 1.1, between PMMs (1.1) and CRFs (1.2). In Section III, we specify the CRFs equivalent to PMCs. We also specify the family of CRFs equivalent to triplet Markov chains (TMCs [START_REF] Pieczynski | Triplet Markov chains in hidden signal restoration[END_REF]). We present conclusions and perspectives in Section IV.

𝑦 1 𝑦 2 𝑦 3 𝑦 4 𝑥 1 𝑥 2 𝑥 3 𝑥 4 (a) 𝑦 1 𝑦 2 𝑦 3 𝑦 4 𝑥 1 𝑥 2 𝑥 3 𝑥 4 (b) 𝑦 1 𝑦 2 𝑦 3 𝑦 4 𝑥 1 𝑥 2 𝑥 3 𝑥 4 (c)

EQUIVALENCE BETWEEN PMMS AND CRFS

In this paragraph, we show that for each CRF (1.2) there exists an equivalent PMM (1.1). More precisely, we specify how to compute all distributions 𝑞(𝑥 1 ), 𝑞(𝑦 Remark 2.1 CRFs and PMMs are equivalent; however, they are parameterized differently and can therefore give different results in real applications. Indeed, in real applications, one has to estimate the parameters in a preliminary step. Since the parameters are different in CRFs and PMMs, the estimators used are different and may be more or less suitable for the intended application. Thus, even in real-life situations where the sequential nature of PMMs is not of importance, there is no reason to prefer one family of models to the other. Therefore, even in situations where the sequential nature of PMMs is not important and where CRFs -or HCRFs, see remark 2.2 below -give excellent results, it is still worth applying PMMs -or TPMMs -for comparison.

Remark 2.2

As mentioned in Introduction, equivalence between CRFs and PMMs implies equivalence between "hidden" CRFs (HCRFs, [START_REF] Quattoni | Conditional random fields for object recognition[END_REF], [START_REF] Quattoni | Hidden conditional random fields[END_REF]) and "triplet" PMMs (TPMMs, [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF]). Indeed, HCRFs (or TPMMs) (𝑋 1:𝑁 , 𝑈 1:𝑁 , 𝑌 1:𝑁 ) also are CRFs (or PMMs) (𝑉 1:𝑁 , 𝑌 1:𝑁 ), with 𝑉 1:𝑁 = (𝑋 1:𝑁 , 𝑈 1:𝑁 ).

CRFS EQUIVALENT TO PMCS

Consider the "pairwise Markov chains" given by (1.7). As PMCs are an extension of HMMs, they are applicable in the same situations and are likely to improve the results provided by HMMs. As we shall see, some CRFs presented as alternative to HMMs are equivalent to PMCs. This is the case for Maximum Entropy Markov Models (MEMMs) [START_REF] Mccallum | Maximum Entropy Markov Models for information extraction and segmentation[END_REF].

Proposition 3.1 below, which we can consider as a corollary of the theorem, characterizes the family of CRFs equivalent to the family of PMCs. . We note that in such a PMM 𝑌 1 , …, 𝑌 𝑁 are independent and equidistributed, which may seem somewhat unusual. However, this has no bearing on the problem of finding 𝑋 1:𝑁 from 𝑌 1:𝑁 with Bayesian classifiers; indeed, they are independent from 𝑝(𝑥 1:𝑁 , 𝑦 1:𝑁 ). We have

𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) = 𝑞(𝑥 1 , 𝑦 1 ) ∏ 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 ) = 𝑁-1 𝑛=1 c 𝑁 𝑞(𝑥 1 |𝑦 1 ) ∏ 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) 𝑁-1 𝑛=1 (3.2)
According to the Theorem, 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) are computable by the following backward recursion:

-𝛽 𝑁 (𝑥 𝑁 ) = 1, 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 );

-𝛽 𝑛 (𝑥 𝑛 ) = c ∑ 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )𝛽 𝑛+1 (𝑥 𝑛+1 ) 𝑥 𝑛+1 = c 𝑁-𝑛 ; -𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) = c 𝑛-𝑁 [∑ 1 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 1:𝑁 ) 𝑦 𝑛+1:𝑁 ] -1 ; -𝑞(𝑥 1 |𝑦 1 ) = [∑ 𝑞(𝑦 2:𝑁 |𝑥 1 ,𝑦 1 ) 𝑝(𝑥 1 |𝑦 1:𝑁 ) 𝑦 2:𝑁 ] -1 .
Knowing that the classifiers only depend on 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ), this shows that any "generative" classifier defined from a PMM can be calculated in a discriminative way, by replacing in the calculations 𝑞(𝑦 2 |𝑥 1 , 𝑦 1 ) , …, 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) by 𝑐 = .

