text:
Helladic chronology

Helladic chronology is a relative dating system used in archaeology and art history. It complements the Minoan chronology scheme devised by Sir Arthur Evans for the categorisation of Bronze Age artefacts from the Minoan civilization within a historical framework. Whereas Minoan chronology is specific to Crete, the cultural and geographical scope of Helladic chronology is confined to mainland Greece during the same timespan (c. 3200 – c. 1050 BC). Similarly, a Cycladic chronology system is used for artifacts found in the Aegean islands. Archaeological evidence has shown that, broadly, civilisation developed concurrently across the whole region and so the three schemes complement each other chronologically. They are grouped together as "Aegean" in terms such as Aegean art and, rather more controversially, Aegean civilization.

The systems derive primarily from changes in the style of pottery, which is a benchmark for relative dating of associated artifacts such as tools and weapons. On the basis of style and technique, Evans divided his Cretan Bronze Age pottery finds into three main periods which he called Early, Middle and Late Minoan. These were sub-divided into phases and some of those into sub-phases. The Helladic and Cycladic schemes were devised later and have similar sub-divisions. Evans' system has stood the test of time remarkably well but his labels do not provide firm dates because change is never constant and some styles were retained in use much longer than others. In fact it is partly this lack of dates that has been the strength of Evans’s system; several of the dates Evans believed have certainly changed, and others remain under discussion, though within fairly narrow ranges, but the scheme just adjusts for such changes. Some pottery can be dated with reasonable precision by reference to Egyptian artifacts whose dates are more certain.

Helladic society and culture have antecedents in Neolithic Greece when most settlements were small villages which subsisted by means of agriculture, farming and hunting. The gradual development of skills such as bronze metallurgy, monumental architecture and construction of fortifications brought about the transition from the Neolithic to the Bronze Age. The Late Helladic (c. 1550 – c. 1050 BC) is sometimes called the Mycenaean Age because Mycenae was then the dominant state in Greece. At the end of the Bronze Age (c. 1050 BC), Aegean culture went into a long period of decline, termed a Dark Age by some historians, as a result of invasion and war.

Etymology
The three terms Cycladic, Helladic, and Minoan refer to location of origin. Thus, Middle Minoan objects might be found in the Cyclades, but they are not on that account Middle Cycladic, just as an Early Helladic pot found in Crete is not Early Minoan. The scheme tends to be less applicable in areas on the periphery of the Aegean, such as the Levant or North Africa. Pottery there might imitate Aegean cultural models and yet be locally manufactured.

Background
Archaeology has found evidence, primarily in the form of pottery, that a broadly similar way of life was spread over mainland Greece, the Cyclades and Crete as the Neolithic (New Stone) Age was superseded by the Bronze Age before 3000 BC. Evidence increases through Bronze Age strata with social and economic development seen to develop more quickly. Unlike the Egyptian and Mesopotamian civilisations, the Aegean peoples were illiterate through the third millennium and so, in the absence of useful written artifacts, any attempt at chronology must be based on the dating of material objects. Pottery was by far the most widespread in terms of everyday use and also the most resistant to destruction even when broken, as the pieces, or "sherds", survive. Given the different styles and techniques used over a long period of time, the surviving pots and shards can be classified according to age. As stratified deposits prove which of similar objects from other sites are contemporary, they can therefore be equated chronologically.

Periodisation
The Early, Middle and Late scheme can be applied at different levels. Rather than use such cumbersome terms as Early Early, archaeologists follow Evans' convention of I, II, III for the second level, A, B, C for the third level, 1, 2, 3 for the fourth level and A, B, C for the fifth. Not all levels are present at every site. If additional levels are required, another Early, Middle or Late can be appended. The Helladic chronology is subdivided as:

Settlements of the Helladic period
These are the estimated populations of hamlets, villages, and towns of the Helladic period over time. Note that there are several problems with estimating the sizes of individual settlements, and the highest estimates for a given settlements, in a given period, may be several times the lowest.

Early Helladic (EH)
The Early Helladic period (or EH) of Bronze Age Greece is generally characterized by the Neolithic agricultural population importing bronze and copper, as well as using rudimentary bronze-working techniques first developed in Anatolia with which they had cultural contacts. The EH period corresponds in time to the Old Kingdom in Egypt. Important EH sites are clustered on the Aegean shores of the mainland in Boeotia and Argolid (Manika, Lerna, Pefkakia, Thebes, Tiryns) or coastal islands such as Aegina (Kolonna) and Euboea (Lefkandi) and are marked by pottery showing influences from western Anatolia and the introduction of the fast-spinning version of the potter's wheel. The large "longhouse" called a megaron was introduced in EHII. The infiltration of Anatolian cultural models (i.e. "Lefkandi I") was not accompanied by widespread site destruction.

Early Helladic I (EHI)
The Early Helladic I period (or EHI), also known as the "Eutresis culture" c. 3200–2650 BC, is characterized by the presence of unslipped and burnished or red slipped and burnished pottery at Korakou and other sites (metal objects, however, were extremely rare during this period). In terms of ceramics and settlement patterns, there is considerable continuity between the EHI period and the preceding Final Neolithic period (or FN); changes in settlement location during the EHI period are attributed to alterations in economic practices.

Early Helladic II (EHII)
The transition from Early Helladic I to the Early Helladic II period (or EHII) or Korakou culture c.2650–2200 BC, occurred rapidly and without disruption where multiple socio-cultural innovations were developed such as metallurgy (i.e. bronze-working), a hierarchical social organization, and monumental architecture and fortifications. Changes in settlement during the EHII period were accompanied with alterations in agricultural practices (i.e. oxen-driven plow).

Early Helladic III (EHIII)

The Early Helladic II period came to an end at Lerna with the destruction of the "House of the Tiles", a corridor house. The nature of the destruction of EHII sites was at first attributed to an invasion of Greeks and/or Indo-Europeans during the Early Helladic III or Tiryns culture period c.2200–2000 BC (or EHIII); however, this is no longer maintained given the lack of uniformity in the destruction of EHII sites and the presence of EHII–EHIII/MH continuity in settlements such as Lithares, Phlius, Manika, etc. Furthermore, the presence of "new/intrusive" cultural elements such as apsidal houses, terracotta anchors, shaft-hole hammer-axes, ritual tumuli, and intramural burials precede the EHIII period in Greece and are in actuality attributed to indigenous developments (i.e. terracotta anchors from Boeotia; ritual tumuli from Ayia Sophia in Neolithic Thessaly), as well as continuous contacts during the EHII–MH period between mainland Greece and various areas such as western Asia Minor, the Cyclades, Albania, and Dalmatia. Changes in climate also appear to have contributed to the significant cultural transformations that occurred in Greece between the EHII period and the EHIII period (c. 2200 BCE).

Middle Helladic (MH)
In Greece, the Middle Helladic period (or MH), c. 2000–1550 BC, was a period of cultural retrogression, which first manifested in the preceding EHIII period. The MH period is characterized by the wide-scale emergence of Minyan ware, which may be directly related to the people whom ancient Greek historians called Minyans; a group of monochrome burnished pottery from Middle Helladic sites was conventionally dubbed "Minyan" ware by Troy's discoverer Heinrich Schliemann.

Gray Minyan ware was first identified as the pottery introduced by a Middle Bronze Age migration; the theory, however, is outdated as excavations at Lerna in the 1950s revealed the development of pottery styles to have been continuous (i.e. the fine gray burnished pottery of the EHIII Tiryns culture was the direct progenitor of Minyan ware). In general, painted pottery decors are rectilinear and abstract until Middle Helladic III, when Cycladic and Minoan influences inspired a variety of curvilinear and even representational motifs.

Pottery is the most abundant object found from the Middle Helladic period, and it is matt-painted pottery that begins to appear during this period as a new style of ceramic works. While Minyan pottery is made on a wheel that produces sharp designs and shapes, matt-painted pottery is sculpted by hand and has dull paint applied.

Matt-painted pottery, from the Middle Helladic period demonstrates many stylistic motifs that can be traced back to the Early Helladic Period and even other Aegean cultures. Patterns like oblique lines, zigzags, lozenges, running dogs, spiraliforme, and circles and triangles alternating one another beneath the rim on Middle Helladic pottery have been found to be inspired by Cycladic pottery motifs. Spiraliforme can also be traced back to Crete, as well as Griffon motifs. However, patterns like lozenges and pendent-style triangles on matt-painted pottery is a continuation of stylistic motifs from the Early Helladic period, and the addition of vertical fringed lines on pottery is an advancement that took place during the Middle Helladic period itself. With the majority of designs and motifs on Middle Helladic pottery being Cycladic in influence it can be assumed that the Middle Helladic culture and the Cycladic culture interacted with one another heavily.

There are four types of graves that are found at sites from the Middle Helladic period; pit graves, tholos graves, cist graves, and shaft graves. A pit grave is self explanatory, as it is simply a pit in the ground, while tholos styled graves are characterized as being more of a chamber like tomb. Cist graves and shaft graves are interesting because they are two styles of burial that originate from the Middle Helladic period itself, and it is believed that migrants who moved to Greece during this period influenced the creation of these new burial styles. Cist graves are deep and rectangular with a tumulus, or mound of earth, placed over top and came about during the beginning of the Middle Helladic period. Shaft graves are larger and deeper than cist graves (measuring on average 6 meters long, 4 meters wide, and 4 meters deep) and came about during the end of the Middle Helladic period. Additionally, infants are buried in special jars, pithoi, that generally measure around 30 in tall. Based on the archaeological evidence, at Middle Helladic burial ceremonies bodies are placed in graves on their sides with their knees bent (women are placed on their left sides, and men on their right ), then those present at the ceremony drink from cups that they then leave at the tomb. Burial customs also included leaving valued items with the bodies like pottery, silver, or bronze.

At the Middle Helladic site Lerna, there are over 200 graves that have been excavated. While roughly a third of these graves are extramural burials (bodies are buried outside of the community), intramural burials (bodies are buried within the community) make up two thirds of the graves found including towns being built around a cist.

