Lydie Naulé

Clarisse Marie-Luce

Caroline Parmentier

Mariangela Martini

Christelle Albac

Anne-Charlotte Trouillet

Matthieu Keller

Hélène Hardin- Pouzet

Sakina Mhaouty-Kodja
email: sakina.mhaouty-kodja@upmc.fr

Revisiting the neural role of estrogen receptor beta in male sexual behavior by conditional mutagenesis

Keywords: Sex steroid hormones, Estrogen receptor beta, Nervous system, Estradiol, Sexual behavior, Conditional mutagenesis, Male reproduction, Sexual dimorphism

Estradiol derived from neural aromatization of gonadal testosterone plays a key role in the perinatal organization of the neural circuitry underlying male sexual behavior. The aim of this study was to investigate the contribution of neural estrogen receptor (ER)  in estradiolinduced effects without interfering with its peripheral functions. For this purpose, male mice lacking ER in the nervous system were generated. Analyses of males in two consecutive tests with a time interval of two weeks showed an effect of experience, but not of genotype, on the latencies to the first mount, intromission, pelvic thrusting and ejaculation. Similarly, there was an effect of experience, but not of genotype, on the number of thrusts and mating length. Neural ER deletion had no effect on the ability of males to adopt a lordosis posture in response to male mounts, after castration and priming with estradiol and progesterone. Indeed, only low percentages of both genotypes exhibited a low lordosis quotient. It also did not affect their olfactory preference. Quantification of tyrosine hydroxylase-and kisspeptinimmunoreactive neurons in the preoptic area showed unaffected sexual dimorphism of both populations in mutants. By contrast, the number of androgen receptor-and ERimmunoreactive cells was significantly increased in the bed nucleus of stria terminalis of mutant males. These data show that neural ER does not play a crucial role in the organization and activation of the neural circuitry underlying male sexual behavior. These discrepancies with the phenotype of global ER knockout models are discussed.

Introduction

In male rodents, sexual behavior is induced by olfactory cues. Pheromonal cues are transmitted from the main olfactory epithelium and vomeronasal organ to, respectively, the main and accessory olfactory bulbs, then to chemosensory responsive nuclei in the medial amygdala (MeA), bed nucleus of stria terminalis (BNST), and medial preoptic area (MPOA) where they are processed in behavioral responses. This neural circuitry is under the tight control of gonadal hormones. Estradiol derived from neural aromatization of perinatal testosterone induces irreversible masculinization and defeminization processes (Schwarz and McCarthy, 2011). Masculinization is the potentiation of neuroanatomical and behavioral patterns that are exhibited to a greater degree by males than females (e.g., preference for receptive females and copulatory behaviors). Defeminization is the loss of the ability to display female-typical behaviors such as preference for males and receptive mating posture (lordosis). The organizational effects of estradiol result in sex differences at the structural, neurochemical and molecular levels along the circuitry involved in the control of sexual behavior and reproductive functions. For instance, a cluster of calbindin-immunoreactive neurons in the MPOA, corresponding to the rat sexually dimorphic nucleus involved in sexual behavior, contains more cells in males than in females [START_REF] Orikasa | Estrogen configures sexual dimorphism in the preoptic area of C57BL/6J and ddN strains of mice[END_REF].

Inversely, neurons expressing tyrosine hydroxylase (TH) or kisspeptin are less numerous in males compared to females in the anteroventral periventricular nucleus (AVPV), a subdivision of the medial preoptic area involved in the ovulatory surge of LH [START_REF] Clarkson | Postnatal development of kisspeptin neurons in mouse hypothalamus; sexual dimorphism and projections to gonadotropin-releasing hormone neurons[END_REF][START_REF] Kauffman | Sexual differentiation of Kiss1 gene expression in the brain of the rat[END_REF][START_REF] Simerly | The distribution of monoaminergic cells and fibers in a periventricular preoptic nucleus involved in the control of gonadotropin release: immunohistochemical evidence for a dopaminergic sexual dimorphism[END_REF].

Estradiol acts mainly through two nuclear receptors (ER)  and  encoded by two different genes. Genetic studies highlighted the role of ER in male reproduction and expression of male sexual behavior since global ER knockout males are infertile and exhibit impaired behavior [START_REF] Ogawa | Behavioral effects of estrogen receptor gene disruption in male mice[END_REF][START_REF] Ogawa | Modifications of testosterone-dependent behaviors by estrogen receptor-alpha gene disruption in male mice[END_REF]Wersinger et al., 1997). The involvement of ER in estradiol-induced effects needs further clarification. The analysis of the first genetic model with global ER deletion [START_REF] Krege | Generation and reproductive phenotypes of mice lacking estrogen receptor beta[END_REF] showed that mutant males are fertile and display normal sexual behavior and olfactory preference [START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF][START_REF] Ogawa | Survival of reproductive behaviors in estrogen receptor beta gene-deficient (betaERKO) male and female mice[END_REF]. A transient effect of ER deletion was observed around the time of puberty since peripubertal mutants displayed delayed ejaculation behaviour [START_REF] Temple | Lack of functional estrogen receptor beta gene disrupts pubertal male sexual behavior[END_REF]. When mutant males were castrated at adulthood and primed with estradiol and progesterone, they displayed a higher lordosis behavior than wild-types [START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF]. At the neuroanatomical level, it was found that the number of TH-immunoreactive cells was increased in the AVPV region of mutant males by comparison to wild-types [START_REF] Bodo | Both estrogen receptor-alpha and -beta are required for sexual differentiation of the anteroventral periventricular area in mice[END_REF]. This suggested that ER mediates the estradiol-induced defeminization of the male brain. Global ER deletion also affected the sexually dimorphic expression of ER in the preoptic area [START_REF] Temple | Estrogen receptor beta regulates sexually dimorphic neural responses to estradiol[END_REF]. By contrast, in the BNST, the volume and neuronal number, which are more important in males than females, were not affected [START_REF] Tsukahara | Effects of aromatase or estrogen receptor gene deletion on masculinization of the principal nucleus of the bed nucleus of the stria terminalis of mice[END_REF]. More recently, a global ER knockout mouse line, devoid of any ER transcript, was generated by using the Cre-loxP system [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF]. These mutant males are infertile and exhibit mildly impaired sexual behavior [START_REF] Antal | Estrogen dependent activation function of ERβ is essential for the sexual behavior of mouse females[END_REF]. They display higher numbers of mounts and intromissions as well as delayed ejaculation, but these deficits were improved by sexual experience. In this mouse model, the involvement of ER in the defeminization processes of the male brain has not been studied.

Although useful, the global genetic models limit the understanding of the neural contribution of ERβ, due to the ubiquitous nature of the gene deletion. Estrogens through ERβ which is expressed in the testis, epididymis and prostate [START_REF] Saunders | Expression of oestrogen receptor beta (ER beta) occurs in multiple cell types, including some germ cells, in the rat testis[END_REF][START_REF] Van Pelt | Ontogeny of estrogen receptor-beta expression in rat testis[END_REF], play also a role in the physiology of the male urogenital tract [START_REF] Imamov | Estrogen receptor beta regulates epithelial cellular differentiation in the mouse ventral prostate[END_REF][START_REF] Sar | Oestrogen receptor alpha and beta in rat prostate and epididymis[END_REF][START_REF] Wahlgren | Estrogen receptor beta selective ligand 5alpha-Androstane-3beta, 17beta-diol stimulates spermatogonial deoxyribonucleic acid synthesis in rat seminiferous epithelium in vitro[END_REF]. The present study was undertaken in order to investigate the neural implication of ER in the masculinization and defeminization of the neural circuitry underlying male sexual behavior, without interference with its peripheral functions. For this purpose, we generated a mouse line lacking ER in the nervous system by using Cre-loxP technology. Male sexual behavior was analyzed in both naïve and sexually experienced males in the presence of receptive females. The ability of males to adopt lordosis posture in response to mounts of stud males was also analyzed. The effects of neural ER mutation on the organization of TH-and kisspeptin-immunoreactive neurons located in the sexually dimorphic rostral periventricular area of the third ventricle (RP3V) were investigated. Finally, the potential impact of neural ER deletion on the expression of androgen receptor (AR) and ER expression was evaluated in brain areas underlying male sexual behavior.