We remark that 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 , 𝑦 𝑛+1 ) = 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ).

As in example 3. , so that 𝑝(𝑥 𝑛+1 , 𝑦 1:𝑛+1 ) can be computed from 𝑝(𝑥 𝑛 , 𝑦 1:𝑛 ) with complexity #Λ , and hence 𝑝(𝑥 𝑛+1 |𝑦 1:𝑛+1 ) is computable from 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ) with complexity 2(#Λ) , independent from 𝑛. Remark 3.2 Consider a HCRF (𝑋 1:𝑁 , 𝑈 1:𝑁 , 𝑌 1:𝑁 ) such that the CRF (𝑉 1:𝑁 , 𝑌 1:𝑁 ) -where 𝑉 1:𝑁 = (𝑋 1:𝑁 , 𝑈 1:𝑁 ) -verifies (3.1). It is then equivalent to PMCs (𝑉 1:𝑁 , 𝑌 1:𝑁 ) = (𝑋 1:𝑁 , 𝑈 1:𝑁 , 𝑌 1:𝑁 ), models know as "triplet Markov chains" (TMCs) [START_REF] Pieczynski | Triplet Markov chains in hidden signal restoration[END_REF]. Here the situation is the opposite to those mentioned in examples 2.1 and 2.2: authors have applied TMCs in different situations, while there was no obvious reason for CRF specialists to be interested in their equivalent CRFs verifying (3.1). More precisely, TMCs have found applications in the modelling of nonstationary time series of normalized vegetation indices [START_REF] Ben Abbes | A non-stationary NDVI time series modelling using triplet Markov chain[END_REF], the segmentation of 3D magnetic resonance brain imaging [START_REF] Brick | Triplet Markov chain for 3D MRI brain segmentation using a probabilistic atlas[END_REF], the modelling of consumer loan repayment behavior [START_REF] Chen | Modeling repayment behavior of consumer loan in portfolio across business cycle: A triplet Markov model approach[END_REF], the activity classification [START_REF] Li | An adaptive and on-line IMU-based locomotion activity classification method using a triplet Markov model[END_REF], or non-stationary sequences or images segmentation [START_REF] Boudaren | Phasic triplet Markov chains[END_REF], [START_REF] Fernandes | Non-stationary data segmentation with hidden evidential semi-Markov chains[END_REF], or [START_REF] Gangloff | Deep parameterizations of pairwise and triplet Markov models for unsupervised classification of sequential data[END_REF] where the third process is continuous. Let us also cite recent TMCs using variational Bayesian inference [START_REF] Morales | Variational bayesian inference for pairwise Markov models[END_REF], [START_REF] Morales | A probabilistic semi-supervised approach with triplet Markov chains[END_REF]. As TMCs are particular TPMMs, they are equivalent to a subfamily of CRFs. This means that in situations where the sequential nature of TMCs is not required, the results obtained with a TMC are likely to be improved with a more general CRF.

CONCLUSIONS AND PERSPECTIVES

We studied the relationships between CRFs and different HMM extensions. The main contribution is the proof of equivalence, within the meaning of definition 1.1, between CRFs [START_REF] Lafferty | Conditional random fields: probabilistic models for segmenting and labeling sequence data[END_REF] and PMMs [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF]. More precisely, for each CRF (1.2), we have given all PMMs (1.1) whose posterior distribution is the given CRF.

In summary, we have the following four equivalences: (iv) Hidden CRFs (HCRFs [START_REF] Quattoni | Conditional random fields for object recognition[END_REF], [START_REF] Quattoni | Hidden conditional random fields[END_REF]) are equivalent to Triplet PMMs (TPMMs [START_REF] Pieczynski | Triplet partially Markov chains and tree[END_REF]). Point (i) was recently established in [START_REF] Azeraf | Equivalence between LC-CRF and HMM, and discriminative computing of HMM-based MPM and MAP[END_REF] (see also [START_REF] Heigold | Equivalence of Generative and Log-Linear Models[END_REF], where a similar result is established under stronger assumptions), points (ii) to (iv) are contributions of the article. The potential application interest of these results is that the HMM, PMC, PMM and TPMM are sequential -within the meaning of remark 3.1 -, while the CRFs are not. This means that in situations where sequential processing is attractive, it is possible to use HMM, PMC, PMM or TPMM instead of equivalent CRFs without losing modeling power. In cases where the sequential nature of treatments is not essential, no family is a priori more interesting than the other; indeed, both families have identical modeling power. However, because their parameters are different, the parameter estimates are different and, therefore, the final yields may be different. In such situations, it would be interesting to test both models, because even if the effectiveness of a model from one family is satisfactory, it can be improved by a model from the other family.