Communities during the Middle Helladic period, specifically Lerna, had irregular layouts with no specific pattern, and houses were tightly packed together. It is theorized that the arrangement of houses may have been based on living close to extended family or close to members of a similar group or faction. Houses were one story tall, built in a "U" shape, and made of clay. Generally houses would feature a porch, with up to three rooms, an inner chamber that would contain a hearth, and spaces for storage and cooking. A larger, free standing house has been identified as a possible home to a chief or leader of the community, and features a separate a storage facility as well as a courtyard with a hearth.

People of the Middle Helladic period grew crops like wheat (which would be ground into flour for baking), barley, flax, peas, chickpeas, lentils, and beans in addition to watching over animals like sheep, goats, swine, oxen, horses, and dogs. Middle Helladic people would spin thread to be woven into fabrics on a loom, and the clothes they made were both fastened and often decorated with pins. The people would also adorn themselves in necklaces and bracelets made of stone and shells.

By studying the remains at sites like Lerna it was evident that men tended to eat more protein than women, and women tended to partake more in softer, more processed foods. Stress marks were identified on the skeletons of people of both genders, but men had them to a higher degree, denoting that men engaged more frequently in heavy physical labor than women did. More often than women, men also had higher level of lesions caused by infectious diseases, meaning they had greater exposure to foreign pathogens through direct contact with outside groups and people. This makes it appear as though there is division in labor between the genders.

The Middle Helladic period corresponds in time to the Middle Kingdom of Egypt. Settlements draw more closely together and tend to be sited on hilltops. Middle Helladic sites are located throughout the Peloponnese and central Greece (including sites in the interior of Aetolia such as Thermon) as far north as the Spercheios River valley. Malthi in Messenia and Lerna V are the only Middle Helladic sites to have been thoroughly excavated.

Late Helladic (LH)
The Late Helladic period (or LH) is the time when Mycenaean Greece flourished, under new influences from Minoan Crete and the Cyclades. Those who made LH pottery sometimes inscribed their work with a syllabic script, Linear B, which has been deciphered as Greek. LH is divided into LHI, LHII, and LHIII; of which LHI and LHII overlap Late Minoan ware and LHIII overtakes it. LHIII is further subdivided into LHIIIA, LHIIIB, and LHIIIC. The table below provides the approximate dates of the Late Helladic phases (LH) on the Greek mainland, based on Knodell (2021) and Manning (2010):

Late Helladic I (LHI)
The LHI pottery is known from the fill of the Shaft Graves of Lerna and the settlements of Voroulia and Nichoria (Messenia), Ayios Stephanos, (Laconia) and Korakou. Furumark divided the LH in phases A and B, but Furumark's LHIB has been reassigned to LHIIA by Oliver Dickinson. Some recent C-14 dates from the Tsoungiza site north of Mycenae indicate LHI there was dated to between 1675/1650 and 1600/1550 BC, which is earlier than the assigned pottery dates by about 100 years. The Thera eruption also occurred during LHI (and LCI and LMIA), variously dated within the 1650–1625 BC span. Alex Knodell (2021), based on Manning (2010), dates Late Helladic I between 1700/1675 and 1635/1600 BC.

Not found at Thera, but extant in late LHI from Messenia, and therefore likely commencing after the eruption, is a material culture known as "Peloponnesian LHI". This is characterised by "tall funnel-like Keftiu cups of Type III"; "small closed shapes such as squat jugs decorated with hatched loops ('rackets') or simplified spirals"; "dark-on-light lustrous-painted motifs", which "include small neat types of simple linked spiral such as varieties of hook-spiral or wave-spiral (with or without small dots in the field), forms of the hatched loop and double-axe, and accessorial rows of small dots and single or double wavy lines"; also, the "ripple pattern" on "Keftiu" cups. These local innovations continued into the LHIIA styles throughout the mainland.

Late Helladic II (LHII)
The description of the LHIIA is mainly based on the material from Kourakou East Alley. Domestic and Palatial shapes are distinguished. There are strong links between LHIIA and LMIB. LHIIB began before the end of LMIB, and sees a lessening of Cretan influences. Pure LHIIB assemblages are rare and originate from Tiryns, Asine and Korakou. C-14 dates from Tsoungiza indicate LHII was dated to between 1600/1550 and 1435/1405 BC, the start of which is earlier than the assigned pottery date by about 100 years, but the end of which nearly corresponds to the pottery phase. In Egypt, both periods of LHII correspond with the beginning of its Theban "Imperial" period, the New Kingdom of Egypt, from pharaohs Hatshepsut to Thutmose III (r. 1479–1425 BC) of the Eighteenth Dynasty. Alex Knodell considers Late Helladic II to be between 1635/1600 and 1420/1410 BC.

Late Helladic III (LHIII)

LHIII and LMIII are contemporary. Toward LMIIIB, non-Helladic ware from the Aegean ceases to be homogeneous; insofar as LMIIIB differs from Helladic, it should at most be considered a "sub-Minoan" variant of LHIIIB.

The uniform and widely spread LHIIIA:1 pottery was defined by the material from the Ramp house at Mycenae, the palace at Thebes (now dated to LHIIIA:2 or LHIIIB by most researchers) and Triada at Rhodes. There is material from Asine, Athens (wells), Sparta (Menelaion), Nichoria and the 'Atreus Bothros', rubbish sealed under the Dromos of the Treasury of Atreus at Mycenae as well. C-14 dates from Tsoungiza indicate LHIIIA:1 should be more nearly 1435/1406 to 1390/1370 BC, slightly earlier than the pottery phase, but by less than 50 years. LHIIIA:1 ware has also been found in Maşat Höyük in Hittite Anatolia.

The LHIIIA:2 pottery marks a Mycenaean expansion covering most of the Eastern Mediterranean. There are many new shapes. The motifs of the painted pottery continue from LHIIIA:1 but show a great deal of standardization. In Egypt, the Amarna site contains LHIIIA:1 ware during the reign of Amenhotep III and LHIIIA:2 ware during that of his son Akhenaten; it also has the barest beginnings of LHIIIB. LHIIIA:2 ware is in the Uluburun shipwreck, which sank in the 14th century BC. Again, Tsoungiza dates are earlier, 1390/1370 to 1360/1325 BC; but LHIIIA:2 ware also exists in a burn layer of Miletus which likely occurred early in the reign of Mursili II and therefore some years prior to Mursili's eclipse in 1312 BC. The transition period between IIIA and IIIB begins after 1320 BC, but not long after (Cemal Pulak thinks before 1295 BC).

The definition of the LHIIIB by Furumark was mainly based on grave finds and the settlement material from Zygouries. It has been divided into two sub-phases by Elizabeth B. French, based on the finds from Mycenae and the West wall at Tiryns. LHIIIB:2 assemblages are sparse, as painted pottery is rare in tombs and many settlements of this period ended by destruction, leaving few complete pots behind.

LHIIIB pottery is associated in the Greek mainland palaces with the Linear B archives. (Linear B had been in use in Crete since Late Minoan II.) Pulak's proposed LHIIIA/B boundary would make LHIIIB contemporary in Anatolia with the resurgent Hittites following Mursili's eclipse; in Egypt with the 19th Dynasty, also known as the Ramessides; and in northern Mesopotamia with Assyria's ascendancy over Mitanni. The end of LHIIIB is associated with the destruction of Ugarit, whose ruins contain the last of that pottery. The Tsoungiza date for the end of LHIIIB is 1200/1190 BC. The beginning of LHIIIC, therefore, is now commonly set into the reign of Queen Twosret. The LHIIIC has been divided into LHIIIC:1 and LHIIIC:2 by Furumark, based on materials from tombs in Mycenae, Asine, Kephalonia, and Rhodes. In the 1960s, the excavations of the citadel at Mycenae and of Lefkandi in Euboea yielded stratified material revealing significant regional variation in LHIIIC, especially in the later phases. Late LHIIIC pottery is found in Troy VIIa and a few pieces in Tarsus. It was also made locally in the Philistine settlements of Ashdod, Ashkelon, Ekron, Gath, and Gaza.

Fortified settlements

During the Helladic period, a number of major advances were developed including fortified urban settlements with monumental buildings such as corridor houses, which may prove the existence of complex societies organized by an elite or at least achieving corporate, proto-city state form. One of these settlements was Manika, located in Euboea, dated to the Early Helladic period II (2800–2200 BC). The settlement covered an area of 70–80 hectares, was inhabited by 6,000–15,000 people, and was one of the largest settlements of the Bronze Age in Greece.

Another settlement was Lerna in the Argolid region, which was perhaps the most important and wealthiest of Early Helladic sites. The settlement has a monumental building known as the House of the Tiles, a "corridor house", notable for several architectural features that were advanced for its time, such as its roof being covered by baked tiles, which gave the building its name. The structure dates to the Early Helladic II period (2500–2300 BC) and is sometimes interpreted as the dwelling of an elite member of the community, a proto-palace, or an administrative center. Alternatively, it has also been considered to be a communal structure or the common property of the townspeople. The exact functions of the building remain unknown due to a lack of small finds indicating the specific uses of the building. The house had a stairway leading to a second story, and was protected by a tiled roof. Debris found at the site contained thousands of terracotta tiles having fallen from the roof. Although such roofs were also found in the Early Helladic site of Akovitika, and later in the Mycenaean towns of Gla and Midea, they only became common in Greek architecture in the 7th century BC. The walls of the House of the Tiles were constructed with sun-dried bricks on stone socles.

Other fortified settlements include Tiryns, which covered an area of 5.9 hectares sustaining 1,180–1,770 people, and had a large tiled two-storeyed "round house" (Rundbau) with a diameter of 28 m on the upper citadel. It may have served as a palace or temple or perhaps it was a communal granary. Other sites include Ayia Irini, which covered an area of 1 hectare and had a population of perhaps up to 1,250; Eutresis covering 8 hectares with an estimated population of 1,600–2,400; Thebes covering 20 hectares with a population of 4,000–6,000; Lefkandi (unknown in size and population); and Kolonna (or Aegina), a densely populated settlement with impressive fortifications, monumental stone buildings and sophisticated town planning.