Material and methods

Animals

The ER NesCre mouse line was obtained, on a C57BL/6J genetic background, by crossing floxed ER females in which exon 3 of ERβ was flanked by loxP sites [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF] with floxed ER males expressing the Cre recombinase under the control of the rat nestin (Nes) promoter and neural-specific enhancer [START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF] as recently described [START_REF] Naulé | Delayed pubertal onset and prepubertal Kiss1 expression in female mice lacking central oestrogen receptor beta[END_REF]. Cre-mediated excision of floxed exon 3 of the ER gene allows the deletion of all ER transcripts [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF]. Mutant mice (ERβ fl/fl carrying the NesCre transgene; ERβ NesCre ) and their control littermates (ERβ fl/fl ) were group-housed under a controlled photoperiod (12:12-h light-dark cyclelights on at 7 am), maintained at 22°C, with free access to food and water. All studies were performed on 2-4 months old animals, in accordance with the European guidelines for use of experimental animals (Decree 87-848, 86/609/ECC). Experiments were performed accordingly, to minimize animal number and discomfort and were approved by the local Department of Animal Protection and Health.

PCR and RT-PCR

Neural ER invalidation was confirmed by both PCR and RT-PCR. The lack of antibodies specific enough against ER receptor [START_REF] Snyder | Multiple ERbeta antisera label in ERbeta knockout and null mouse tissues[END_REF] did not allow analyses at the protein level. For PCR, detection of the Cre recombinase and ER alleles in DNA extracts from adult and neonatal brains was performed as previously described [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF][START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF]. For RT-PCR, total RNAs were extracted from the brain and epididymis using Trizol reagent (Invitrogen, Carlsbad, USA). RNA (2 μg) was reverse transcribed using the Superscript III first strand Synthesis System (In vitrogen). PCR reactions were performed using the resulting cDNA, Taq DNA pol (In vitrogen), dNTPs (10 nM each), forward (5'-CAGAGAGACCCTGAAGAGGA-3') and reverse (5'-CCTTGAATGCTTCTTTTAAA-3')

Urogenital tract, hormone levels and fertility

Intact animals were sacrificed to collect blood and to weigh seminal vesicles. Sera were extracted and circulating levels of testosterone were measured by RIA at the hormonal assay platform of the laboratory of behavioral and reproductive physiology (UMR 7247

INRA/CNRS/Université François Rabelais) using 3 H-T, as previously described [START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. The mean intra-assay coefficient of variation was 7% and assay sensitivity was 125 pg/ml.

To evaluate fertility, three months-old males (4 per genotype) were mated for 4 months. Each male was individually housed with two age-matched females. The number of pups and the interval from mating to the first litter were recorded.

Behavioral analyses

Tests were conducted under red-light illumination 2 hours after lights-off and videotaped for analyses.

Male-typical behaviors of intact males

Male sexual behavior

Intact animals were individually housed 3 days before the first test. Each male was tested in its home cage for 10 h after the introduction of an estrus female. They were tested twice with a time interval of two weeks. Male sexual behavior was analyzed by scoring the latency and the frequency of mounts, intromissions, thrusts and ejaculation as previously described [START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF]. Estrus C57BL/6J females used as stimuli were ovariectomized under general anesthesia (xylazine 10 mg/kg / ketamine 100 mg/kg), implanted with SILASTIC implants filled with 50 μg of estradiol-benzoate (Sigma-Aldrich, Saint Louis, United States) in 30 μl of sesame oil and subcutaneously treated with 1 mg of progesterone (Sigma-Aldrich) in 100 μl of sesame oil four to five hours before the tests, as previously reported [START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF]. Female receptivity was verified before the beginning of experiments as following.

Each female was put in the presence of a sexually experienced male, which was not in contact with a female for at least 1 week. The female was considered receptive when she displayed a lordosis posture with the four paws grounded, the hind region lifted and the back arched in response to male mounts.

For each male, the latencies from female introduction to the first mount, intromission, thrusting and to ejaculation were measured. The total number of mounts, without and with intromissions, and the total number of thrusts were measured. Mating length was defined as the time from the first mount to ejaculation.

Olfactory preference

Sexually experienced males were placed into an enclosed Plexiglas Y-maze without any stimuli, for 5 min on two consecutive days, to allow them to adapt to the apparatus. Animals were tested for mate preference on the third day by placing an anesthetized receptive female and gonadally intact male in boxes with perforated partitions at the end of each distal arm as previously described [START_REF] Keller | Sexual experience does not compensate for the disruptive effects of zinc sulfate--lesioning of the main olfactory epithelium on sexual behavior in male mice[END_REF]. The time spent sniffing at each partition was scored over the five-minute test. Results are expressed as a percentage of total time spent sniffing male or female cues. The maze was cleaned with 10% ethanol between trials [START_REF] Naulé | Neuroendocrine and behavioral effects of maternal exposure to oral bisphenol A in female mice[END_REF].

Female-typical behaviors of castrated males primed with estradiol and progesterone

Lordosis behavior

Males (22 ER fl/fl and 21 ER NesCre mice) were castrated under general anesthesia (xylazine / ketamine). Four weeks later, they were tested for female sexual behavior in three consecutive tests conducted at one-week interval as previously described [START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. Briefly, subjects were subcutaneously injected with estradiol-benzoate (10 μg dissolved in 100 μl of sesame oil) 48 h prior to the test and progesterone (1 mg in 100 l of sesame oil) four hours before the tests. Experimental males were put in the presence of sexually experienced C57BL/6J male mice serving as stimulus animals. Tests ended when the subject received 20 mounts or after 20 minutes of test. The lordosis posture in response to stud male mounting was determined as mentioned above. The lordosis quotient was calculated only for the males, which received 20 mounts, as the number of times the male adopts a lordosis-like posture in response to stimulus male mount. A group of females (n = 10) used as controls for the lordosis behavior test was ovariectomized, implanted with estradiol and primed with progesterone as described above. They were tested twice in the presence of stud males at oneweek interval.

Olfactory preference

Tests were performed as described above for intact males. Males castrated and primed with estradiol and progesterone were tested 1 week after lordosis behavior tests.

Immunohistochemistry

Intact males were sacrificed and transcardially perfused with a solution of 4% paraformaldehyde (PFA) in phosphate buffer (PB). Brains were post-fixed overnight in 4%

PFA-PB, cryoprotected in sucrose and stored until analyses. They were sliced into coronal sections of 30 µm using a cryotome (Leica CM 3000). Kisspeptin, AR-and ERαimmunostaining were carried as previously described [START_REF] Naulé | Neuroendocrine and behavioral effects of maternal exposure to oral bisphenol A in female mice[END_REF][START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF].

For TH-immunostaining, the sections were blocked for 2 h with 2% normal donkey serum (Sigma-Aldrich) in PB saline (PBS) containing 0.1% Triton-X100 and 0.25% human albumin, then incubated with polyclonal anti-TH antibody (1:5000; Chemicon, Temecula, United States) overnight. Immunofluorescence was performed with a CY3 donkey anti-rabbit secondary antibody (1:500, Jackson Immunoresearch, Montlucon, France) for 2.5 h at room temperature. After several rinses in PBS, sections were rinsed in water, dried, mounted in Fluoromount-G (Southern Biotechnology, Birmingham, AL, USA) under a coverslip and stored at 4°C in the dark.

The numbers of kisspeptin-, TH-, AR and ER-immunoreactive cells were counted in anatomically matched sections identified using the Mouse Brain Atlas of [START_REF] Paxinos | The mouse brain in stereotaxic coordinates[END_REF] as previously described [START_REF] Naulé | Neuroendocrine and behavioral effects of maternal exposure to oral bisphenol A in female mice[END_REF][START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. Kisspeptinimmunoreactive cells were analyzed within each of the three subdivisions of the rostral periventricular area of the third ventricle within an area of 0.24 mm 2 including the AVPV nucleus (plates 28-29) and the preoptic periventricular nucleus, divided into rostral (plate 30)

and caudal regions (plates 31-32). TH-immunoreactive cells were counted in the AVPV within an area of 0.24 mm 2 (plates 27-31). AR and ER-immunoreactive cells were analyzed in the MPOA within an area of 0.68 mm 2 , in the BNST within an area of 0.70 mm 2 (plate 30), and in the MeA within an area of 0.56 mm 2 (plate 47).