Definition 1 . 1 (

 11 1:𝑁 , 𝑦 1:𝑁 ) = (1.1) 𝑞(𝑥 1 )𝑞(𝑦 1 |𝑥 1 ) ∏ 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 ) paper, we adopt the following definitions: i) a distribution 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) is "equivalent" to a distribution 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) if and only if (iff) 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ); (1.3) (ii) two distributions 𝑟(𝑥 1:𝑁 , 𝑦 1:𝑁 ) , 𝑠(𝑥 1:𝑁 , 𝑦 1:𝑁 ) are equivalent iff 𝑟(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑠(𝑥 1:𝑁 |𝑦 1:𝑁 ). (1.4) Let us consider the following popular CRFs distribution: 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑐𝑒 -[∑ 𝜑(𝑥 𝑛 ,𝑥 𝑛+1 ,𝑦 1:𝑁 )

Figure .

 . Figure. Dependence graphs. (a): hidden Markov model (HMM); (b) pairwise Markov chain (PMC); (c) partially Markov model (PMM).

Proof. 1 ,

 1 Since 𝑟(𝑣, 𝑤) = 𝑟(𝑤|𝑣)𝑞(𝑣) = 𝑟(𝑣|𝑤)𝑝(𝑤) , we have 𝑟(which gives (2.22) and completes the proof. 

1 =Example 3 . 1

 131 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 )𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 )𝛽 𝑛+1 (𝑥 𝑛+1 ) 𝛽 𝑛 (𝑥 𝑛 ) = 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1:𝑁 ). Thus 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1:𝑁 ) , which is (3.1); 2. Let 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) be a CRF (1.2) verifying (3.1). We have 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ). Conversely, let us show that 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) = 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ) implies 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) = 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 𝑛 ) . We look for a PMC in which 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ) = 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 𝑛 ). Then (2.15) gives 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) = [∑ 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 ,𝑦 1:𝑛 ) 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 1𝑛+1:𝑁 |𝑥 𝑛 ,𝑦 𝑛 ) 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 𝑛:𝑁 ) 𝑦 𝑛+1:𝑁 ] -𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 𝑛:𝑁 ) ∑ 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 ,𝑦 𝑛 ) 𝑦 𝑛+1:𝑁 = 𝑝(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 𝑛:𝑁 ) , which completes the proof. Let 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) be a CRF defined with 𝑝(𝑥 1 |𝑦 1:𝑁 ), 𝑝(𝑥 2 |𝑥 1 , 𝑦 1:𝑁 ), …, 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ). What is the simplest equivalent PMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) ? Since 𝑞(𝑦 1 ) , 𝑞(𝑦 2 |𝑥 1 , 𝑦 1 ) , …, 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) are arbitrary, let us take them all equal to the constant 𝑐 = 1 #Λ

Example 3 . 2 3 )

 323 PMMs the results related to Maximum a posteriori (MAP) and Maximum posterior mode (MPM) classifiers presented in the framework of HMMs in [3]. In classic HMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) we have 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) = 𝑞(𝑥 𝑛+1 |𝑥 𝑛 )𝑞(𝑦 𝑛+1 |𝑥 𝑛+1 ). Then 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) = ∑ 𝑞(𝑥 𝑛+1 |𝑥 𝑛 )𝑞(𝑦 𝑛+1 |𝑥 𝑛+1 ) 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 , 𝑦 𝑛+1 ) = 𝑞(𝑥 𝑛+1 |𝑥 𝑛 )𝑞(𝑦 𝑛+1 |𝑥 𝑛+1 ) ∑ 𝑞(𝑥 𝑛+1 |𝑥 𝑛 )𝑞(𝑦 𝑛+1 |𝑥 𝑛+1 ) 𝑥 𝑛+1

  (i) Simple CRFs (1.10) are equivalent to HMMs (1.8); (ii) CRFs satisfying (3.1) are equivalent to pairwise Markov chains (1.7); (iii) General CRFs (1.2) are equivalent to PMMs (1.1);

  If so, can probabilities 𝑞(𝑥 1 )𝑞(𝑦 1 |𝑥 1 ), 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 ) in (1.1) be calculated from the probabilities 𝑝(𝑥 1 |𝑦 1:𝑁 ), 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) in (1.2) ? Can we specify all the solutions of the problem? We answer these questions in the theorem in the next section. Let us remark that one immediate practical consequence is that if we wish to estimate of 𝑋 1:𝑁 from 𝑌 1:𝑁 by a Bayesian classifier, we can use either CRFs or any equivalent PMMs. Indeed, Bayesian estimators only depend on the a posteriori distribution 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ).