Already before 2500–2400 BC, Kolonna experienced remarkable economic growth and had its own administrative "Corridor House", the so-called "Haus am Felsrand". During the phase Aegina III 2400–2300 BC, which corresponds to the transition phase Lefkandi I-Kastri, the evidence of the economic structure and administrative and social organization of the community become more clear. The "White House" (Weisses Haus; 165 square metres) constitutes the monumental community building that succeeds the "Haus am Felsrand", which had the same function. Kolonna may constitute the Aegean's first state as it appears to be the earliest ranked society in the area outside Minoan Crete and perhaps a political center in the Middle Helladic period where it achieved state-level after the Minoans but before the Mycenaeans.

title: Helladic chronology


text:
Talk:Exponential family

Must H be a pdf
Question: must the reference distribution H be a probability distribution, or will a positive measure do? Note that the normalization condition applies to F, not to H. I am thinking of cases where the reference "distribution" is Lebesgue measure (to wit: the Normal distribution) or counting measure. — Miguel 11:08, 2005 Apr 16 (UTC)

 * Certainly it is Lebesgue measure in some cases. And counting measure on positive integers -- clearly not assigning finite measure to the whole space -- in some cases.  Which causes me to notice that this article is woefully deficient in examples.  I'll be back.... Michael Hardy 20:36, 16 Apr 2005 (UTC)

Article reversion
Would you mind explaining the reversion of my edits on the article on the Exponential family? — Miguel 07:14, 2005 Apr 18 (UTC)

It said:

 * A is important in its own right, as it the cumulant-generating function of the probability distribution of the sufficient statistic T(X) when the distribution of X is H.

You changed it to:

 * A is important in its own right, as it is the cumulant-generating function of the probability distribution of the sufficient statistic T(X).

The edit consisted of deleting the words "when the distribution of X is H. The statement doesn't make sense without those words.  A cumulant-generating function is always a cumulant-generating function of some particular probability distribution.

 * Well, actually the derivatives of A(η) evaluated at η instead of at zero give you the cumulants of dF(x|η), which is what I meant. The cumulants of dH are actually irrelevant to dF, what is interesting is that the cumulants of the entire family of exponential distributions with the same dH and T are encoded in A. Miguel 09:34, 2005 Apr 19 (UTC)

I see now that you also changed some other things. I haven't looked at those closely, but I now see that you changed "cdf" to "Lebesgue-Stieltjes integrator". I don't think that change makes sense either. That it is the Lebesgue-Stieltjes integrator is true, but the fact that it is the cdf is more to the point in this context. Michael Hardy 21:59, 18 Apr 2005 (UTC)

 * Except that, as you agree above, dH need not be a probability distribution, and hence it need not have a cdf. It is a positive measure, and dH is the integrated measure. I have never seen x on the whole real line called a cdf. It is fine if you want to call it a cdf, but then you'll have to explain somewhere else that the corresponding probability distribution may be "non-normalizable", and that will raise some eyebrows (not mine, though).

 * You also reverted a lot of valid content about the relationship between the exponential family and information entropy, as well as a reorganization of the existing information into sections, plus placeholders for discussing estimation and testing. The two edits that so bothered you were the last of a long series spanning two days. You could have been a little more careful. The appropriate thing would have been to discuss these things on this page. Miguel 09:34, 2005 Apr 19 (UTC)

Disputed
The current revision says that the Weibull distributions do not form an exponential family. This seems to ignore non-canonical exponential families (where the natural parameter may be transformed by another function). What am I missing? --MarkSweep (call me collect) 23:28, 14 November 2005 (UTC)

I may have lifted that from another source without checking the pdf. This is not open to interpretation. Either the Weibull distribution is exponential or it isn't according to the definition. Transforming the natural parameter is not the issue: the normal distribution's natural parameters are not the mean and variance. Miguel 09:31, 15 November 2005 (UTC)

 * AFAIK, it's not an exponential family according to the definition given in this article. So I guess I put the dubious tag in the wrong place. What I was trying to point out is that there is a more general definition of exponential family in common use, see e.g. .  The difference between these definition becomes apparent when one considers the Weibull distribution: it's not an exponential family according to the definition used here, but it is according to the more general definition (unless I'm missing something). --MarkSweep (call me collect) 19:49, 15 November 2005 (UTC)

If you "transform" the natural parameter, you're using a different parametrization of the family of probability distributions involved, but you're not looking at a different family of probability distributions. Is that what you're talking about? Michael Hardy 22:46, 15 November 2005 (UTC)

The Weibull distribution is not in the exponential family according to the definition given here, which is one that can also be found in widely-used textbooks such as Casella and Berger, Statistical Inference (2nd edition), 2002, page 114. Later in the article it is noted that you NEED this specific definition for a distribution to have sufficient statistics (a result of Darmois, Koopman and Pitman from the 1930s), so you can't generalize the definition any further without penalty. To shorten the controversy about Weibull, I propose that that ALL MENTION of Weibull be dropped from the article, leaving only Cauchy as the agreed-upon non-member of the exponential family. Ed 02:01, 16 June 2006 (UTC)

OK, further study does not find any references supporting Weibull in the exponential family, so I suggest that we remove the "dispute" tag and let the original wording stand, the one where Cauchy and Weibull are both left out of the family. I consulted books on the exponential family by Lawrence Brown and Ole Barndorff-Nielsen. EdJohnston 17:14, 23 June 2006 (UTC)

 * It is an exponential distribution and is one according to the definition given in this article. The cdf defines a measure. It does not depend on the particular parameterization of the family of distributions. The natural parameters of the Weibull are (λ^-k,k-1) where λ and k are the parameters given in Wikipedia's defn of the Weibull. H is then Lebesque measure, and you can work out A yourself. Note that the Weibull is a generalization of the exponential distribution which is Weibull with k=1. CWoo on Planetmath gets this right (though his article seems to simplify some things). Odometer 05:34, 27 December 2006 (UTC)
 * To put Weibull in the exponential family needs a reference, in my opinion. Membership in the family is not expected to be invariant under transformation of the parameters. I left a question about this at User_talk:Odometer but have not received a response. I found that I was unable to 'work out A myself'. PlanetMath is interesting but it is not a reliable source for our purposes. EdJohnston 18:06, 8 January 2007 (UTC)

 * Look at the planetmath definition of exponential family. It's an equivalent definition when density functions exist, and it's the one that's often used because it's much easier to comprehend and pattern match on, so it's probably the more appropriate definition to use on wikipedia. Casella and Berger use that defn in their book. You just factor the density into a parameter part, a "data" part, and an interaction part. Generally speaking the 'A' doesn't matter that much when determining if it's an exponential family. 'A' is just the normalizing constant to make the density integrate to 1. It's sometimes called the log partition function while the rest of the density is sometimes called the kernel. In this case it's -k log(λ) + log(k) using the parameterization in wikipedia. You can stick that in the natural parameterization if you want. Also membership in the family is definitely invariant under transformation of the parameters. A family of distributions is just a collection of probability measures which are indexed by some parameters. You can change the indices, and it's still the same collection. Odometer 00:33, 16 February 2007 (UTC)
 * Can you provide the functions a, b, c and d needed at for the Weibull distribution? I would be happy if a simpler pattern for the exponential family could be used in this article. It's not crystal clear that measure theory is essential for explaining this stuff. It does set a high prerequisite for understanding the article.  EdJohnston 02:15, 16 February 2007 (UTC)

It's worth noting that McCullagh and Nelder's "Generalized Linear Models" talks about knowing the dispersion parameter (i.e. $$\sigma^2 $$ in the normal case) as being akin to having a one-parameter ($$\mu $$) distribution. But this implies that the distribution could be placed in exponential family form before the dispersion parameter was fixed. They also talk about fitting with a Weibull distribution and mention that the fitting procedure isn't entirely inside the framework of a GLM--meaning it can't be placed in exponential family form except when it's coincident with the exponential distribution (pp 423). O18 06:54, 9 July 2007 (UTC)

Question
Questions: Is the "prior" mentioned in the opening section the same as the "prior distribution" mentioned later? If so isn't the convention that the first mention of something links to the wikipedia article on it? (And calling it the "prior distribution there would be good too - people like me don't actually know any stats)

Also shouldn't "cdf" appear in brackets after the words "Cumulative distribution function" for clarity rather than just straight in the text? 20:25, 18 May 2006 (BST)

Technical(expert) tag
I just added a technical(expert) tag to this page. I understand and have used exponential family forms of distributions many times, but I find this article extremely difficult to read and almost willfully obscure. I can understand why one might want to use Lebesgue-Stieltjes integration, but the examples I have read have managed to avoid the topic completely and still dealt with continuous as well as discrete probability distributions. See for example, McCullagh and Nelder, "Generalize Linear Models" pp 28, or Givens and Hoeting, "Computational Statistics" pp 5. A read though the mathematics Wikiproject page, specifically, the section titled, "Some issues to think about" and the Wikipedia page on making technical articles accessible.

The general idea is to start simple (i.e. without invoking complicated concepts), which is definitely possible as I have noted two examples from textbooks, and then to get to more rigorous treatments later on. Thanks in advance to anyone who helps with this! O18 17:17, 1 June 2007 (UTC)
 * Yes, this article would be more useful without the measure theory and the Lebesgue-Stieltjes integration. No books on my shelf seem to do a patient exposition of the exponential family. Maybe the treatment you found in McCullagh and Nelder would provide inspiration for our article? Also there's quite a bit of matrix notation used in the article and that adds to the reader's burden. EdJohnston 04:31, 24 June 2007 (UTC)

 * I think the MC&N method is too focused on GLM, I think others might be better. O18 14:43, 24 June 2007 (UTC)

minus sign mismatch? and no previous mention of $$K(u)$$
The definition of the exponential family is given at the top of the page by $$dF(x|\eta) = e^{-\eta^{\top} T(x) - A(\eta)}\, dH(x)$$, which has a minus sign before the first $$\eta$$. However, section "Differential identities: an example" claims the "exponential family with canonical parameter" is not preceded by a minus sign. Is the minus absorbed into $$\eta$$ in the "Differential identities" in order for the Expectation and Variance formulas to work? If so, perhaps the form of the exponential family should remain the same and this result be formulated consistent with the single form.

Also, section "Differential identities: an example" claims that "As mentioned above $$\scriptstyle K(u) = A(u + \eta) - A(\eta) $$", but $$\scriptstyle K(u)$$ is not previously mentioned.