Statistical analysis

Data were expressed as mean ± S.E.M. Student's t-tests were used to determine the effect of genotype on circulating levels of testosterone, weight of seminal vesicles and fertility. Effect sizes were further estimated by calculating the Cohen's d (d = M/SD, where M is the mean of differences and SD is the standard deviation of differences; d = 0.2 is considered as a small effect size, d = 0.5 as a medium effect size and d = 0.8 as a large effect size). Two-way ANOVA was used to analyze the main effects of genotype and experience on male sexual behavior and lordosis quotient or genotype and stimulus on olfactory preference. Tukey posthoc tests were used to determine group differences. Effect sizes were further estimated by calculating the eta-squared η 2 (η 2 = SS effect / SS total , where SS effect is the sums of squares for the effect of interest and SS total is the total sums of squares for all effects, interactions and errors, η 2 = 0.02 is considered as a small effect size, η 2 = 0.13 as a medium effect size and η 2 = 0.26 as a large effect size). As variances were not homogeneous between groups, TH-, kisspeptin-ERα-and AR immnunoreactivity was analyzed with Mann-Whitney nonparametric test. P values of less than 0.05 were considered to be significant.

Results

General characterization of the ER NesCre mouse line

The

Behavioral effects of neural ER invalidation

Male sexual behavior and olfactory preference of intact males

Comparable percentages of naïve males (86% of ERβ fl/fl and 81% of ER NesCre genotypes) displayed a full range of sexual behavior and reached ejaculation. Analyses of the latencies to the first behaviors showed a significant effect of experience for the latencies to the first mount (F (1.28) = 11.240, p = 0.002, η 2 = 0.141), pelvic thrusting (F (1,28) = 12.400, p = 0.001, η 2 = 0.165), intromission (F (1.28) = 13.450, p = 0.002, η 2 = 0.156) and latency to ejaculation (F (1.28) = 14.730, p < 0.001, η 2 = 0.186) as illustrated in Fig. 2A. There was no significant effect of genotype on the latencies of these behaviors (mount: F (1.28) = 2.100, p = 0.160, η 2 = 0.035; intromission: F (1.28) = 2.650, p = 0.115, η 2 = 0.042; thrusting: F (1.28) = 2.890, p = 0.100, η 2 = 0.046; ejaculation: F (1.28) = 2.960, p = 0.096, η 2 = 0.043). The number of mounts without (M)

or with intromission (MI), the total number of thrusts and mating length for both naïve and experienced males were also quantified (Table 1). Two-way ANOVA showed a significant effect of experience for the number of thrusts (F (1,28) = 6.920, p = 0.014, η 2 = 0.105) and mating length (F (1,28) = 4.520, p = 0.042, η 2 = 0.070) but not of genotype (F (1,28) = 0.610, p = 0.441, η 2 = 0.010 and F (1,28) = 0.710, p = 0.408, η 2 = 0.012, respectively). There was no significant effect of experience (M: F (1,28) = 0.250, p = 0.622, η 2 = 0.005; MI: F (1,28) = 0.850, p = 0.360, η 2 = 0.012) or genotype (M: F (1,28) = 0.070, p = 0.799, η 2 = 0.001; MI: F (1,28) = 1.410, p = 0.240, η 2 = 0.027) on the other components of mating.

The ability of males to discriminate between male and female pheromones in tests using gonad-intact male versus estrus female was tested. There was a significant effect of stimulus

Lordosis posture and olfactory preference of castrated males primed with estradiol and progesterone

The ability of males to exhibit a typical female posture after castration and priming with estradiol and progesterone was measured. The percentage of males receiving 20 mounts from experienced intact males over the three tests averaged 50% in ERβ The males were then subjected to olfactory preference tests. There was no significant effect of stimulus (F (1,33) = 0.050, p = 0.825, η 2 = 0.002) or genotype (F (1,33) = 1.210, p = 0.290, η 2 = 0.033), indicating that males of the two genotypes displayed no olfactory preference (Fig. 2D). The total time spent investigating the two cues was similar between the two genotypes (117.97  9.03 sec versus 106.99  4.98 sec, respectively; t = 1.098, p = 0.290, d = 0.570).

Neuroanatomical organization of the medial preoptic area

The medial preoptic area, a key target of perinatal estradiol, contains known sexually dimorphic neuronal populations. Global deletion of ER was shown to increase the number of TH neurons in the AVPV, a subdivision of the medial preoptic area, suggesting that ER is involved in brain perinatal feminization of this region [START_REF] Bodo | Both estrogen receptor-alpha and -beta are required for sexual differentiation of the anteroventral periventricular area in mice[END_REF]. Thus, it was assessed whether neural ER invalidation alters the neuronatomical organization of THimmunoreactive cells. In accordance with previous studies [START_REF] Simerly | The distribution of monoaminergic cells and fibers in a periventricular preoptic nucleus involved in the control of gonadotropin release: immunohistochemical evidence for a dopaminergic sexual dimorphism[END_REF], the number of TH-immunoreactive cells was greater (2.4-fold) in females than in ERβ fl/fl males (p = 0.034; Fig. 3A andC). ER NesCre males showed a male pattern since no significant differences were seen with their ERβ fl/fl littermates (p = 0.885). The number of kisspeptinimmunoreactive cells, another sexually dimorphic population, was then quantified in the three subdivisions of the rostral periventricular area of the third ventricle (RP3V). Data show sex differences, with females exhibiting 24 to 225-fold higher number of kisspeptin-ir neurons in the AVPV (p = 0.026), rostral (p = 0.034) and caudal (p = 0.028) periventricular nuclei than ERβ fl/fl males (Fig. 3B andD). Again, no significant differences were observed between ERβ fl/fl and ER NesCre males (p = 0.317, p = 0.102 and p = 0.278 for the AVPV, rPeN and cPeN, respectively; Fig. 3D). genotype; Fig. 5A-B).

Quantification of the number of ER-and AR-immunoreactive cells

Discussion

In order to determine the relative contribution of neural sex steroid receptors in reproductive behaviors, a mouse line lacking neural ER was characterized. This genetic model was generated by using the same strategy and NesCre transgene previously described for the mouse line lacking neural AR gene [START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF].

In naïve ER NesCre males, the latencies and frequencies to perform the various components of copulatory behavior were not statistically different from those observed in their control littermates. Sexual experience ameliorated mating in both ERβ fl/fl and ER NesCre males by reducing the latencies to behaviors and mating length, and by increasing the number of thrusts. ER NesCre males exhibited also normal preference towards female olfactory cues.

These results are in agreement with the lack of ER involvement in the masculinization of sexual behavior and olfactory preference previously reported for the initial global ER knockout model [START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF][START_REF] Ogawa | Survival of reproductive behaviors in estrogen receptor beta gene-deficient (betaERKO) male and female mice[END_REF]. They contrast with the recent global ER invalidation [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF], which resulted in increased number of mounts and intromissions and delayed ejaculation, although sexual experience progressively restored these behavioral differences [START_REF] Antal | Estrogen dependent activation function of ERβ is essential for the sexual behavior of mouse females[END_REF]. As these mice were obtained from the same floxed model and similar genetic background as the present conditional model, we suggest that the mild behavioral deficiency induced by this global mutation was probably due to peripheral effects of ER deletion. The present conditional mutation did not alter male fertility and circulating levels of testosterone while the global ER deletion generated by Chambon's laboratory resulted in an infertile phenotype of unknown origin [START_REF] Antal | Sterility and absence of histopathological defects in nonreproductive organs of a mouse ERbeta-null mutant[END_REF]. Whether or not these global ER knockout males exhibit altered regulation of the hypothalamus-pituitary-gonad axis, which may in turn interfere with reproductive behaviors, has not been reported. In males, ERβ is expressed in somatic and germ cells of the testis [START_REF] Van Pelt | Ontogeny of estrogen receptor-beta expression in rat testis[END_REF] and seems to be involved in testosterone production [START_REF] Dumasia | Effect of estrogen receptorsubtype-specific ligands on fertility in adult male rats[END_REF].

It has been shown that ER is important in the defeminization processes induced perinatally by estradiol in the neural circuitry underlying male sexual behavior [START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF]. Mutant males from the initial global ER knockout model [START_REF] Krege | Generation and reproductive phenotypes of mice lacking estrogen receptor beta[END_REF], exhibited a higher lordosis quotient than the wild-type males, after adult castration and priming with estradiol and progesterone. In our conditional model, only a small percentage of ER NesCre males castrated and primed with estradiol and progesterone, showed a lordosis posture. Furthermore, ER NesCre males exhibiting lordosis behavior displayed a low LQ (5%)

in Test 3 while a mean value of 25% was reported for global ER knockout mice [START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF]. Moreover, this low expression of lordosis behavior by ER NesCre males was random across tests, indicating that there was no effect of sexual experience. In similar conditions, a group of control females exhibited a high LQ reaching 71% since Test 2. This demonstrates that the low behavior displayed by ER NesCre males was not due to experimental limitations.