Azeraf is with Watson Department, IBM France. E-mail elie.azeraf@gmail.com  W. Pieczynski is with Samovar Telecom Sudparis Institut Polytechnique de Paris, Palaiseau, France. E-mail: wojciech.pieczynski@telecom-sudparis.eu L that 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) ?

  Proof. 1. Let 𝑊 1:𝑁 be Markov : 𝑝(𝑤 1 , … , 𝑤 𝑁 ) = 𝑝(𝑤 1 )𝑝(𝑤 2 | 𝑤 1 )𝑝(𝑤 3 | 𝑤 2 ) … 𝑝(𝑤 𝑁 | 𝑤 𝑁-1 ). Then (2.1) is verified by 𝜑 1 (𝑤 1 , 𝑤 2 ) = 𝑝(𝑤 1 )𝑝(𝑤 2 | 𝑤 1 ), 𝜑 2 (𝑤 2 , 𝑤 3 ) = 𝑝(𝑤 3 | 𝑤 2 ), …, 𝜑 𝑁-1 (𝑤 𝑁-1 , 𝑤 𝑁 ) = 𝑝(𝑤 𝑁 | 𝑤 𝑁-1 ). Let 𝑋 1:𝑁 = (𝑋 1 , … , 𝑋 𝑁 ) , 𝑌 1:𝑁 = (𝑌 1 , … , 𝑌 𝑁 ) be two stochastic sequences taking their values in finite sets 𝛬, 𝛺, respectively.Proof. The proof is immediate applying Lemma 1 to𝑋 1:𝑁 = 𝑊 1:𝑁 , 𝜑 1 (𝑥 1 , 𝑥 2 ) = 𝑞(𝑥 1 , 𝑦 1 )𝑞(𝑥 2 , 𝑦 2 |𝑥 1 , 𝑦 1:2 ) , 𝜑 2 (𝑥 2 , 𝑥 3 ) = 𝑞(𝑥 3 , 𝑦 3 |𝑥 2 , 𝑦 1:2 ) , …, 𝜑 𝑁-1 (𝑥 𝑁-1 , 𝑥 𝑁 ) = 𝑞(𝑥 𝑁 , 𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) The theorem below specify the reverse proposition, which is more difficult to prove and is the core contribution of the paper.𝛽 𝑛 (𝑥 𝑛 ) = 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ) for 𝑛 = 𝑁 -1, …, 1. 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 );We search a PMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) (2.12) verifying 𝑞(𝑥 1:𝑁 | 𝑦 1:𝑁 ) = 𝑝(𝑥 1:𝑁 | 𝑦 1:𝑁 ). This equality is equivalent to 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ), 𝑞(𝑥 𝑁-1 |𝑥 𝑁-2 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁-1 |𝑥 𝑁-2 , 𝑦 1:𝑁 ), ) for the one in the first line, equation (2) for the one in the second line, …, equation (N-1) for the last equation. To show that all the equations are true with 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) , …, 𝑞(𝑥 1 |𝑦 1:𝑁 ) verifying (2.15), we show that the first equation in (2.17) is true, and we show that if the first 𝑁 -𝑛 equations (for 1 ≤ 𝑛 ≤ 𝑁 -1 ) are true with 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) and 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) , …, 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 ) , the latter sequence being arbitrary but fixed in the proof), can we find 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) such that (2.20) holds with 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ) ? Consider 𝑟(. ) distributions on Λ conditional on (𝑥 𝑛-1 , 𝑦 1:𝑛 ), and let us set 𝑣 = 𝑥 𝑛 , 𝑤 = 𝑦 𝑛+1:𝑁 . The question above is equivalent to: can we find 𝑟(𝑣) such that