Erik Barry Erhardt 21:10, 23 June 2007 (UTC)

 * I personally prefer the definition with the plus sign, but then there is an inconsistency among the three definitions given at the beginning. The elementary single-parameter definition and the vector-parameter definitions have a + sign in front of the η but a - sign in front of the 'A', whereas the measure-theoretic definition has a minus signs on all terms. Miguel (talk) 17:17, 14 April 2010 (UTC)

uniform dist
This used to say that the uniform distribution is not in the exponential family. But the uniform dist is a special kind of beta dist, and the beta dist is in the family. Benwing 06:02, 22 August 2007 (UTC)

 * I've never seen anyone miss the point of a definition so completely. Yes the uniform distribution is a beta distribution, yes, the beta distributions form an exponential family.  To suggest that that in some way implies that the uniform distributions form an exponential family is ridiculous.  You really need to read and understand the definition of "exponential family" before you can understand such things.  An exponential family is a not just a distribution; it is a family of distributions.  Every uniform distribution belongs to some exponential family, and in fact to more than one exponential family.  That does not mean that the family of distributions that includes only the uniform distributions is an exponential family.  It obviously is not, since the support of the various distributions in the family varies. Michael Hardy 00:16, 23 August 2007 (UTC)

 * OK, fine, I made a mistake, but you have too -- you've forgotten to be civil, see WP:CIVIL. Benwing 07:40, 23 August 2007 (UTC)

"since the support of the various distributions in the family varies. "- what does this mean? — Preceding unsigned comment added by Wikiusers1 (talk • contribs) 19:54, 2 March 2011 (UTC)

far too technical
This page is a very good example of how a math page should *NOT* be. It's totally opaque to someone who doesn't already have a PhD in statistics, and such a person has no need for this page anyway. It would be far, far better if this page dispensed with all this Lebesgue measure business and gave a simple explanation, plus simple, clearly explained examples, plus a simple derivation of the relation to maximum entropy, etc. Then -- maybe -- include a completely, techically correct discussion at the bottom.

It should go something like this:

First, explain the simple case:

 * Explain the simple case of one parameter, f(x|t) = exp(a(t) + b(x) + c(t)*d(x)) = A(t)*h(x)*exp(c(t)*d(x)) = B(u)*h(x)*exp(u*d(x)).
 * Show briefly why these different definitions are equivalent, and how you can reparameterize from t to u to eliminate c(t), and describe that u is a "natural parameter".
 * Explain how A(t) is just a normalization factor.
 * Explain briefly that d(x) is the basis of a sufficient statistic.

Then expand the definition to cover vector-valued x and t.

Then rewrite sections 1-4 to eliminate the discussion in terms of CDF's, Lebesgue measure, Radon-Nikodym derivatives, Einstein's summation convention, and anything else that an undergrad is likely to find opaque. Show how the maxent relationship is derived, rather than just saying "it's a simple matter of variational calculus" -- and do it *without* invoking any calculus of variations. (If you're not sure how to do this, look up standard NLP papers on maximum entropy.)

Then, finally, if you want, include the full gory advanced details.

If you are a statistics whiz, you may think that doing what I suggested is intolerably stupid or obvious, etc. But keep in mind that Wikipedia articles are *NOT* addressed to fellow experts, but to a general audience. See WP:WPM and WP:MOSDEF for more discussion.

The corresponding article on PlanetMath would be a good place to start.

Benwing 07:40, 23 August 2007 (UTC)

 * Actually, I am no slouch at statistics, yet what you propose (aside for details to be fleshed out later) is neither stupid nor intolerably obvious.  I hesitate to dive right in though, as my tendency is to keep as much intrinsic material if possible and appropriate but change the writing style, yet my comfort level with these topics would make my work quite drudging and painful.

 * A good start might be to adapt the section from generalized linear model; I should note I had a hand at writing that section. Check it out and let us know if that is closer to what you had in mind (I know it would have to be adapted somewhat away from GLM context...). Baccyak4H (Yak!) 14:29, 23 August 2007 (UTC)

 * I agree with every word Benwing said. In fact, I have been meaning to work on this but never seem to get around to it. I think there is no escaping a major rewrite here. --Zvika 18:55, 7 September 2007 (UTC)

 * "It's totally opaque to someone who doesn't already have a PhD in statistics"

That is nonsense. Any mathematics graduate student who knows the basic definitions in probability theory would understand it. Many mathematicians who do not already know this material would understand it readily. Michael Hardy 19:06, 7 September 2007 (UTC)
 * Perhaps nonsense in letter, but not in spirit. The thrust of this talk section is dicussion to improve the article, not to point out true but minor pedagogical points which will not help improve the article, and may as a side effect disparage well meaning editors.  That aside, I know you have a good handle on these topics, your participation could be very helpful.  Would you care to contribute to this effort? Baccyak4H (Yak!) 19:21, 7 September 2007 (UTC)

 * (edit conflict) The point is that a basically simple idea is explained in overly technical terms. You should think not of a mathematics graduate but of an engineer or scientist, who has taken a basic course in probability. These courses very often do not include any measure theory at all; many readers will certainly never have heard of a Lebesgue-Stieltjes integral. Yet the idea of an exponential family -- a class of distributions having a particular probability function -- can be understood both at the technical and at the intuitive level by such a reader. All we have to do is initially talk only about continuous and discrete distributions, and defer the general treatment to a later section. "Things should be made as simple as possible, but not any simpler." --Zvika 19:33, 7 September 2007 (UTC)

I agree that it could be made accessible to a wider audience. But there's no reason to dismiss as worthless the many mathematicians not familiar with this concept who could learn it by reading this article. Michael Hardy 23:21, 7 September 2007 (UTC)

 * (edit conflict) Michael, do you agree with Benwing thought that maybe this is the least important audience (add text, "Make_technical_articles_accessible suggests that this topic should be accessible to the widest possible audience?")
 * Would anyone else be willing to work on a (sandbox version)? O18 00:14, 8 September 2007 (UTC) (edit O18 00:41, 8 September 2007 (UTC)) -- edited to remove broken link O18 04:10, 30 September 2007 (UTC)

What, specifically, is the least important audience? I don't see anything above where Benwing says any particular audience is the least important one. Michael Hardy 00:21, 8 September 2007 (UTC)

proposed overhaul
Several editors have written an alternative version of this page that is intended to be more readable for readers with out Ph.D.s in maths. This effort resulted form the above conversation, and is located at (User:Pdbailey/Sandbox/Exponential_family). I intend to replace the body of this article in a few days, if there are no objections. Please also feel free to edit the linked page before it is moved or after it is moved. O18 19:51, 22 September 2007 (UTC) -- edited to remove broken sandbox link

 * Update: The rewrite has now been carried out. We hope you like it. Feel free to directly edit the exponential family page. --Zvika 08:38, 27 September 2007 (UTC)

Conjugate priors
Hello, the article says that "In the case of a likelihood which belongs to the exponential family there exists a conjugate prior, which is often also in the exponential family." This suggests that exponential families exist with conjugate priors that are not in the exponential family, is that true? But the form given for the conjugate prior, $$\pi(\eta) \propto \exp(-\eta^{\top} \alpha - \beta\, A(\eta))$$, looks like it is in the exponential family (for instance if we append to the vector $$\eta$$ a component $$A(\eta)$$). Thanks in advance. A5 10:05, 1 November 2007 (UTC)

isn't it rather darmois koopman pitman theorem ? letters... +maybe the dates —Preceding unsigned comment added by 129.175.52.7 (talk) 13:45, 14 February 2008 (UTC)

Still too technical ? (and inaccurate)
This article is probably still too technical. While many basic books speak about "THE exponential family", this article confuses the lay (though mathematically educated) reader by suggesting that there are many such families. (The first sentence is very confusing as it suggests that any group of distribution functions that share any arbitrary property is an exponential family, e.g. the set of distributions uniform over a domain. Unless the word "class" implies the use of the exponential function, of course... but again this would be arcane stuff.) While there is certainly room for a presentation of the exponential famil(y/ies) at a more technical level in the body of this article, it should also clearly start at the level of graduated scientists and engineers who didn't have advanced courses in prob/stat. Belonging to the last category (and in summary) I found this article of little use. —Preceding unsigned comment added by 130.223.123.54 (talk • contribs) 19 February 2008, 08:52 (UTC)
 * I changed the first sentence and I hope you find it clearer. Regarding the use of the exponential family vs. an exponential family, two standard textbooks that I have with me both use the an option:
 * Lehmann and Casella, Theory of Point Estimation, 2nd ed., p.22.
 * Shao, Mathematical Statistics, 2nd ed., p.96.
 * In both cases, they define an exponential family as a parametric family of distributions; each such family is a different exponential family. So there is no "the" (in the sense of "one and only") exponential family. --Zvika (talk) 09:14, 19 February 2008 (UTC)
 * Previous talk on this page has tried to figure out if an exponential family is just a set of functions, or if any of the parameters also separate one family from another. I don't recall the result, but certainly the premise was that there are many families. Even if 130.223.123.54 can not produce a reference, it might be worth considering having a section that addresses exactly this topic--notably, I am not qualified to write it. O18 (talk) 21:05, 20 February 2008 (UTC)
 * There may be a confusion here with the exponential distribution. The exponential distributions do form an exponential family. Miguel (talk) 18:08, 14 April 2010 (UTC)
 * No, that is not the confusion. The question is, what is the definition of a family? 018 (talk) 18:27, 14 April 2010 (UTC)
 * Right. I see you linked to natural exponential family below. I wonder whether the content of that article shouldn't be merged into this one. In fact, the opening paragraph of the article as it now stands claims
 * an exponential family is a class of probability distributions sharing a certain form, specified below. It is said that such distributions belong to the exponential class of density functions
 * So we have exponential function, exponential distribution, exponential family, natural exponential family, and exponential class... No wonder people get confused. — Miguel (talk) 19:07, 14 April 2010 (UTC)
 * There should be a clear definition of exponential family at the beginning of the definition section, above the examples. 75.32.190.223 (talk) 18:43, 1 April 2019 (UTC)

Redirection from Pitman-Koopman theorem
Pitman-Koopman theorem redirects to here; it deserves its own page. The theorem is also not listed on this page under its own name. Agalmic (talk) 01:05, 4 July 2008 (UTC)

Negative Binomial
Mention of the negative binomial should be qualified, too. As far as I can tell, it wouldn't be in the an exponential family unless the parameter, r, is fixed. Schomerus (talk) 23:29, 12 October 2009 (UTC)

 * I think you're right, and I changed the article accordingly. P.S. In the future please add new discussions at the bottom of the talk page. --Zvika (talk) 05:45, 13 October 2009 (UTC)

Normal distribution
No objections to the article and no typos found.