At the neuroanatomical level, the number of TH-ir neurons was not modified in the AVPV of ER NesCre males. These results contrast with previous studies showing increased number of TH-ir neurons in the AVPV of global ER knockout males [START_REF] Bodo | Both estrogen receptor-alpha and -beta are required for sexual differentiation of the anteroventral periventricular area in mice[END_REF]. However, analysis of kisspeptin neurons, another sexually dimorphic population of the RP3V, confirmed the lack of ER NesCre mutation effect on the sexual differentiation of the AVPV.

The discrepancy between the effects of global versus conditional ER mutation on the behavioral and neuroanatomical defeminization of the AVPV can not be attributed to differences in the genetic background as both models were studied on a C57BL6 background [START_REF] Bodo | Both estrogen receptor-alpha and -beta are required for sexual differentiation of the anteroventral periventricular area in mice[END_REF][START_REF] Kudwa | A previously uncharacterized role for estrogen receptor beta: defeminization of male brain and behavior[END_REF]. A possible explanation could be that global ER invalidation altered somehow other important pathways such as neural ER or downstream regulated neurotransmitters or neuropeptides, which in turn interfered with sexual brain differentiation. Increased number of TH-ir neurons has been, indeed, reported in the preoptic area of ER knockout males [START_REF] Simerly | Estrogen receptor-dependent sexual differentiation of dopaminergic neurons in the preoptic region of the mouse[END_REF]. Alternatively, cell types other than neuronal and glial cells targeted by ER NesCre could be responsible of the phenotype observed in global ER knockout mice. Microglia cells were recently shown to be important for estradiol-induced sexual differentiation of the preoptic area and copulatory behavior [START_REF] Lenz | Microglia are essential to masculinization of brain and behavior[END_REF].

Neural invalidation of ER resulted in increased number of AR-and ER-immunoreactive neurons specifically in the BNST. Previous studies reported that ER modulates ER expression in hypothalamic cells [START_REF] Malikov | Regulation of ERα protein expression by 17β-estradiol in cultured neurons of hypothalamic ventromedial nucleus[END_REF]. Whether ER modulates the expression of both AR and ER in the BNST needs further investigation. Nevertheless, such cross-regulations between sex steroid receptor signaling pathways are not uncommon since a similar increase of ER-immunoreactive cell number was noticed in the MeA and MPOA of males lacking neural AR [START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. It is unlikely that the increased amount of AR and ER proteins in the BNST compensates for the lack of ER in the expression of sexual behavior. Indeed, ER was deleted along the neural circuitry underlying sexual behavior and no changes in AR-or ER-immunoreactivity were observed in the MeA or MPOA. The BNST is involved in other behaviors such as anxiety-like behavior [START_REF] Daniel | Stress Modulation of Opposing Circuits in the Bed Nucleus of the Stria Terminalis[END_REF]. Administration of a selective ERβ agonist to ovariectomized female rats has an anxiolytic effect [START_REF] Lund | Novel actions of estrogen receptorbeta on anxiety-related behaviors[END_REF][START_REF] Weiser | Estrogen receptor-beta agonist diarylpropionitrile: biological activities of R-and S-enantiomers on behavior and hormonal response to stress[END_REF]. This anxiolytic-like effect was observed in wild-type female mice but not in global ERβ knockouts [START_REF] Oyola | Anxiolytic effects and neuroanatomical targets of estrogen receptorβ (ERβ) activation by a selective ERβ agonist in female mice[END_REF][START_REF] Walf | Antisense oligodeoxynucleotides for estrogen receptor-beta and alpha attenuate estradiol's modulation of affective and sexual behavior, respectively[END_REF], which exhibit increased anxiety-like behavior [START_REF] Krezel | Increased anxiety and synaptic plasticity in estrogen receptor beta -deficient mice[END_REF]. In agreement with these observations, neural deletion of ER results in increased anxiety-state level during the follicular phase in female mice [START_REF] Naulé | Delayed pubertal onset and prepubertal Kiss1 expression in female mice lacking central oestrogen receptor beta[END_REF]. In male mice, the involvement of ERβ in estrogen-modulated anxiety state still needs to be documented. Minor effects of gene invalidation were reported for global ERβ knockout males [START_REF] Krezel | Increased anxiety and synaptic plasticity in estrogen receptor beta -deficient mice[END_REF], while chronic administration of androgen metabolites with actions at ERβ decreased the anxiety state level in rats [START_REF] Osborne | Chronic administration of androgens with actions at estrogen receptor beta have anti-anxiety and cognitive-enhancing effects in male rats[END_REF]. Future studies will address the effects of neural ER deletion on anxiety-like behavior and aggression, another BNST-linked behavior altered in global ER knockout males [START_REF] Nomura | Genotype/age interactions on aggressive behavior in gonadally intact estrogen receptor beta knockout (betaERKO) male mice[END_REF].

These data together with previous studies suggest that testosterone might regulate male sexual behavior mainly through ER-and AR-signaling pathways. First, global ER knockout males exhibit a severe sexual deficiency as evidenced by their lack of olfactory cues discrimination and partner preference [START_REF] Wersinger | Oestrogen receptor alpha is essential for femaledirected chemo-investigatory behaviour but is not required for the pheromone-induced luteinizing hormone surge in male mice[END_REF], increased latencies to mount, thrust and intromit and inability to ejaculate [START_REF] Ogawa | Behavioral effects of estrogen receptor gene disruption in male mice[END_REF][START_REF] Ogawa | Modifications of testosterone-dependent behaviors by estrogen receptor-alpha gene disruption in male mice[END_REF]Wersinger et al., 1997). It remains however to clarify whether the lack of ejaculation can be attributed solely to central effects of ER mutation since this receptor plays also a role in the physiology of the male urogenital tract [START_REF] Hess | A role for oestrogens in the male reproductive system[END_REF][START_REF] Joseph | Absence of estrogen receptor alpha leads to physiological alterations in the mouse epididymis and consequent defects in sperm function[END_REF]. Second, neural invalidation of AR results also in sexual deficiency [START_REF] Raskin | Conditional inactivation of androgen receptor gene in the nervous system: effects on male behavioral and neuroendocrine responses[END_REF]. Unlike global ER knockout males, males lacking the neural AR exhibit normal olfactory preference and are able to reach ejaculation in the C57BL6/J background, despite longer latencies to initiate the mounting and thrusting behaviors and reduced number of efficient mounts even after a first sexual experience [START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. Neuroanatomical analyses of sexually dimorphic populations in brain areas underlying reproductive behaviors strongly suggest that the neural AR is not involved in their perinatal organization, but can rather mediate their activation during adulthood [START_REF] Marie-Luce | Effects of neural androgen receptor disruption on aggressive behavior, arginine vasopressin and galanin systems in the bed nucleus of stria terminalis and lateral septum[END_REF][START_REF] Picot | Vulnerability of the neural circuitry underlying sexual behavior to chronic adult exposure to oral bisphenol a in male mice[END_REF]. In the spinal sites involved in erection and ejaculation, the neural AR plays a key role in postnatal differentiation and adult maintenance of the spinal nucleus of the bulbocavernosus and gastrin-releasing peptide neuron systems [START_REF] Raskin | Characterization of the spinal nucleus of the bulbocavernosus neuromuscular system in male mice lacking androgen receptor in the nervous system[END_REF][START_REF] Sakamoto | Androgen regulates development of the sexually dimorphic gastrin-releasing peptide neuron system in the lumbar spinal cord: evidence from a mouse line lacking androgen receptor in the nervous system[END_REF]. These data are in good agreement with the ontogeny of AR expression showing that this receptor is expressed after the perinatal period in both brain and spinal areas underlying male-typical behavior [START_REF] Juntti | The androgen receptor governs the execution, but not programming, of male sexual and territorial behaviors[END_REF][START_REF] Smith | Ontogeny of androgen receptor expression in spinal nucleus of the bulbocavernosus motoneurons and their target muscles in male mice[END_REF].