							𝛽 1 (𝑤 1 ) ∑ 𝛽 1 (𝑤 1 ) 𝑤 1	;
	For a given PMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) , let us consider the "backward" probabilities 𝛽 𝑁 (𝑥 𝑁 ) = 1, and 𝑟(𝑣|𝑤) = 𝑟(𝑣)𝑟(𝑤|𝑣) ? (2.21) 𝑟(𝑤) In other words, we have 𝑟(𝑣|𝑤) and 𝑟(𝑤|𝑣), and we (2.8) search 𝑟(𝑣) verifying (2.21). According to Lemma 2 Similarly to what is true in classic HMMs, they verify (2.7), rewritten as 𝛽 𝑁 (𝑥 𝑁 ) = 1, and (2.9) 𝛽 𝑛 (𝑥 𝑛 ) = (2.10) ∑ 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 )𝛽 𝑛+1 (𝑥 𝑛+1 ) 𝑥 𝑛+1 be a CRF. Then the PMM 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) = (2.12) 𝑞(𝑦 1 )𝑞(𝑥 1 |𝑦 1 ) ∏ 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) 𝑁-1 𝑛=1 below, the solution is 𝑟(𝑣) = [∑ 𝑟(𝑤|𝑣) 𝑟(𝑣|𝑤) 𝑤 ] -1 . (2.22) Since 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ) = 𝛽 𝑛 (𝑥 𝑛 ), (2.22) is equivalent to (2.15). Finally, (2.16) is obtained applying Lemma 2 to 𝑟(.	𝑝(𝑤 𝑛+1 |𝑤 𝑛 ) = …, where 𝛽 1 (𝑤 1 ), …, 𝛽 𝑁 (𝑤 𝑁 ) verify the backward recursion 𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 )𝛽 𝑛+1 (𝑤 𝑛+1 ) , (2.17) (2.2) 𝛽 𝑛 (𝑤 𝑛 ) 𝛽 𝑁 (𝑤 𝑁 ) = 1, 𝛽 𝑛 (𝑤 𝑛 ) = ∑ 𝜑 𝑛 (𝑤 𝑛 , 𝑤 𝑛+1 )𝛽 𝑛+1 (𝑤 𝑛+1 ) 𝑤 𝑛+1 . 𝑞(𝑥 2 |𝑥 1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 2 |𝑥 1 , 𝑦 1:𝑁 ), 𝑞(𝑥 1 |𝑦 1:𝑁 ) = 𝑝(𝑥 1 |𝑦 1:𝑁 ). Let's number the equations (2.17) as follows: equation (2.3) 𝑝(𝑤 1 ,…,𝑤 𝑛 ,𝑤 𝑛+1 ) 𝑝(𝑤 1 ,…,𝑤 𝑛 ) = (2.4) On the one hand, we see that 𝛽 𝑛 (𝑤 𝑛 ) = ∑ 𝜑 𝑛 (𝑤 𝑛 , 𝑤 𝑛+1 )𝛽 𝑛+1 (𝑤 𝑛+1 ) 𝑤 𝑛+1 . On the other hand, according to (2.4) we 𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 )𝛽 𝑛+1 (𝑤 𝑛+1 ) 𝛽 𝑛 (𝑤 𝑛 ) . As 𝑝(𝑤 1 ) = 𝛽 1 (𝑤 1 ) ∑ 𝛽 1 (𝑤 1 ) 𝑤 1 𝑝(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ). , (2.2) and (N -𝑛 -1) th equation 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ) = have 𝑝(𝑤 𝑛+1 |𝑤 𝑛 ) = (1verified for
							(2.3) are verified, which ends the proof.  We have
							Proposition 2.1 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ) = 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 , 𝑦 𝑛+1:𝑁 ) =
	for 𝑛 = 𝑁 -1, …, 2:					(2.14)	Let 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) be a PMM of form (1.1). Then 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ) is a of form (1.2), with 𝑞(𝑥 𝑛 , 𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑦 1:𝑛 ,) = (2.18) 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 )𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑥 𝑛 , 𝑦 1:𝑛 ) 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) .
	𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) = [∑ where 𝛽 𝑛 (𝑥 𝑛 ) is computed from 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 ) , 𝛽 𝑛 (𝑥 𝑛 ) 𝑦 𝑛+1:𝑁 ] -1 , (2.15) 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 1:𝑁 ) 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ), and 𝛽 𝑛+1 (𝑥 𝑛+1 ) with (2.7);	𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = where 𝛽 𝑛 (𝑥 𝑛 ) are defined with 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )𝛽 𝑛+1 (𝑥 𝑛+1 ) 𝛽 𝑛 (𝑥 𝑛 ) As 𝑞 is a PMM, we can write 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑥 𝑛 , 𝑦 1:𝑛 ) = 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ), (2.19) ,(2.5) 𝛽 𝑛 (𝑥 𝑛 ) = 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ), (2.6) so that finally we have
	𝑞(𝑥 1 |𝑦 1 ) = [∑	𝑦 2:𝑁	𝑞(𝑦 2:𝑁 |𝑥 1 ,𝑦 1 ) 𝑝(𝑥 1 |𝑦 1:𝑁 )	] -1	.	(2.16)	and can be computed recursively with 𝛽 𝑁 (𝑥 𝑁 ) = 1; 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑁 ) = 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 )𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 )
							𝛽 𝑛 (𝑥 𝑛 ) = ∑	𝑥 𝑛+1	𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )𝛽 𝑛+1 (𝑥 𝑛+1 )	(2.7)