I came across this article looking for inspiration on the following question. Having not gained any, I pose the question here:

The family of distributions N(0, sigma^2) (or any normal distribution with known mean), is evidently not complete, and thus also not an exponential family. Where does the definition of an exponential family fail? As the article indicates, should the mean be unknown, things follow through. —Preceding unsigned comment added by 131.155.71.49 (talk) 12:23, 20 October 2009 (UTC)

 * Actually, unless I'm not seeing straight today, if the mean is known to be $$\mu$$, the definition is satisfied as
 * $$f_\mu(x;\theta)dx = e^{{-1\over 2}\bigl({x-\mu\over\sigma}\bigr)^2}{dx\over\sqrt{2\pi\sigma^2}}= h(x) \exp(\eta(\theta) T_\mu(x) - A(\theta))dx$$
 * if we let $$\theta = \sigma^2$$ and
 * $$h(x)={1\over\sqrt{2\pi}}$$
 * $$\eta(\theta)={-1\over 2\theta}$$
 * $$T_\mu(x)=(x-\mu)^2$$
 * $$A(\theta) = {1\over 2}\log\theta$$
 * See Cramér–Rao bound. --- Miguel (talk) 13:35, 29 November 2009 (UTC)

 * I wonder if this editor was thinking of Natural exponential family. 018 (talk) 18:32, 14 April 2010 (UTC)

 * A natural exponential family is just an exponential family for which the sample mean is a sufficient statistic... — Miguel (talk) 19:53, 14 April 2010 (UTC)

Canonical parameters
I believe the exposition would be clearer and easier to understand if we start with the exponential family in its canonical form, where θ is the natural parameter of the family. In that way we would be able to talk about the dual parameters, and about estimation of the parameters (e.g. the theorem that the exponential family is the only family where parameters can be estimated efficiently). …  st pasha  » 19:34, 24 December 2009 (UTC)

Measure-theoretic formulation excludes dx !
I edited
 * Suppose H is a non-decreasing function of a real variable and H(x) approaches 0 as x approaches −∞.

into
 * Suppose H is a non-decreasing function of a real variable.

Clearly we want to include the case where H(x) = x. — Miguel (talk) 16:53, 14 April 2010 (UTC)

Newbie question. . . but it really bugs me! Math Notation, read as . ..
How do we read the expressions fx(x|θ) and fσ(x;μ)? While I'm on the topic, wouldn't it be better if every new symbol introduced into a mathematics article were also posted in the Table of Mathematical Symbols []? As it is, whenever I read a mathematical article, I am then forced to look up the symbols elsewhere to get the pronunciation, rendering the otherwise fine math content of Wikipedia pretty much redundunt. Wikipedia is, after all, supposed to be a resource for the non-expert as well as the expert. — Preceding unsigned comment added by Romanicus (talk • contribs) 17:14, 16 December 2010 (UTC)
 * I guess you are looking for the Greek alphabet article? //  st pasha  » 21:41, 16 December 2010 (UTC)

I mean, how does one read the entire expression? For instance, this - "2+3" - is not read as "two and a little cross and three". It's read as, "two plus three". So, how are these two expressions - "fx(x|θ)" and "fσ(x;μ)" - read? — Preceding unsigned comment added by Romanicus (talk • contribs) 21:12, 14 January 2011 (UTC)
 * They're not read, they're written. Higher-level maths isn't a spoken language, it's a written one. Even when face-to-face, mathematicians need pencil and paper, or blackboard and chalk, or whiteboard and marker, to communicate. --Qwfp (talk) 08:57, 15 January 2011 (UTC)

In the definition, h(x) is not any arbitrary function- there are conditions imposed on it like it is differentiable everywhere. Right? —Preceding unsigned comment added by 91.125.78.233 (talk) 12:03, 16 March 2011 (UTC)
 * Since x can be discrete, differentiability of h(x) can play no role. Melcombe (talk) 12:54, 16 March 2011 (UTC)

Dear Qwfp. I've been told that:

"fx(x|θ)" can be read as, "f of x, at x, restricted to the possible choices of parameter θ".

"fσ(x;μ)" can be read as, "f of x, at σ, across the possible choices of parameter μ".

But then, I don't know "higher math". (your math class must have had a zen-like silence).

Query on Correctness of Moment generating function section
The MGF given is stated to apply for all exponential families... I believe this might be an error and that it only applies to Natural Exponential Families. If someone can refute this please do so. Otherwise I will try to find a reference that clears up the matter one way or the other.... —Preceding unsigned comment added by 137.194.233.21 (talk) 12:41, 22 April 2011 (UTC)

OK... I think I see what is happening the distribution of the sufficient statistic has a MGF with the given form, but the MGF of the distribution of x will also be in that form if it is natural exponential family (but not for some exponential families)...

As this confused me for some time maybe something could be added.... —Preceding unsigned comment added by 137.194.233.21 (talk) 13:07, 22 April 2011 (UTC) --137.194.233.21 (talk) 14:49, 22 April 2011 (UTC)

Table of distributions
I think this article would benefit from a table of commonly encountered exponential family distributions and their corresponding representations as such. There are a couple in the article now, but I think having a table with many more would be useful. I've created an initial version of what I have in mind (below) which no doubt contains at least a few mistakes and omissions. Thoughts?

Nippashish (talk) 08:20, 7 March 2012 (UTC)

 * For the Categorical distribution (variant 2 ) in the table, the constant C was added, i.e. [log p1 + C,..., log pk + C). Shouldn't instead be [log p1 + log C, ..., log pk + log C]? If so, then the column h(x) should the be modified accordingly such that instead of 1, it should be 1/C. As in the article it is said:
 * "The canonical form is non-unique, since η(θ) can be multiplied by any nonzero constant, provided that T(x) is multiplied by that constant's reciprocal, or a constant c can be added to η(θ) and h(x) multiplied by exp(-c.T(x))" Averroes786 (talk) 18:06, 22 January 2023 (UTC)

Here's an updated version of the table. I've fixed a few mistakes and added a column which gives $$A$$ in terms of the natural parameter, in addition to what I had before. I'll leave this here for a week or two and if there are still no objections I'll add it to the article.

Nippashish (talk) 01:09, 17 March 2012 (UTC)

 * Very strange! I was obviously thinking the same thing, and went ahead and added a table.  But I had no idea until just now that you already made one; I populated the table myself, and it took a fair amount of math, since I couldn't always find sources or they had errors in them.  I'll cross-check my results against yours and maybe incorporate the column where you write A in turns of the standard parameters rather than the natural parameters. Benwing (talk) 08:02, 1 April 2012 (UTC)

 * I did the math to populate my version myself also, since I had the same experience looking for sources as you did. Hopefully our combined effort has lead to something with few mistakes.  Nippashish (talk) 19:25, 1 April 2012 (UTC)

I think the table on the page can benefit from column with moments of the sufficient statistics. What do you think?

BorisYangel (talk) 21:43, 17 March 2013 (UTC)

In the discussion just before the table, the scalar variable-scalar parameter form uses bold (vector notation) for the variable and parameter. I was going to change to non-bold, but it occurred to me that I might be missing someone's reason for doing it that way. Is there a reason? Clay Spence (talk) 14:49, 25 November 2014 (UTC)

Table of distributions
In the following table the support of the density is denoted $$x$$. The symbol $$\theta$$ stands for the distribution parameters as they appear in their own wiki page, for example in the entry for the Bernoulli, $$\theta=p$$ whereas in the entry for the Normal distribution, we use $$\theta = \{\mu, \sigma\}$$.

What is the meaning of the prime notation here: $$\eta(\theta)'\cdot T(x)$$
In the subsection "scalar parameter", there is:
 * More generally, η(θ) and T(x) can each be vector-valued such that $$\eta(\theta)'\cdot T(x)$$ is real-valued.

What is the meaning of $$\eta(\theta)'\cdot T(x)$$? Is it supposed to be $$\eta(\theta)^T T(x)$$? --Bongilles (talk) 15:54, 13 April 2016 (UTC)

Regularity conditions for Pitman-Koopman-Darmois theorem
The current statement of the Pitman-Koopman-Darmois theorem seems to be missing some regularity conditions. The latest development I am aware of is the proof by Barndorff-Nielsen and Pederson (1968) in the paper "Sufficient data reduction and exponential families." There, beyond the "support does not change across the family" assumption, in all cases it is required that the probability densities for the family are continuous and that the statistic is continuous. Moreover, when the dimension of the sufficient statistic is greater than 1, it also is required that the densities have continuous partial derivatives.

I can make a (suitably simple-to-read) version including the appropriate regularity conditions, but before that I wanted to see if anyone knows if any of the regularity conditions can be relaxed; i.e. are there any results beyond those from Barndorff-Nielsen and Pederson? Tekhnofiend (talk) 21:29, 15 May 2016 (UTC)

Definition of vector exponential family
I think that the description of exponential families with vector parameters are a bit unclear in terms of dimensions. First $$ \theta $$ is of dimension s, which is also the number of elements in the sum. Later it becomes of dimension d. So I guess that $$ \eta $$ should be a mapping from  $$ R^d $$ to $$ R^s $$ right from the beginning? But maybe there are problems related to this choice afterwards? I am not specialized enough to change this contents, which is flawed.

Table of distributions needs more distributions
This is far from complete listing. The most useful information is the log-partition function. 68.134.243.51 (talk) 20:37, 13 September 2022 (UTC)

Section entitled "Factorization of the variables involved"
The section Factorization of the variables involved appears to be original research. It has no citations, and it is not very helpful. It seems to be an inconclusive attempt by an editor to provide a recipe for recognizing when an expression can be put into the form of an exponential family.

It is abstract, but not complete. It is a rule of thumb that immediately has exceptions, not a systematic method.