Therefore, the ER may play the main role in the perinatal organization of the brain circuitry underlying sexual behavior. The AR may act postnatally in the spinal cord and lately during pubertal/adult periods at both spinal and brain sites to activate the sexual circuitry. In this context, ER is not required for the organization and activation of sexual behavior. Previous studies suggested a role of this receptor in the timing of male sexual behavior at puberty [START_REF] Temple | Lack of functional estrogen receptor beta gene disrupts pubertal male sexual behavior[END_REF]. This together with our recent work, showing that neural ER deletion alters the timing of pubertal maturation in females [START_REF] Naulé | Delayed pubertal onset and prepubertal Kiss1 expression in female mice lacking central oestrogen receptor beta[END_REF], suggest transient prepubertal functions for ER in both sexes. Further studies will characterize the pubertal phenotype of ER NesCre males.

In conclusion, the evaluation of neural effects of ER by using a conditional knockout model indicates that this receptor is not involved in the masculinization and defeminization of sexual behavior and related brain areas. ER appears then as the dominant estrogen receptor mediating perinatal effects of estradiol, and AR and ER might play complementary roles in the full expression of male sexual behavior. Since the ER NesCre mouse line displayed modifications in ER and AR in the BNST, it will be therefore very useful for the investigation of the mechanisms underlying neural ER involvement in mood and aggressive behaviors without interference with male reproductive functions.

  selective neural deletion of ERβ was confirmed by RT-PCR. A 177 bp-amplified fragment was present at comparable levels of expression in the epididymis of ERβ fl/fl and ER NesCre males (t = -2.543, p = 0.064, d = -2.540). This signal was present in the brain of ERβ fl/fl males, and highly reduced in their ER NesCre littermates (-98%, t = 7.716, p = 0.001, d = 7.660; Fig.1A-B). The NesCre transgene used triggers gene deletion in neural precursor cells by embryonic day 10.5, before gonadal differentiation. To ensure that excision of neural ERβ exon 3 was indeed efficient during the perinatal organization of the male brain, we performed a PCR analysis for ERβ alleles and Cre recombinase on DNA extracted from neonatal and adult brains. A small amplicon of 250 bp indicating Cre-mediated excision of ERβ exon 3 was found in the brain of both neonatal and adult males expressing the Cre recombinase (Fig.1C). By comparison, a 850 bp signal corresponding to the floxed allele was seen in the brain of control littermates lacking the NesCre transgene.Student's t tests showed no significant effect of the ER NesCre mutation on circulating levels of testosterone (t = 0.731, p = 0.470, d = 0.250 versus ERβ fl/fl ; Fig.1D). This result was corroborated by the unchanged weight of the androgen-dependent seminal vesicles in ER NesCre males (t = -1.312, p = 0.198, d = -0.650 versus controls; Fig.1E). In fertility tests, ER NesCre males produced a total number of 45 ± 5.0 pups versus 46 ± 3.0 for ERβ fl/fl mice (t = 0.083, p = 0.936, d = 0.070). The interval from mating to first litter was also similar between the two genotypes (22 ± 0.4 days in ER NesCre versus 21 ± 1.0 days in ERβ fl/fl ; t = 0.600, p = 0.570, d = 0.490).

(F

  (1,25) = 7.020, p = 0.023, η 2 = 0.211) but not of genotype (F (1,25) = 0.350, p = 0.565, η 2 = 0.011; Fig.2B). The total time devoted to chemoinvestigation was not significantly different between sexually experienced ERβ fl/fl and ERβ NesCre males (128.06  10.35 sec versus 150.38  12.93 sec, respectively; t = 0.601, p = 0.560, d = 0.360).

  ER and AR signaling pathways play an important role in the expression of male sexual behavior. It was thus evaluated whether neural ER invalidation altered the number of ERαand AR-immunoreactive cells in the neural circuitry underlying this behavior. The number of ERα -immunoreactive cells was unchanged in the MeA (p = 0.275) and MPOA (p = 0.513) of ER NesCre males by comparison to ERβ fl/fl males (Fig.4A-B). It was, however, significantly increased by 37% in the BNST (p = 0.050 versus ERβ fl/fl genotype). Similarly, the number of AR-immunoreactive cells was unaltered in the MeA (p = 0.827) and MPOA (p = 0.513) and significantly increased in the BNST of ER NesCre mice (+38%, p = 0.050 versus ERβ fl/fl

Fig. 1 .

 1 Fig. 1. Characterization of the ER NesCre mouse line. (A) RT-PCR of total RNAs obtained

Fig. 2 .

 2 Fig. 2. Effects of neural ER invalidation on sexual behavior and olfactory preference in

Fig. 3 .Fig. 4 .

 34 Fig. 3. Tyrosine hydroxylase (TH) and kisspeptin immunoreactivity in ER fl/fl and

Fig. 5 .

 5 Fig. 5. Quantification of AR-immunoreactive cell number in brain areas of ER fl/fl and

  fl/fl males and 57% in ERβ NesCre mice. Only 14% of ERβ fl/fl and 16% of ERβ NesCre genotypes exhibited a lordosis posture. Statistical analysis of the lordosis quotient (LQ) of these males across the three tests

	showed a significant effect of time (F (2,62) = 4.560, p = 0.014, η 2 = 0.113), but not of genotype
	(F (1,62) = 0.690, p = 0.410, η 2 = 0.009; Fig. 2C). The mean LQ of ERβ fl/fl males averaged 15%
	at Test 2 but then decreased to 1% at Test 3 while it was comprised between 2.5% and 5% in
	ERβ

NesCre mice. To make sure that these low LQ were not due to experimental limitations, we assessed a group of control females in similar conditions. Females exhibited a lordosis behavior with an LQ equivalent to 33.8 ± 7.6% since Test 1; it increased to reach 71.0 ± 8.2% in Test 2 (paired Student's t test, t = -3.510, p = 0.006, d = -4.966).

Table legends .Table 1 . Quantification of the sexual behavior displayed by naïve and sexually experienced ER fl/fl and ER NesCre males.

 legends1 The number of mounts without (M) or with intromission (MI), the total number of thrusts (Th) and mating length are shown for males (n = 13-17 per genotype) tested in Tests 1 and Test 2. a p < 0.05 versus Test 1.

	686
	687
	688
	689
	684 690
	685 691
	33 34 35 36 37

Acknowledgements

We thank Prof. Pierre Chambon (Institut de Génétique et de Biologie Moléculaire et Cellulaire, Illkirch 67404, France) for providing the floxed ERβ mouse line. This work was supported by the "Contaminants-Ecosystèmes-Santé" program of the "Agence Nationale de la Recherche", "Réseau Santé Environnement Toxicologie" of the "Région Ile de France" and by the CNRS, INSERM and UPMC. Wersinger, S.R., Sannen, K., Villalba, C., Lubahn, D.B., Rissman, E.F., De Vries, G.J., 1997. Masculine sexual behavior is disrupted in male and female mice lacking a functional estrogen receptor alpha gene. Horm Behav 32,[176][177][178][179][180][181][182][183]


G Biau

E Scornet

Sylvain Arlot
email: sylvain.arlot@math.u-psud.fr

Comments on: "A Random Forest Guided Tour" by

 about random forests. We focus on the problem of quantifying the impact of each ingredient of random forests on their performance. We show that such a quantification is possible for a simple pure forest, leading to conclusions that could apply more generally. Then, we consider "hold-out" random forests, which are a good middle point between "toy" pure forests and Breiman's original random forests.

. It shows that for understanding better random forests mechanisms, we must go beyond consistency results and quantify the impact of each ingredient of random forests on their performance.

In this comment, we prove that such a quantification is possible for a simple pure forest, leading to conclusions that could apply more generally (Section 1). Then, we consider in Section 2 "hold-out" random forests, which are a good middle point between "toy" pure forests and Breiman's original random forests. 1

1 Aggregation, two kinds of randomization and resampling

As clearly shown by [START_REF] Biau | A random forest guided tour[END_REF], two key ingredients of random forests are the aggregation process (several trees are combined to get the final estimator) and the diversity of the trees that are aggregated. We can distinguish two sources of diversity:

• randomization of the partition P final ,

• randomization of the labels, that is, of the predicted value in each cell of P final , given P final .

In Breiman's random forests, resampling acts on the two kinds of randomization, while the choice of M try at each node of each tree acts on the randomization of the partitions only. In purely random forests, partitions are built independently from the data, so resampling (if any) only acts on the randomization of the labels. Therefore, the role of each kind of randomization is easier to understand, and to quantify separately, for purely random forests. In this section, we propose to do so for the one-dimensional toy forest introduced by Arlot and Genuer (2014, Section 4).