1 |𝑦 1 ), 𝑞(𝑥 2 , 𝑦 2 |𝑥 1 , 𝑦 1 ), 𝑞(𝑥 3 , 𝑦 3 |𝑥 1 , 𝑦 1:2 ),…, 𝑞(𝑥 𝑁 , 𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ), defining all equivalent PMMs (1.1), from 𝑝(𝑥 1 |𝑦 1:𝑁 ) , 𝑝(𝑥 2 |𝑥 1 , 𝑦 1:𝑁 ), …, 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) defining the CRF (1.2) under consideration.

We will use the following Lemma 1.

Lemma 1.

Let 𝑊 1:𝑁 = (𝑊 1 , … , 𝑊 𝑁 ) be random sequence, taking its values in a finite set 𝛥. Then (i) 𝑊 1:𝑁 is a Markov chain iff there exist 𝑁 -1 functions 𝜑 1 , … , 𝜑 𝑁-1 from 𝛥 2 to ℝ + such that

𝑝(𝑤 1 , … , 𝑤 𝑁 ) = 𝐾𝜑 1 (𝑤

1 , 𝑤 2 ) … 𝜑 𝑁-1 (𝑤 𝑁-1 , 𝑤 𝑁 ), (2.1) with 𝐾 constant; (ii) for HMM defined with 𝜑 1 , …, 𝜑 𝑁-1 verifying (2.1), 𝑝(𝑤 1 ) , and 𝑝(𝑤 𝑛+1 |𝑤 𝑛 ) for 𝑛 = 2, …, 𝑁 -1 , are given with 𝑝(𝑤 1 ) = 2. Conversely, let 𝑝(𝑤 1 , … , 𝑤 𝑁 ) verifies (2.1). This implies that for each 𝑛 = 1, …, 𝑁 -1 we have

𝑝(𝑤 𝑛+1 |𝑤 1 , … , 𝑤 𝑛 ) = ∑ 𝜑 1 (𝑤 1 ,𝑤 2 ) …𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 )𝜑 𝑛+1 (𝑤 𝑛+1 ,𝑤 𝑛+2 )…𝜑 𝑁-1 (𝑤 𝑁-1 ,𝑤 𝑁 ) (𝑤 𝑛+2 ,…,𝑤 𝑁 ,) ∑ 𝜑 1 (𝑤 1 ,𝑤 2 ) …𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 )𝜑 𝑛+1 (𝑤 𝑛+1 ,𝑤 𝑛+2 )…𝜑 𝑁-1 (𝑤 𝑁-1 ,𝑤 𝑁 ) (𝑤 𝑛+1 ,…,𝑤 𝑁 ,) = 𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 ) ∑ 𝜑 𝑛+1 (𝑤 𝑛+1 ,𝑤 𝑛+2 )…𝜑 𝑁-1 (𝑤 𝑁-1 ,𝑤 𝑁 ) (𝑤 𝑛+2 ,…,𝑤 𝑁 ,) ∑ 𝜑 𝑛 (𝑤 𝑛 ,𝑤 𝑛+1 )𝜑 𝑛+1 (𝑤 𝑛+1 ,𝑤 𝑛+2 )…𝜑 𝑁-1 (𝑤 𝑁-1 ,𝑤 𝑁 ) (𝑤 𝑛+1 ,𝑤 𝑛+2 ,…,𝑤 𝑁 ,) = 𝑝(𝑤 𝑛+1 | 𝑤 𝑛 ),

which shows that 𝑝(𝑤 1 , … , 𝑤 𝑁 ) is Markov. Besides, let us set, for 𝑛 = 1, …, 𝑁 -1: 𝛽 𝑛 (𝑤 𝑛 ) = ∑ 𝜑 𝑛 (𝑤 𝑛 , 𝑤 𝑛+1 ) … 𝜑 𝑁-1 (𝑤 𝑁-1 , 𝑤 𝑁 ) (,𝑤 𝑛+1 ,…,𝑤 𝑁 )

.