It would be more useful to replace this section by some examples. Or rewrite it so that it does not make a claim to comprehensiveness.

2A02:1210:2642:4A00:5846:AA29:F37C:805C (talk) 11:57, 23 June 2023 (UTC)

Is the Tweedie distribution family an exponential family in the sense of this article?
Biggerj1 (talk) 18:48, 29 October 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Exponential family


text:
Johann Georg Gichtel

Johann Georg Gichtel (14 March 1638 – 21 January 1710) was a German mystic and religious leader who was a critic of Lutheranism. His followers ultimately separated from this faith.

Biography
Gichtel was born at Regensburg, where his father was a member of senate. Having acquired at school an acquaintance with Greek, Hebrew, Syriac and even Arabic, he proceeded to Strasbourg to study theology; but finding the theological prelections of J. S. Schmidt and P. J. Spener distasteful, he entered the faculty of law. He was admitted an advocate, first at Speyer, and then at Regensburg; but having become acquainted with the baron Justinianus von Weltz (1621–1668), a Hungarian nobleman who cherished schemes for the reunion of Christendom and the conversion of the world, and having himself become acquainted with another world in dreams and visions, he abandoned all interest in his profession, and became an energetic promoter of the Christerbauliche Jesusgesellschaft (Christian Edification Society of Jesus).

The movement in its beginnings provoked at least no active hostility; but when Gichtel began to attack the teaching of the Lutheran clergy and church, especially upon the fundamental doctrine of justification by faith, he exposed himself to a prosecution which resulted in sentence of banishment and confiscation (1665). After many months of wandering and occasionally romantic adventure, he reached the Netherlands in January 1667, and settled at Zwolle, where he co-operated with Friedrich Breckling (1629–1711), who shared his views and aspirations.

Having become involved in the troubles of this friend, Gichtel, after a period of imprisonment, was banished for a term of years from Zwolle, but finally in 1668 found a home in Amsterdam, where he made the acquaintance of Antoinette Bourignon, and in a state of poverty (which, however, never became destitution) lived out his life of visions and day-dreams, of prophecy and prayer. He gathered a community of the "Brethren of the Angelic Life".

He became an ardent disciple of Jakob Böhme, whose works he published in 1682 (Amsterdam, 2 vols); but before the time of his death, he had attracted to himself a small band of followers known as "Gichtelians" or "Brethren of the Angels," who propagated certain views at which he had arrived independently of Böhme. Seeking ever to hear the authoritative voice of God within them, and endeavouring to attain to a life altogether free from carnal desires, like that of "the angels in heaven, who neither marry nor are given in marriage," they claimed to exercise a priesthood "after the order of Melchizedek," appeasing the wrath of God, and ransoming the souls of the lost by sufferings endured vicariously after the example of Christ.

While, however, Böhme "desired to remain a faithful son of the Church," the Gichtelians became separatists.

Works
Gichtel's correspondence was published without his knowledge by Gottfried Arnold, a disciple, in 1701 (2 vols.), and again in 1708 (3 vols.). It has been frequently reprinted under the title Theosophia practica. The seventh volume of the Berlin edition (1768) contains a notice of Gichtel's life.

title: Johann Georg Gichtel


text:
Beer in Mexico

History of beer in Mexico dates from the Spanish conquest of the Aztec Empire. While Mesoamerican cultures knew of fermented alcoholic beverages, including a corn beer, long before the 16th century, European style beer brewed with barley was introduced with the Spanish invasion soon after Hernán Cortés's arrival. Production of this beer here was limited during the colonial period due to the lack of materials and severe restrictions and taxes placed on the product by Spanish authorities. After the Mexican War of Independence, these restrictions disappeared, and the industry was permitted to develop. Furthermore, the arrival of German immigrants during the ephemeral Second Mexican Empire of elected Maximilian I of Mexico, born an Austrian archduke, in the 19th century provided the impetus for the opening of many breweries in various parts of the country.

By 1918, there were 36 brewing companies, but over the 20th century, the industry consolidated until today, only two corporations, Grupo Modelo (now owned by AB InBev) and Cervecería Cuauhtémoc Moctezuma formerly known as FEMSA Cerveza (now owned by Heineken N.V.) control 90% of the Mexican beer market. This industry is one of the most prevalent in the country, with over 63% of the population buying one brand or another. Beer is also a major export for the country, with most going to the United States, but is available in over 150 countries worldwide. In 2022, Mexico was the largest beer exporter in the world.

16th century to 19th century
Prior to the Spanish conquest of what is now Mexico, there had been fermented alcoholic beverages in Mexico. The best known of these is pulque, which is the fermented sap of the maguey or agave plant. More similar to beer is a lesser-known beverage, called tesgüino or izquiate, brewed by various cultures. This is made from fermented corn, and creates light, amber-colored liquid which is whisked before drinking. Tesgüino can still be found in Mexico today, mostly homemade, in the north and west of Mexico in states such as Chihuahua, Sonora and Colima. Among the Tarahumaras, the drink is used for rituals. A similar beverage, called pozol, is made in Oaxaca, Chiapas and Tabasco with corn and cocoa beans. Beer brewed with grain such as barley was produced in small quantities by Hernán Cortés’ soldiers, but it was limited due to the lack of supplies. The first official concession to brew European-style beer was granted to Alfonso de Herrero in 1543 or 1544. Its exact location is unknown, but it is thought to have been located in the south of Mexico City (where Metro Portales is today) or in Amecameca, Mexico State. Herrera's brewery struggled during its first years, as alcohol consumption was highly regulated by authorities, and the new brew had to compete with native beverages. It was also more expensive due to the lack of ingredients. However, the beverage caught on, as it was drunk by colonial authorities, leading others to want it as well. Herrera worked to expand his brewery and the land on which wheat and barley were raised. In the long run, Herrero's brewery did not survive, and the production of European-style beverages such as beer and wine were heavily taxed and heavily regulated by Spain to protect home markets. The purpose of this was to make colonials import these products from Europe. While the policy mostly worked, beer brewing never entirely ceased. In the years just before independence, beer consumption was becoming established in Mexico, leading to disputes over the rights to produce it. Englishmen Gillons and Mairet, Miguel Ramos Arizpe and Justino Tuallion all claimed exclusive rights to produce beer in Mexico. After the end of the war, the beer produced by the Tuallion brewery was the most popular. After the war, colonial restrictions were gone and the industry was allowed to develop, starting in the 1820s. In 1845, a barley beer flavored with piloncillo was introduced with the names of Pila Seca and La Candelaria by Swiss Bernhard Boldgard and Bavarian Federico Herzog.

The industry truly began to develop in the latter half of the 19th century, due to an influx of German immigrants to Mexico and the short-lived Second Mexican Empire headed by emperor Maximilian I of Mexico of the House of Habsburg, an Austro-Germanic ruling family. The emperor had his own brewer, who produced Vienna-style dark beers. This influence can be seen in two popular brands of Negra Modelo and Dos Equis Ambar.

From the mid-19th to the early 20th century, a large number of breweries were opened in various regions of the country. Some of these include the Cervecería Toluca y México by Swiss Agustín Marendaz in 1865, Cervecería Cruz Blanca in Mexico City by Emil Dercher in 1869 (which produced the first lager), the San Diego Brewery by Carlos Fredenbaen in 1860, the Cervecería Yucateca by José Ponce Solis in Mérida in 1869 (with German equipment and a German brewer), the Gran Cervecería de San Luis in 1882 in San Luis Potosí by Jose M. Otahegui and Juan Fouilloux (a French brewer who imported equipment from Paris), the Fábrica de Cerveza y Hielo in Monterrey in 1886, the Cevercería Piazzini in Monterrey in 1889, the Cervecería Estrella by Juan E. Ohrner in 1890, the Cervecería Guillermo Hasse y Compañia in Orizaba by Henry Manthey, William Hasse, Cuno Van Alten and Adolph Burhard (which started production at 15,000 bottles daily and would later change its name to Cervecería Moctezuma), the Cervecería Chihuahua by Juan Terrazas in Chihuahua in 1896, the Cervecería Sonora by George Gruning, Dr. Albert Hoeffer, and Jacob Schuele in 1896, the Cervecería del Pacífico by Jorge Claussen, Germán Evers and Emilio Philippy in 1900.

The establishment of a railroad system in Mexico at the end of the 19th century allowed the import of machinery and malt from the United States, but it also forced Mexican breweries to compete against U.S. beer, which began to be distributed in the country. By 1890, the first substantial, industrial brewing facility in the country was built in Monterrey by Cevercería Cuauhtémoc. Four years later, another large brewery, Cervecería Moctezuma, began in Orizaba.

20th century
By 1918, there were 36 beer producers in Mexico. Prohibition in the United States during the 1920s helped the Mexican beer industry, with Americans crossing the border to drink. This spurred breweries along the border, such as Mexicali Brewery and the Aztec Brewing Company, both in Baja California. Beer became big business by the early 20th century. By 1925, despite the strong preference still for pulque in the center of the country, Mexico was producing 50,000 liters of beer per year. To promote their product further, European immigrant beer brewers in the first part of the 20th century campaigned against native drinks such as pulque. They claimed such drinks were produced by unsanitary methods (including the use of feces as a fermenting agent) and promoted beer as "rigorously hygienic and modern". The strategy proved successful, with pulque now generally looked-down-upon and imbibed by relatively few people, with Mexican-brewed beer ubiquitous and extremely popular. However, competition soon forced the consolidation of the industry. Cervecería Cuauhtémoc in Monterrey first went national when it bought the Tecate brewery. Cervecería Toluca became Cervecería Modelo in 1925, and soon began buying smaller competitors. During the rest of the 20th century, larger companies bought out smaller companies, assuming their brands, until only two major producers were left, Cervecería Modelo (or Grupo Modelo) and Cervecería Cuauhtémoc-Moctezuma owned by FEMSA. Many smaller breweries, such as Cervecería Mexicali opened in 1920, which were not bought by these giants were forced to close. Most of the brands known today were creations made by the smaller breweries of the past that were absorbed into the stock of these two giants, who distribute these products in both Mexico and abroad. These producers have seventeen plants located in eleven states with a capacity of 46 million hectoliters annually and support 92 centers of barley production in Mexico. The industry employs 90,000 people and 800,000 jobs are related to it indirectly. The Mexican beer industry is one of the economy's most prolific with 63% of the domestic population consuming one or more of the brands, and Mexico ranks third in global exports of beer. In 2004, exports of beer were valued at US$1.2 billion. Domestic sales were $6 billion.