Toy forest

We first define the toy forest model, assuming that X = [0, 1]. Let 2 ≤ k ≤ a ≤ n be some integers. The partition associated to each tree is given by 0

, 1 -T k , 1 -T k , 2 -T k , . . . , k -1 -T k , k -T k , k -T k , 1
with T ∼ U([0, 1]). Then, for each tree, a subsample (X i , Y i ) i∈I of size a is chosen (uniformly over the subsamples of size a), independently from T . The tree estimator is defined as usual: the predicted value at x is the average of the Y i such that X i belongs to A n (x; T ), the cell of the partition that contains x. Finally, the forest estimator is obtained by averaging M ≥ 1 trees, where the trees are independent conditionally to D n .

Assume that X i ∼ U([0, 1]), the (X i , Y i ) are independent with the same distribution, the noise-level

E[(Y -m(X)) 2 | X] = σ 2 is constant, m is of class C 3 , n 1 and a k log(n). Let x ∈ [k -1 , 1 -k -1
] (to avoid border effects, see [START_REF] Arlot | Analysis of purely random forests bias[END_REF] for details) be fixed. Then, Table 1 provides the order of magnitude of

E m toy M,n (x) -m(x) 2 ,
Table 1: Order of magnitude of the quadratic risk at x (approximation error + estimation error) for the one-dimensional toy random forest combined with subsampling without replacement; a denotes the subsample size, 1/k is the size of each cell in a partition, see Appendix B for precise assumptions and approximations. The impact of subsampling can be quantified by comparing a = n with a < n in each formula.

Single tree Infinite forest

No randomization of partitions

c 1 (m, x) k 2 + σ 2 k a c 1 (m, x) k 2 + σ 2 k n Randomization of partitions c 1 (m, x) k 2 + σ 2 k a c 2 (m, x) k 4 + 2σ 2 k 3n where c 1 (m, x) = m (x) 2 12 and c 2 (m, x) = m (x) 2 144 .
the quadratic risk at x of a toy forest m toy M,n with M trees, in various situations: with or without aggregation (M = +∞ or M = 1), with or without randomization of the partitions (we remove randomization of partitions by putting T = 0 for all trees). Subsampling can also be removed by taking a = n in all formulas. Note that in Table 1, the risk is written as the sum of an approximation error and an estimation error, see Appendix A. The main lines of the proof are given in Appendix B.

Table 1 allows to quantify the impact of aggregation, randomization of the partitions, randomization of the labels-and their combinations-on the performance of toy forests:

• aggregation: comparing the two columns of Table 1 shows that aggregation always improve the performance (which is true for any forest model, by Jensen's inequality). The improvement can be huge: when partitions and labels are randomized, a n and k 1, both approximation and estimation errors decrease by an order of magnitude.

• randomization of the partitions: comparing the two lines of Table 1 shows that randomizing the partitions strongly improves the performance of the infinite forest (there is no change for a single tree, of course). The approximation error decreases by an order of magnitude, as previously showed by [START_REF] Arlot | Analysis of purely random forests bias[END_REF]. The estimation error also decreases-as showed by [START_REF] Genuer | Variance reduction in purely random forests[END_REF] for another pure forest-, but by a factor 3/2 only.

• randomization of the labels: comparing a = n with a n in the formulas of Table 1 shows the influence of subsampling, which is the only randomization mechanism for the labels. Single trees perform worse when a < n (as expected since the sample size is lowered). The performance of infinite forests does not change with subsampling, which might seem a bit surprising given several results mentioned by [START_REF] Biau | A random forest guided tour[END_REF]. This phenomenon corresponds to the fact that subagging does not improve a stable estimator [START_REF] Bühlmann | Analyzing bagging[END_REF], and that a regular histogram is stable. Section 1.2 below explains why there is no contradiction with the random forests literature.

Discussion

Section 1.1 sheds some light on previous theoretical results on random forests, and suggests a few conjectures which deserve to be investigated.

Parametrization of the trees. The end of Section 3.1 of Biau and Scornet's survey might seem contradictory with the above results for the toy forest. According to most papers in the literature, "random forests reduce the estimation error of a single tree, while maintaining the same approximation error". Moreover, an infinite forest can be consistent even when a single tree (grown with a sample of size n) is not. Table 1 precisely shows the opposite situation: the estimation error is almost the same for a single tree and for an infinite forest, while the approximation error is dramatically smaller for an infinite forest. In addition, when an infinite forest is consistent, 1 k n hence a single tree trained with a = n points is also consistent.

The point is that these results and ours consider different parametrizations of the trees. In Section 1.1, trees are parametrized by the number of leaves k + 1; so, when comparing a tree with a forest, we think fair to compare (i) a tree of k + 1 leaves trained with n data points, with (ii) a forest where each tree has k + 1 leaves and is trained with a data points. In the literature, trees are often parametrized by the number nodesize of data points per cell. Then, comparisons are done between (i) a tree of ≈ n/nodesize leaves trained with n data points, and (ii) a forest where each tree has ≈ a/nodesize leaves and is trained with a data points. So, if we take k ≈ a/nodesize, the two approaches consider (approximately) the same forest (ii), but the reference trees (i) are quite different.

We do not mean that one of these two parametrizations is definitely better than the other: nodesize is a natural parameter for Breiman's random forests, while toy forests are naturally parametrized by their number of leaves. Nevertheless, one must keep in mind that any comparison between a forest and a single tree trained with the full sample does depend on the parametrization.

The parametrization by nodesize can also hide some difficulties. For the toy forest model, k has to be chosen, and this is not an easy task. One could think that this problem is solved by taking the nodesize parametrization with, say, nodesize = 1. This is wrong because we then have to choose the subsample size a, which is equivalent to the original problem since k ≈ a/nodesize.

What about Breiman's forests? Section 1.1 suggests that for the toy forest, the most important ingredient in the tree diversity is the randomization of the partitions. We conjecture that this holds true for general random forests.

Nevertheless, we do not mean that the resampling step should always be discarded, since for Breiman's random forests (for instance), resampling also acts on the randomization of the partitions. A key open problem is to quantify the relative roles of resampling and of the choice of M try on the randomization of the partitions. Section 2 below shows that "hold-out random forests" can be a good playground for such investigations.

Bootstrap or subsampling? Another important question is the choice between the bootstrap and subsampling, which remains an open problem according to [START_REF] Biau | A random forest guided tour[END_REF].

We conjecture that Table 1 is also valid for the a out of n bootstrap, which would mean that bootstrap and subsampling are fully equivalent with respect to the randomization of the labels, with no impact on the performance of pure forests. Assuming this holds true, subsampling (with or without replacement) with a n remains interesting for reducing the computational cost-Table 1

only requires that a k log(n).

A key open problem remains: compare bootstrap and subsampling with respect to the randomization of the partitions for Breiman's random forests. Again, "hold-out random forests" described in Section 2 should be a good starting point for such a comparison.

Hold-out random forests

We now consider a more complex forest model, called hold-out random forests, which is close to Breiman's random forests while being simpler to analyze. Hold-out random forests have been proposed by Biau (2012, Section 3) and appear in the experiments of Arlot and Genuer (2014, Section 7).

Definition

Hold-out random forests can be defined as follows. First, the data set D n is split, once and for all, into two subsamples D 1 n 1 and D 2 n 2 , of respective sizes n 1 and n 2 , satisfying n = n 1 + n 2 . This split is done independently from D n . Then, conditionally to (D 1 n 1 , D 2 n 2 ), the M trees are built independently as follows. The partition associated with the j-th tree is built as for Breiman's random forests with D 1 n 1 as data set. In other words, it is the partition P final defined by Biau and Scornet (2016, Algorithm 1) with training set D 1 n 1 as an input. The j-th tree estimate at x is defined by

m hoRF n x; Θ j , D 1 n 1 , D 2 n 2 := (X i ,Y i )∈D 2 n 2 1 X i ∈An 1 (x;Θ j ,D 1 n 1 ) Y i N n 2 (x; Θ j , D 1 n 1 , D 2 n 2 )
where A n 1 (x; Θ j , D 1 n 1 ) is the cell of this partition that contains x and

N n 2 (x; Θ j , D 1 n 1 , D 2 n 2 ) is the number of points (X i , Y i ) ∈ D 2 n 2 such that X i ∈ A n 1 (x; Θ j , D 1 n 1
). Finally, the hold-out forest estimate is defined by

m hoRF M,n x; Θ 1...M , D 1 n 1 , D 2 n 2 = 1 M M j=1 m hoRF n x; Θ j , D 1 n 1 , D 2 n 2 .
In the definition of m hoRF M,n , the building of the partitions depends on the same parameters as Breiman's random forests: mtry, nodesize, the fact that resampling can be done with or without replacement, and the resample size a n 1 . It is also possible to add another resampling step when assigning labels to the leaves of each tree with D 2 n 2 ; we do not consider it here since Section 1 suggests that this would not change much the performance (at least for forests).