We specify the converse proposition in Theorem below:

Theorem. Let 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) = (2.

11) 𝑝(𝑥 1 |𝑦 1:𝑁 )𝑝(𝑥 2 |𝑥 1 , 𝑦 1:𝑁 ) … 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ). is equivalent to 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) if and only if (i) sequence 𝑞(𝑦 1 ), 𝑞(𝑦 2 |𝑥 1 , 𝑦 1 ), …, 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) (2.13) is arbitrary; (ii) for given sequence (2.13), 𝑞(𝑥 1 |𝑦 1 ), 𝑞(𝑥 2 |𝑥 1 , 𝑦 1:2 ), …, 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) is defined from 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) and (2.13) with the following backward recursion: Proof. Let 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) be a given CRF (2.11), and 𝑞(𝑦 1 ), 𝑞(𝑦 2 |𝑥 1 , 𝑦 1 ), …, 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) arbitrary sequence. 1:𝑁 ) , …, 𝑞(𝑥 1 |𝑦 1:𝑁 ) verifying (2.15), then the (𝑁 -𝑛 -1)th equation also is. As 𝑞(𝑥 𝑁 , 𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ) = 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 )𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ), we see that the first equality in (2.17) is verified for 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) and for any 𝑞(𝑦 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁-1 ). Let us assume that 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) , …, 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛+1 ) verify (2.16), and verify (2.17) for 𝑁 -𝑛 first equations. Let us search all 𝑞(𝑥 𝑛 , 𝑦 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛-1 ) such that (2.16) is 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) (2.20) Then the question is: knowing that 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ) in (2.20) is given (recall that it is equal to 𝛽 𝑛 (𝑥 𝑛 ), given with recursion (2.7) from 𝑞(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 1:𝑁 ) , …, ) distributions on Λ conditional on 𝑦 1 , and setting 𝑣 = 𝑥 1 , 𝑤 = 𝑦 2:𝑁 , which completes the proof. Lemma 2. Let 𝑉, 𝑊 be two discrete finite random variables taking their values in 𝛬 . Let 𝑟(𝑤|𝑣) , 𝑟(𝑣|𝑤) be two conditional distributions. Then 𝑟(𝑣) is given by (2.22).

  Proof. 1. Let 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) be a PMC verifying 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ). Thus for 𝑛 = 1, …, 𝑁 -1, 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) . Setting 𝛽 𝑛 (𝑥 𝑛 ) = 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 , 𝑦 1:𝑛 ) for 𝑛 = 1, …, 𝑁 -1, and 𝛽 𝑁 (𝑥 𝑁 ) = 1, we have, according to the classic properties of PMCs, 𝛽 𝑛 (𝑥 𝑛 ) = ∑ 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 )𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 )𝛽 𝑛+1 (𝑥 𝑛+1 )

	Proposition 3.1 A CRF (1.2) is equivalent to a PMC (1.6) if and
	only if for 𝑛 = 1, …, 𝑁 -1,
	𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1:𝑁 ).	(3.1)
	𝑥 𝑛+1