21st century
Mexico displaced the Netherlands in 2003 as the largest worldwide beer exporter, selling 1.39 million tonnes, with sales, primarily to the U.S., continuing to increase. Grupo Modelo and FEMSA send more than 80% of their exports to the United States. Mexico's growth is coming largely at the expense of U.S. brands. The two main Mexican producers reported increases in export volume of 42% and 20.5% in 2006, compared to less than five percent for Anheuser-Busch and Molson Coors in the same year. Mexican beer has done so well in the United States that Miller SAB tried selling citrus and salt-flavored Miller Chill and Anheuser Busch attempted Bud Light Lime. The best-known and best-selling Mexican beer in the United States by far is Corona, produced by Grupo Modelo and distributed by Anheuser Busch. FEMSA entered the US market later, but has paired with Dutch enterprise Heineken USA to promote and distribute its brands, especially Dos Equis and Tecate. Some Mexican beers, such as Modelo Especial and Negra Modelo, are available in limited quantities on tap in cities such as New York, Houston, Raleigh and Phoenix.

Breweries and brands
In Mexico, beer is primarily produced by two large conglomerates, Cervecería Modelo/Grupo Modelo and Cervecería Cuauhtémoc-Moctezuma/FEMSA. Cervecería Modelo was founded in 1925 in Mexico City, with its first two brands, Modelo and Corona, exporting eight million bottles a year to various countries. First exports to the United States were realized as early as 1933. The first of the company's many acquisitions was the Cervecería Toluca y México, absorbing its Victoria and Pilsener brands in 1935. Modelo continued buying smaller local breweries in various parts of the country, absorbing most of the brands produced and making many of them available nationwide. Starting in the 1980s, the enterprise began new businesses, such as INAMEX, which produces malt, which led to the name change to Grupo Modelo. During the same period, the company began exports of Corona beer to the United States, becoming the second most imbibed imported beer there by 1986. Exports to other countries followed, and Corona became the number one premium imported beer in the United States in 1997. Half of Grupo Modelo's stock is owned by Anheuser Busch.

The beer-brewing division of FEMSA was created when this entity bought Cervecería Cuauhtémoc-Moctezuma, which itself was created when Cervecería Cuauhtémoc bought Cervecería Moctezuma. Cervecería Cuauhtémoc was founded in 1890 by Issac Garza, José Muguerza, Joseph M. Schnaider and Francisco Sada, selling their first beer, Carta Blanca. Cervecería Cuauhtémoc grew in size in Monterrey, and like Cervecería Modelo, went national as it began to buy smaller breweries in other parts of the country, absorbing many of the local brands and making them available nationally. The biggest acquisition was that of Cervecería Moctezuma in Orizaba in the 1980s. Cervecería Moctezuma started out as the Cervecería Guillermo Hasse y Compañia in 1893, and eventually changed its name to Cervecería Moctezuma. It was a major producer of beer since the early 20th century, and was one of the largest brewing companies in the world with the merger of the two, but the new company controls over twelve brand names. FEMSA bought the combined breweries to add to its other businesses, such as bottling and packaging enterprises. FEMSA's brands today include Tecate, Sol, Dos Equis, Carta Blanca, Superior, Indio, and Bohemia. Grupo Modelo's brands include Corona, Corona Light, Negra Modelo, Modelo Especial, Victoria, Estrella, Léon, Montejo, and Pacifico. Most of these beers are lagers brewed in large industrial plants, and made with minimal malt. Except for some dark beers, such as Dos Equis Ambar, León Negra, Negra Modelo, and Noche Buena, which are Vienna-style beers, almost all beer produced in Mexico is pilsner. Beers with top fermentation had been produced in Mexico. The Cervecería Toluca was founded in 1865 by a Swiss especially to produce this type of beer, but the introduction of Bohemian-style beers through the giant brewery Cuauhtémoc would define Mexican beer as pilsner. Globally, one of the best known Mexican beers is Corona, which is the flagship beer of Grupo Modelo. Corona is the best-selling beer produced by Mexico, and the best-selling non-domestic beer in both the U.S., U.K and Australia. It is one of the five most-consumed beers in the world, available in more than 150 countries. It is a lager, and was created in 1925 to celebrate Cervecería Modelo's tenth anniversary. Corona is light straw in color and has a very mild flavor, with little hop bitterness and 4.6% alcohol by volume. It is produced by eight facilities with a total of 4.6 billion litres per year capacity. Corona beer is available in a variety of bottled presentations, ranging from the 250-ml ampolleta (labeled Coronita and just referred as the cuartito (little quarter)) up to the 940-ml Corona Familiar (known as the caguama (sea turtle) or ballena (whale)). Unlike most beers, Corona is bottled in a clear bottle, increasing the opportunity for spoilage from sunlight, which can affect the hop oils in the beer. A draught version also exists, as does canned Corona in some markets. The Corona brand trademark in the United States was initially owned by Puerto Rico's Cerveceria Corona, which eventually sold the trademark rights to Cerveceria Modelo.

The oldest and most traditional pilsner in Mexico is Bohemia, which has a significant hops flavor and is quite dense given its clarity. Bohemia has reached the respected distinction of being one of the finest beers of the world. The name comes from the Bohemia region in the Czech Republic that is known for beer. It is one of the longest-aged products from Cervecería Cuauhtémoc, and is the only one to use Lepa Styrian hops. There is a dark version of this brand, which is a Vienna-style beer. In 2009, the company introduced a wheat version of this beer called Bohemia Weizen made with wheat, Mount Hood hops, coriander and orange peel. It is the first wheat beer to be produced by a major beer company in Mexico. Dos Equis was first brewed in Mexico by the German brewer Wilhelm Hasse in 1897. The original name of the beer was "Siglo XX" (20th Century) with the double X standing for the number 20, and it commemorated the arrival of that century; the moniker "Dos Equis" (Double X) was originally just a nickname, but it became widely known by said nickname, as a result the "Siglo XX" name was dropped in favor of "Dos Equis" The original version is the Dos Equis Ambar, a Vienna-style dark beer. This was Cervecería Moctezuma's best selling beer in the 1940s and 1950s. Demand for the beer has resurged, especially in the United States, where it is now the best-selling imported dark beer. The clear version of this brand is a lager derived from the Ambar.

Sol was introduced in the 1890s as El Sol. The name (Spanish for The Sun or Sun) came from a ray of sunshine that fell on a pot while preparing the mash. After being off the market for many years, this brand was reintroduced in 1993, and is now exported to many countries in Latin America, Europe, Asia and Australia. It is a very light-coloured beer with little hops flavour, and considered to be a beer for the young people and the working class. Sol is known for its sexy advertising. Sol comes in a number of varieties. Sol 2 is a stronger flavored beer, Sol limón and sal have lime and salt flavors already added, and there is a Sol Cero, a nonalcoholic beer in regular and lime and salt versions.

Tecate was originally brewed by Cervecería Tecate, and is named after the city of Tecate, Baja California. The local brewery there was bought by Cuauhtémoc-Moctezuma in 1955. It was the first beer to be canned for retail sales in Mexico, with Tecate Light launched in Mexico in 1992. Tecate is one of the best-known brands in Mexico due to its patronage and sponsorship of Mexican sports teams and sporting events.

Noche Buena (literally Good Night, referring to Christmas Eve as the good night) is generally only available around Christmas. Many people wait for this beer's availability each year between the months of October and December. Noche Buena is a strong-flavored, dark beer named after the poinsettia plant or noche buena in Spanish, which decorates the beer's bottles and cartons.

Carta Blanca was Cervecería Cuauhtémoc's first premium beer, first marketed in 1890, and is technically a pilsner. The name means "white card" in Spanish, which at the time was given to people as a sign of respect. Carta Blanca was successful when it debuted at Chicago's Columbian Expo of 1893. Since then, the beer has won a number of other awards.

Negra Modelo is one of Cervecería Modelo's original beers, and was first sold as a draft in 1926. While it has been classed as a Vienna-style beer, the company's website now classifies it as a Munich dunkel (dark).

Pacífico, a Mexican pilsner beer originally brewed in Mazatlán, Sinaloa, is named after the Pacific Ocean. The picture on the bottles and cans is the Deer Islands located off the coast of Mazatlán surrounded by a lifesaver. Pacífico is Modelo's best-selling beer in northwest Mexico, and it is exported to the southwest U.S. A light version of this beer was launched in 2008.

Estrella (Star) was originally brewed by Cervecería Estrella of Guadalajara at the end of the 19th century. This brewery was bought by Grupo Modelo in 1954. The beer is still brewed only in Guadalajara and is a regional brand, mostly sold in Jalisco state and other areas in western Mexico.

Indio was originally named Cuauhtémoc by Cervecería Cuauhtémoc. However, consumers soon began to refer to it as "Indian" for the image of an Indian on what was originally a clay bottle, now glass. The beer still has an image of Cuauhtémoc on the label.

Modelo Especial is Grupo Modelo's second brand after Corona, and was first brewed in 1925. It is a pilsner-style beer that is available in both bottle and cans since 1966. It is second in popularity in Mexico and the company's third best seller in the USA. A light version, called Modelo Light, has been available since 1994.

Superior, made by Cuauhtémoc Moctezuma, was originally brewed to be a premium beer. Recently, interest in this beer has reappeared, and it received a gold medal at the Monde Selection in Brussels, Belgium. The design of the label has not changed in the fifty years this brand has been available.

Victoria was first brewed by Cervecería Toluca y México starting in 1865, but Modelo acquired the brand when they bought this company in 1935. It is sold in bottles, both the standard 325 ml and the large 950ml. The beer is a Vienna-style, but is an amber color and lighter than the other Vienna beers brewed in Mexico.

León, Carta Clara and Montejo brands were originally brewed in Mérida, Yucatán by the Cervecería Yucateca, which was bought by Modelo in 1979. León is a Munich-style dark beer, which was initially brewed at the beginning of the 20th century in the southeast of the country. Montejo was first introduced in 1960 to celebrate the 60th anniversary of the Yucateca brewery. This brewery was bought by Grupo Modelo in 1979, and the packaging was changed to the current one in 1999.