A key property of hold-out random forests is that they are purely random: the subsamples D 1 n 1 and D 2 n 2 are independent, hence the partition associated with the trees are independent from D 2 n 2 . Note however that each partition still depends on some data (through D 1 n 1 ), hence it could adapt to some features of the data, such as the "sparsity" of the regression function, the Table 2: Numerical estimation of the quadratic risk (approximation error + estimation error) for the hold-out random forest; k is the number of cells in the partition; X = [0, 1] p with p = 5. There is no randomization of the labels.

Single tree Large forest

No bootstrap mtry = p 0.13

k 0.17 + 1.04σ 2 k n 2 0.13 k 0.17 + 1.04σ 2 k n 2 Bootstrap mtry = p 0.14 k 0.17 + 1.06σ 2 k n 2 0.15 k 0.29 + 0.08σ 2 k n 2 No bootstrap mtry = p/3 0.23 k 0.19 + 1.01σ 2 k n 2 0.06 k 0.31 + 0.06σ 2 k n 2 Bootstrap mtry = p/3 0.25 k 0.20 + 1.02σ 2 k n 2 0.06 k 0.34 + 0.05σ 2 k n 2
non-uniformity over X of the smoothness of m, or the non-uniformity of the distribution of X. Therefore, hold-out random forests can capture much of the complexity of Breiman's random forests, while being easier to analyze since they are purely random. In particular, we can apply the results proved in Appendix A (where our D 2 n 2 is written D n , and our D 1 n 1 is hidden in the relationship between Θ j and the partition of the j-th tree). Then, the quadratic risk of m hoRF M,n can be (approximately) decomposed into the sum of an approximation error and an estimation error, and these two terms can be studied separately. For instance, the results of [START_REF] Arlot | Analysis of purely random forests bias[END_REF] can be applied in order to analyze the approximation error.

For now, we only study the behaviour of these two terms in a short numerical experiment. The results are summarized by Table 2, where estimated values of the approximation and estimation errors are reported as a function of n 2 , σ 2 and of the parameters of the partition building process (k, mtry and bootstrap). Detailed information about this experiment can be found in Appendix C. Let us emphasize here that we consider a single data generation setting, hence these results must be interpreted with care.

Discussion

Based on Table 2 and our experience about Breiman's random forests, we can make the following comments.

Choice of mtry. As illustrated in Table 2, choosing mtry = p/3 instead of p decreases the risk of an infinite forest. When there is no bootstrap, the performance gain is significant and the reason is that it is the only source of randomization of partitions. But, even in presence of bootstrap, it allows to slightly reduce the approximation error. In the same experiments with p = 10, the gain of decreasing mtry in the bootstrap case is larger (see the supplementary material).

Our belief is that when there is some bootstrap, the additional randomization given by taking mtry < p can reduce the risk in some cases, where typically n ≥ p (which holds true in our experiments). This is supported by the experiments of Genuer et al. (2008, Section 2), where small values of mtry give significantly lower risk than mtry = p for some classification problems. For regression, Genuer et al. (2008, Section 2) obtain similar performance when decreasing mtry, which is consistent with Table 2 since these experiments are done in the bootstrap case.

When n p and only a small proportion of the coordinates of x are informative, we conjecture that the optimal mtry is close to p (provided that there is some bootstrap step for randomizing the partitions). Indeed, if mtry is significantly smaller than p, then, the probability to choose at least one informative coordinate in M try is not close to 1, hence the randomization of the partitions might be too strong.

Bootstrap, mtry and randomization of the partitions. When mtry = p, according to Table 2, the bootstrap helps to significantly reduce the risk, compared with the no randomization case. Overall, we get a significantly smaller risk when there is at least one source of randomization of the partitions.

Comparing the three combinations of parameters (bootstrap, mtry < p, or both) for which the partitions are randomized is more difficult: the differences observed in Table 2 might not be significant. Nevertheless, Table 2 suggests that the lowest risk might be obtained when two sources of randomization are present (mtry < p and bootstrap). And if we have to choose only one source of randomization, it seems that randomizing with mtry = p/3 only yields a smaller risk than bootstrapping only.

A Approximation and estimation errors

We state a general decomposition of the risk of a forest having the Xproperty (that is, when partitions are built independently from (Y i ) 1≤i≤n ), that we need for proving the results of Section 1, but can be useful more generally. We assume that E[Y 2 i ] < +∞ for all i. For any random forest m M,n having the X-property, following Biau and Scornet (2016, Sections 2 and 3.2), we can write

m M,n (x; Θ 1...M , D n ) = n i=1 W ni (x)Y i where W ni (x) = W ni (x; Θ 1...M , X 1...n ) = 1 M M j=1 C i (Θ j )1 X i ∈An(x;Θ j ;X 1...n ) N n (x; Θ j ; X 1...n ) , (1)
C i (Θ j ) is the number of times (X i , Y i ) appears in the j-th resample, A n (x; Θ j ; X 1...n ) is the cell containing x in the j-th tree, and

N n (x; Θ j ; X 1...n ) = n i=1 C i (Θ j )1 X i ∈An(x;Θ j ;X 1...n ) .

Now, let us define

m M,n (x; Θ 1...M , X 1...n ) = E m M,n (x; Θ 1...M , D n ) X 1...n , Θ 1...M = n i=1 W ni (x; Θ 1...M , X 1...n )m(X i ) and m M,n (x; Θ 1...M ) = E m M,n (x; Θ 1...M , X 1...n ) Θ 1...M .
By definition of the conditional expectation, we can decompose the risk of m M,n at x into three terms

E m M,n (x) -m(x) 2 = E m M,n (x) -m(x) 2 A=approximation error + E m M,n (x) -m M,n (x) 2 ∆ + E m M,n (x) -m M,n (x) 2 E=estimation error
.

(2)

In the fixed-design regression setting (where the X i are deterministic), A is called approximation error, ∆ = 0, and E is called estimation error.

Things are a bit more complicated in the random-design setting-when (X i , Y i ) 1≤i≤n are independent and identically distributed-since ∆ = 0 in general. Up to minor differences related to how m n is defined on empty cells, A is still the approximation error, and the estimation error is ∆ + E.

Let us finally assume that (X i , Y i ) 1≤i≤n are independent and define

σ 2 (X i ) = E m(X i ) -Y i 2 X i .
Then, since the weights W ni (x) only depend on D n through X 1...n , we have the following formula for the estimation error

E = E   n i=1 W ni (x) m(X i ) -Y i 2   = E n i=1 W ni (x) 2 σ 2 (X i ) .
For instance, in the homoscedastic case, σ 2 (X i ) ≡ σ 2 and

E = E   n i=1 W ni (x) m(X i ) -Y i 2   = σ 2 E n i=1 W ni (x) 2 .
(3)

B Analysis of the toy forest: proofs

We prove the results stated in Section 1 for the one-dimensional toy forest.

Since the toy forest is purely random, all results of Appendix A apply, with Θ = (T, I) and C i (Θ) = 1 i∈I . It remains to compute the three terms of Eq. ( 2).

Since we assume m is of class C 3 , we can use the results of Arlot and Genuer (2014, Section 4) for the approximation error A (up to minor differences in the definition of m M,n (x), due to event where A n (x; Θ) is empty, which has a small probability since a k). We assume that m (x) = 0 and m (x) = 0 for simplicity, so the quantities appearing in Table 1 indeed provide the order of magnitude of A.

The middle term ∆ in decomposition ( 2) is negligible in front of E for a single tree, which can be proved using results from [START_REF] Arlot | V -fold cross-validation improved: V -fold penalization[END_REF], as soon as m (x)/k σ and a k. We assume that it can also be neglected for an infinite forest.