  1, replacing 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) with 𝑐 = 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) . As above, we arrive at a model 𝑟(𝑥 1:𝑁 , 𝑦 1:𝑁 ), which may seem odd, in which 𝑌 1 , …, 𝑌 𝑁 are independent and uniformly distributed on Λ, with 𝑟(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 , 𝑦 𝑛+1 ) transitions of a classic HMM. Of course, such a PMM is not an HMM. 𝑥 𝑛 |𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ), whereas in the HMM 𝑝(𝑥 𝑛 |𝑦 1:𝑁 ) depends on all components of 𝑦 1:𝑁 . In other respects the CRF given with HMM is less general then the MEMM. For example, in MEMM 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ) are of any form, while in HMM, according to(1.4), they are proportional to the product 𝑝(𝑥 𝑛+1 |𝑥 𝑛 )𝑝(𝑦 𝑛+1 |𝑥 𝑛+1 ), which is therefore a particular form. Moreover, we can notice that MEMM is a CRF verifying 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑁 ) = 𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ). Therefore, according to Proposition 2, MEMM is equivalent to a PMC 𝑞(𝑥 1:𝑁 , 𝑦 1:𝑁 ) given with transitions 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) = 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 )𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 , 𝑦 𝑛+1 ) , where 𝑞(𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) are arbitrary. Taking them constant as in example 1, we have𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 1:𝑛 ) = 𝑞(𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 𝑛-1 , 𝑦 𝑛 ) = 𝑥 𝑛 |𝑥 𝑛-1 , 𝑦 𝑛 ),and so MEMM is equivalent to PMC given with transitions 𝑞(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛 ) = c𝑝(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ). The advantage of PMM over the CRF form can be seen in the filtering problem. Let 𝑝(𝑥 1:𝑛 |𝑦 1:𝑛 ) be a CRF, and 𝑞(𝑥 1:𝑛 , 𝑦 1:𝑛 ) an equivalent PMM. The is problem is to estimate 𝑋 𝑛 from 𝑌 1:𝑛 , which requires the calculation 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ) . In filtering, it is interesting to calculate 𝑝(𝑥 1 |𝑦 1 ), 𝑝(𝑥 2 |𝑦 1:2 ), …, 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ), 𝑝(𝑥 𝑛+1 |𝑦 1:𝑛+1 ), … as quickly as possible. Let us assume that 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ) = 𝑞(𝑥 𝑛 |𝑦 1:𝑛 ) are given, each with its own parametrization. When using the equivalent CRF form, there is no direct link between 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ) and 𝑝(𝑥 𝑛+1 |𝑦 1:𝑛+1 ), so 𝑝(𝑥 𝑛 |𝑦 1:𝑛 ) cannot be used directly and 𝑝(𝑥 𝑛+1 |𝑦 1:𝑛+1 ) is calculated as the marginal distribution of the Markov chain 𝑝(𝑥 1 |𝑦 1:𝑛+1 ), 𝑝(𝑥 2 |𝑦 1:𝑛+1 ),…, 𝑝(𝑥 𝑛+1 |𝑦 1:𝑛+1 ). This requires 𝑛 sums over Λ, so its complexity is 𝑛(#Λ), which can be a problem when 𝑛 increases. Using the PMM form, we have 𝑝(𝑥 𝑛+1 , 𝑦 1:𝑛+1 ) = ∑ 𝑝(𝑥 𝑛 , 𝑥 𝑛+1 , 𝑦 1:𝑛+1 ) 𝑥 𝑛 , 𝑦 1:𝑛 )𝑝(𝑥 𝑛+1 , 𝑦 𝑛+1 |𝑥 𝑛 , 𝑦 1:𝑛 )

	1 #Λ will not change 𝑞(𝑥 1:𝑁 |𝑦 1:𝑁 ), so PMM 𝑟(𝑥 1:𝑁 , 𝑦 1:𝑁 ) defined with 𝑟(𝑥 1:𝑁 , 𝑦 1:𝑁 ) = 𝑐 𝑁 𝑞(𝑥 1 |𝑦 1 ) ∏ 𝑞(𝑥 𝑛+1 |𝑥 𝑛 , 𝑦 𝑛+1 ) 𝑁-1 𝑛=1 (3.5) is equivalent to the HMM Example 3.3 Let us consider the Maximum Entropy Markov Models (MEMMs) 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) introduced in [33], which can be seen as ancestor of CRFs. They verify 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 ) = 𝑝(𝑥 1 |𝑦 1 )𝑝(𝑥 2 |𝑥 1 , 𝑦 2 ) … 𝑝(𝑥 𝑁 |𝑥 𝑁-1 , 𝑦 𝑁 ) (3.6) MEMMs have been proposed as alternative to HMMs. MEMMs are neither extensions of HMMS nor particular HMMs. Indeed, in some respects the CRF given with HMM (simply its posterior distribution 𝑝(𝑥 1:𝑁 |𝑦 1:𝑁 )) is more general than MEMM: for example, in MEMM 𝑝([∑ [ 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 𝑛 ) 𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 ,𝑦 1:𝑛 ) 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 1:𝑁 ) 𝑦 𝑛+1:𝑁 1 ] -1 = 𝑝(Remark 3.1 𝑥 𝑛 ∑ 𝑝(𝑥 𝑛	] -1	= [∑	𝑦 𝑛+1:𝑁	𝑞(𝑦 𝑛+1:𝑁 |𝑥 𝑛 ,𝑦 1:𝑛 ) 𝑝(𝑥 𝑛 |𝑥 𝑛-1 ,𝑦 𝑛 ) =	] -1	=

We also noticed that likely in HMMs studied in [3], in PMMs Bayesian generative classifiers can be calculated in a discriminative manner.

Funding: this research was partly funded by the French Government Agency Association Nationale Recherche technologie (ANRT).