Microbreweries
Microbrews are a new and still rare phenomenon in Mexico, especially compared to countries such as the U.S., Canada and those in Europe, but they do exist. The first was established in the 1990s in Mazatlán and called Pepe's y Joe's. Later in the decade, others, such as Cervecería San Ángel and Cervecería Santa Fe Beer Factory, appeared in Mexico City.

Commemorative beers
In addition to microbrews, some enterprises have introduced niche brands based more on name or marketing, rather than on brewing style. One company by the name of Minerva, based in Guadalajara, introduced a beer named "Malverde" after the supposed "saint" of drug traffickers, Jesús Malverde. The beer was first introduced in Culiacán, Sinaloa. It is also available in Guadalajara. This same company had earlier launched "Duff" beer, modeled after the beer on the television show "The Simpsons" but ran into copyright issues. More recently, group of Mexican businessmen introduced a beer called "10 Maradó," which is named in honor of Diego Maradona. This beer is the first is a series dedicated to various football/soccer legends and teams which the enterprise called "La Liga Cervecera" (The Beer League). The beer was introduced at the II Festival de Cerveza de Guadalajara and according to its producers is "light, with body and with a flavor similar to the most premium beers of Argentina." Maradona's face does not appear on the label, only the word Maradó and the number 10, with a background of blue and white stripes. This same group created the company Cervecería Revolución with beers dedicated to Che Guevara, Maquiavelo and Emiliano Zapata.

Beer drinking habits
Mexican beer is distinctive for its lager-like properties, being generally light-bodied with a mild taste and is meant to be consumed cold. Most beer is sold in 325 ml bottles called "medias" in the popular slang, with certain brands, such as Tecate and Modelo Especial often sold in cans. In Mexico, most bottled beer is sold in returnable bottles, the deposit for which can be almost the price of the beer itself. but certain brands, such as Corona and Victoria, are available in large bottles of 925 or 940ml. The slang term for these is caguamas (sea turtles) or in some parts, like Mazatlán, ballenas (whales), but usually referring to the Pacifico brand. This type of bottle was first introduced in Mexico in 1960 along with the six-pack, often referred to in Mexico as a "six". Very little beer in Mexico is served on tap.

In Mexico, beer is commonly served with lime juice.

A beer cocktail called a michelada consists of light beer with lime juice, salt, and sometimes chili powder, Worcestershire sauce, soy sauce or tomato juice. As with a margarita, the salt is generally placed on the rim of glass. A less common version of the drink adds Clamato (clam/tomato juice). The name michelada comes from the phrase mi chela helada (my beer, ice cold). In many places also are found variants which include shrimp or seafood, pineapple and so on.

While many trends in the United States have become popular in Mexico, beer on tap has not. Another trend that has not proven successful in Mexico is that of light beer. Despite Corona Light being the best-selling imported light beer in the United States, sales of Corona Light in Mexico have been dismal. Grupo Modelo introduced the product after having success with exports. The beer was introduced with television ads and featured in two-for-one specials at many bars. However, light beer is generally not favored in Mexico, with the exception of Tecate Light in the north. With poor sales, Corona Light is rarely found in Mexico other than a few select places.

title: Beer in Mexico


text:
Inha University

Inha University is a private research university located in Incheon, South Korea. Known traditionally for research and education in the engineering and physical sciences, the university was established by the first president of South Korea, Syngman Rhee. Inha is a Korean-American collaboration school, even in its name: the Morpheme "In" (인, 仁) comes from the city of Incheon and "Ha" (하, 荷) from Hawaii, USA. Started as a polytechnic university in 1954, named Inha Institute of Technology (Acronym: IIT; Korean, 인하공과대학, Inha Gonggwa Daehak, colloquially Inhagongdae), the institute has been achieving national recognition and a strong reputation as a technological research university thereafter.

Inha University is the most well-known and #1 university in Incheon area. Inha was ranked top 10 nationwide through decades according to Joongang Ilbo's annual rankings of South Korean universities; ranked 8th in 2017. Also, Inha is a member of GU8.

Establishment
In 1952, in the midst of the Korean War, the first president of Korea, Syngman Rhee proposed the foundation of an educational institute that would provide expertise and hope to a lagging industrial sector. The driving force behind the proposal was the will and determination of a group of Koreans who had emigrated to Hawaii 50 years earlier.

The name 'Inha' comes from the name of 'Incheon' and 'Ha' in Hawaii. In March 1954, with the goal of establishing a world-class engineering university like MIT in the US, Dr. Lee Won Chul, an astronomer who obtained his Ph.D. degree, was the inaugural vice president Lee Ki-bong. On April 24, 1954, he had the entrance ceremony for 179 freshmen. On March 14, 1958, the establishment of a graduate school was approved.

Financial resources for the foundation came from the proceeds of the sale of the Korean Christian Institute, an organization founded and managed by Dr. Rhee for the purpose of educating the children of the original emigrants, donations from Korean emigrants in Hawaii and domestic supporters, and a government subsidy. Along with school site donation from Incheon city, the Ministry of Culture and Education approved the foundation of the Inha Institute in February, 1954.

Institute building construction began in earnest, to finally have the Inha Institute of Technology opened later the same year, in the scheme of the parent 6 Departments of Metallurgical, Mechanical, Mining, Electric, Shipbuilding and Chemical Engineering.

Time of expansion and development
The Inha Institute of Technology, with its unique history and great mission to foster excellent students who would contribute to the development of the nation's scientific and technological industries, was funded to do so by the help of huge government subsidies and foreign aid from UNESCO and the German government.

In a move to improve the management of the Inha Institute, the Board of Trustees was reorganized and Dr. Choong-Hoon Jo and Dr. Jwa-Kyung Seong assumed the positions of chairman of the board and the dean, respectively. That change brought more energy to the management, and resulted in the introduction of state-of-the-art experimental and research equipment, construction of new school buildings, research centers, practice factories, experimental tanks, library, and gymnasium, and secured excellent professors, contributing to the development of the Korean industry. With the rapid development of the Inha Institute of Technology, its size and facilities surpassed those of other colleges.

It was inevitable for the institute to be elevated to the status of university, thus in December, 1971, the Ministry of Culture and Education approved the elevation and Dr. Jwa-Kyung Seong took office as the first president, solidifying the basis toward one of the prestigious universities in Korea.

Ultimately on Oct. 2, 2014, in collaboration between Inha University and the Government of Uzbekistan, Inha University in Tashkent, the first South Korean university abroad, has been established, with a view to training professional global IT leaders in the country. The curriculum and academic program of this newly emerging international university comprise the same as those of Inha University in Korea.

Academic structure
Inha has 12 colleges in Engineering, Natural Sciences, Economics and Trade, Business Administration, Education, Law, Social Science, Humanities, Human Ecology, and Medicine, and 1 affiliation, 20 departments, 9 major fields of study. Inha University's Graduate school offers 47 fields of study for the Master's program and 41 for the Doctoral program. The Graduate School of Education has 31 majors for its Master's program and research course. The Graduate School of Business Administration offers 1 Master's program and 2 certificate courses and a part-time MBA program. The Graduate School of Engineering offers 14 majors in its Master's program and research courses. The Graduate School of Public Administration has 3 majors in its Master's program and research courses. Founded in 2000, the Graduate School of Information Technology and Telecommunications offers 1 major in its Master's program and the expanded and reorganized Graduate School of International Trade and Logistics now has 2 departments with 5 majors.

Undergraduate colleges
Inha university consists of 11 undergraduate colleges:
 * Frontier College
 * College of Engineering
 * College of Natural Sciences
 * College of Business Administration
 * College of Education
 * College of Social Sciences
 * College of Humanities
 * College of Medicine
 * College of Future Convergence
 * Arts and Sports
 * Global Studies

Graduate schools
Inha has 9 graduate schools: Graduate School, Graduate School of Information Technology and Telecommunications, Graduate School of Logistics, Law School, School of Medicine, Graduate School of Engineering, Graduate School of Education, Graduate School of Business Administration, Graduate School of Public Administration.

Attached agencies
Inha university has 7 attached agencies which are listed below:

 * Jung-Seok Memorial Library
 * Language Training Center
 * Center for Continuing Education
 * Inha Museum
 * Teachers Training Center
 * Infirmary
 * Jungseok Research Institute of International Logistics and Trade

Research institutes
Inha university has about 60 research institutes attached to it. These institutes are listed below.

Acceptance rate and Ranking
INHA University is one of the top private universities in Incheon, South Korea. It is ranked #531-540 in QS World University Rankings 2023, and #120 in QS Asian University Rankings 2023.

In 2022, 43,062 applicants applied to 2,631 recruitments for early admission to INHA University, recording a competition rate of 16.37:1 (6.1%). This is the result of an increase of 2.10 points in competition rate from 14.27:1 (7.0%) and the increase in number of applicants by 4,936. Despite the continuous decline in school-age population, it showed the highest competition rate in the pasts four years.

Taejoon Jeon, the director of the Admissions Office, said “Despite various difficult conditions such as a decrease in the school age population and a decrease in the number of early admissions, INHA University's competition for early admissions has risen sharply. This is the result of the recent recognition of INHA University's potential, which has shown excellent business orders in various education and research fields."

Notable alumni
• Changmin (TVXQ)

• Choi Min-gi (NU'EST)

• Choi Si-won (Super Junior)

• Lee Minhyuk (BTOB)

• Chun Ho-jin

• Chae Sang-woo

• Han Bo-bae

• Afzal Husain

• Hwang Min-hyun (NU'EST/Wanna One)

• Jeon Hyoseong (Secret)

• Kim Gura

• Kim Ki-Tae

• Kim Ryeowook (Super Junior)

• Kim Sung-ryung

• Lee Hyun-seung

• Lim Kyung-Wan

• No Min-woo (TRAX)

• Park Chanyeol (Exo)

• Kim Jun-myeon (Exo)

• Park Cho-a (AOA)

• Seo Jun-young

• Song Ji-Man

• Song Yoo-geun

• Isak Rajjak Shaikh

• Park Kyung (Block B)

title: Inha University