For the estimation error, we can use Eq. ( 3) and the following arguments. First, for every i ∈ {1, . . . , n}, X i belongs to A n (x; Θ) with probability 1/k. Combined with the subsampling process, we get that

N n (x; Θ; X 1...n ) ∼ B n, a nk
is close to its expectation a/k with probability almost one if a/k log(n). Assuming that this holds simultaneously for a huge fraction of the subsamples, we get the approximation

W toy ni (x) = 1 M M j=1 1 i∈I j 1 X i ∈An(x;Θ j ) N n (x; Θ j ; X 1...n ) ≈ k a 1 M M j=1 1 i∈I j 1 X i ∈An(x;Θ j ) =: W toy ni (x) . (4)
Now, we note that conditionally to X 1...n , the variables 1 i∈I j 1 X i ∈An(x;Θ j ) , j = 1, . . . , M are independent and follow a Bernoulli distribution with the same parameter

a n × 1 -k|X i -x| + .
Therefore,

E W toy ni (x) 2 X 1...n = k 2 na 1 - 1 M a n 1 -k|X i -x| + 2 + 1 M 1 -k|X i -x| + hence E W toy ni (x) 2 = k na 1 - 1 M 2a 3n + 1 M .
By Eq. ( 3), this ends the proof of the results in the bottom line of Table 1. Similar arguments apply for justifying the top line of Table 1, where T j = 0 almost surely. Note that we have not given a full rigorous proof of the results shown in Table 1, because of the approximation (4) and of the term ∆ that we have neglected. We are convinced that the parts of the proof that we have skipped might only require to add some technical assumptions, which would not help to reach our goal of understanding better random forests in general.

C Details about the experiments

This section describes the experiments whose results are shown in Section 2. Data generation process. We take X = [0, 1] p , with p ∈ {5, 10}. Table 2 only shows the results for p = 5. Results for p = 10 are shown in supplementary material.

The data (X i , Y i ) 1≤i≤n 1 +n 2 are independent with the same distribution:

X i ∼ U([0, 1] p ), Y i = m(X i ) + ε i with i ∼ N (0, σ 2 ) independent from X i , σ 2 = 1/16
, and the regression function m is defined by

m : x ∈ [0, 1] p → 1/10 × 10 sin(πx 1 x 2 ) + 20(x 3 -0.5) 2 + 10x 4 + 5x 5 .
The function m is proportional to the Friedman1 function which was introduced by [START_REF] Jerome H Friedman | Multivariate adaptive regression splines[END_REF]. Note that when p > 5, m only depends on the 5 first coordinates of x.

Then, the two subsamples are defined by

D 1 n 1 = (X i , Y i ) 1≤i≤n 1 and D 2 n 2 = (X i , Y i ) n 1 +1≤i≤n 1 +n 2 .
We always take n 1 = 1 280 and n 2 = 25 600.

Trees and forests. For each k ∈ {2 5 , 2 6 , 2 7 , 2 8 }, each experimental condition (bootstrap or not, mtry = p or p/3 ), we build some hold-out random trees and forests as defined in Section 2. These are built with the randomForest R package [START_REF] Liaw | Classification and regression by randomforest[END_REF][START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF], with appropriate parameters (k is controlled by maxnodes, while nodesize = 1).

Resampling within D 1 n 1 (when there is some resampling) is done with a bootstrap sample of size n 1 (that is, with replacement and a n 1 = n 1 ).

"Large" forests are made of M = k trees, a number of trees suggested by [START_REF] Arlot | Analysis of purely random forests bias[END_REF].

Estimates of approximation and estimation error. Estimating approximation and estimation errors (as defined by Eq. ( 2)) requires to estimate some expectations over Θ (which includes the randomness of D 1 n 1 as well as the randomness of the choice of bootstrap subsamples of D 1 n 1 and of the repeated choices of a subset M try ). This is done with a Monte-Carlo approximation, with 500 replicates for trees and 10 replicates for forests. This number might seem small, but we observe that large forests are quite stable, hence expectations can be evaluated precisely from a small number of replicates.

We estimate the approximation error (integrated over x) as follows. For each partition that we build, we compute the corresponding "ideal" tree, which maps each piece of the partition to the average of m over it (this average can be computed almost exactly from the definition of m). Then, to each forest we associate the "ideal" forest m M,n which is the average of the ideal trees. We can thus compute (m M,n (x) -m(x)) 2 for any x ∈ X , and estimate its expectation with respect to Θ. Averaging these estimates Table 3: Numerical estimation of the quadratic risk (approximation error + estimation error) for the hold-out random forest; k is the number of cells in the partition; X = [0, 1] p with p = 10. There is no randomization of the labels.

Single tree

Large Cardinal of partitions (log-scale) q q q q q q tree (r=-0.168, 2^C=0. Cardinal of partitions (log-scale) q q q q q q tree (r=-0.187, 2^C=0.231) forest (r=-0.306, 2^C=0.063) (c) no bootstrap, mtry = 1, p = 5 q q q q 5 6 7 8 -4.5 -4.0 -3.5 -3.0

Bias (log-scale)

Cardinal of partitions (log-scale) q q q q q q tree (r=-0.199, 2^C=0.248) forest (r=-0.341, 2^C=0.057) (d) bootstrap, mtry = 1, p = 5

Figure 1: Estimated values of the approximation error of hold-out trees and "large" forests (in log 2 -scale) as a function of the number of leaves (in log 2 -scale), for the Friedman 1 regression function in dimension p = 5, with various values of the parameters (bootstrap or not, mtry ∈ {p, p/3 }).

The coefficients r and C respectively denote the slope and the intercept of a linear model fitted to the scatter plot. Cardinal of partitions q q q q q q tree (r=1.057, C=-2.326) forest (r=0.082, C=7.36) (b) bootstrap, mtry = 5, p = 5 Cardinal of partitions q q q q q q tree (r=1.016, C=0.12) forest (r=0.053, C=3.259) Cardinal of partitions (log-scale) q q q q q q tree (r=-0.11, 2^C=0.113) forest (r=-0.187, 2^C=0.094) (b) bootstrap, mtry = 10, p = 10 q q q q 5 6 7 8 -4.8 Cardinal of partitions (log-scale) q q q q q q tree (r=-0.179, 2^C=0.207) forest (r=-0.254, 2^C=0.082) (c) no bootstrap, mtry = 3, p = 10 q q q q 5 6 7 8 Cardinal of partitions (log-scale) q q q q q q tree (r=-0.165, 2^C=0. Cardinal of partitions q q q q q q tree (r=1.084, C=-2.348) forest (r=0.04, C=5.309) (c) no bootstrap, mtry = 3, p = 10

Figure 2 :

 2 Figure2: Estimated values of the estimation error (multiplied by n 2 /σ 2 ) of hold-out trees and "large" forests as a function of the number of leaves, for the Friedman 1 regression function in dimension p = 5, with various values of the parameters (bootstrap or not, mtry ∈ {p, p/3 }). The coefficients r and C respectively denote the slope and the intercept of a linear model fitted to the scatter plot.

Figure 3 :

 3 Figure3: Estimated values of the approximation error of hold-out trees and "large" forests (in log 2 -scale) as a function of the number of leaves (in log 2 -scale), for the Friedman 1 regression function in dimension p = 10, with various values of the parameters (bootstrap or not, mtry ∈ {p, p/3 }). The coefficients r and C respectively denote the slope and the intercept of a linear model fitted to the scatter plot.

Figure 4 :

 4 Figure4: Estimated values of the estimation error (multiplied by n 2 /σ 2 ) of hold-out trees and "large" forests as a function of the number of leaves, for the Friedman 1 regression function in dimension p = 10, with various values of the parameters (bootstrap or not, mtry ∈ {p, p/3 }). The coefficients r and C respectively denote the slope and the intercept of a linear model fitted to the scatter plot.

Acknowledgements

The research of the authors is partly supported by the French Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration). S. Arlot is also partly supported by Institut des Hautes Études Scientifiques (IHES, Le Bois-Marie, 35, route de Chartres, 91440 Bures-Sur-Yvette, France).

over 1000 uniform random points x ∈ X provides our estimate of the approximation error.

We estimate the estimation error (integrated over x) from Eq. (3); since σ 2 is known, we focus on the remaining term. Given some hold-out random forest, for any x ∈ X and i ∈ {1, . . . , n}, we can compute

.

Then, averaging i W ni (x) 2 over several replicate trees/forests and over 1 000 uniform random points x ∈ X , we get an estimate of the estimation error (divided by σ 2 ).

Summarizing the results in Table 2. Given the estimates of the (integrated) approximation and estimation errors that we obtain for every k ∈ {2 5 , 2 6 , 2 7 , 2 8 }, we plot each kind of error as a function of k (in log 2 -log 2 scale for the approximation error), and we fit a simple linear model (with an intercept). The estimated parameters of the model directly give the results shown in Table 2 (in which the value of the intercept for the estimation error is omitted for simplicity). The corresponding graphs are shown in supplementary material.