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In [[statistics]], '''multinomial logistic regression''' is a [[statistical classification|classification]] method that generalizes [[logistic regression]] to [[multiclass classification|multiclass problems]], i.e. with more than two possible discrete outcomes.<ref>{{cite book |last=Greene |first=William H. |author-link=William Greene (economist) |title=Econometric Analysis |edition=Seventh |___location=Boston |publisher=Pearson Education |year=2012 |isbn=978-0-273-75356-8 |pages=803–806 }}</ref> That is, it is a model that is used to predict the probabilities of the different possible outcomes of a [[categorical distribution|categorically distributed]] [[dependent variable]], given a set of [[independent variable]]s (which may be real-valued, binary-valued, categorical-valued, etc.).
Multinomial logistic regression is known by a variety of other names, including '''polytomous LR''',<ref>{{Cite journal | doi = 10.1111/j.1467-9574.1988.tb01238.x| title = Polytomous logistic regression| journal = Statistica Neerlandica| volume = 42| issue = 4| pages = 233–252| year = 1988| last1 = Engel | first1 = J.}}</ref><ref>{{cite book |title=Applied Logistic Regression Analysis |url=https://archive.org/details/appliedlogisticr00mena |url-access=limited |first=Scott |last=Menard |publisher=SAGE |year=2002 |page=[https://archive.org/details/appliedlogisticr00mena/page/n99 91]|isbn=9780761922087 }}</ref> '''multiclass LR''', '''[[Softmax activation function|softmax]] regression''', '''multinomial logit''' ('''mlogit'''), the '''maximum entropy''' ('''MaxEnt''') classifier, and the '''conditional maximum entropy model'''.<ref name="malouf">{{cite conference |first=Robert |last=Malouf |year=2002 |url=http://aclweb.org/anthology/W/W02/W02-2018.pdf |title=A comparison of algorithms for maximum entropy parameter estimation |conference=Sixth Conf. on Natural Language Learning (CoNLL) |pages=49–55}}</ref>
==Background==
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If the multinomial logit is used to model choices, it relies on the assumption of [[independence of irrelevant alternatives]] (IIA), which is not always desirable. This assumption states that the odds of preferring one class over another do not depend on the presence or absence of other "irrelevant" alternatives. For example, the relative probabilities of taking a car or bus to work do not change if a bicycle is added as an additional possibility. This allows the choice of ''K'' alternatives to be modeled as a set of ''K''-1 independent binary choices, in which one alternative is chosen as a "pivot" and the other ''K''-1 compared against it, one at a time. The IIA hypothesis is a core hypothesis in rational choice theory; however numerous studies in psychology show that individuals often violate this assumption when making choices. An example of a problem case arises if choices include a car and a blue bus. Suppose the odds ratio between the two is 1 : 1. Now if the option of a red bus is introduced, a person may be indifferent between a red and a blue bus, and hence may exhibit a car : blue bus : red bus odds ratio of 1 : 0.5 : 0.5, thus maintaining a 1 : 1 ratio of car : any bus while adopting a changed car : blue bus ratio of 1 : 0.5. Here the red bus option was not in fact irrelevant, because a red bus was a [[perfect substitute]] for a blue bus.
If the multinomial logit is used to model choices, it may in some situations impose too much constraint on the relative preferences between the different alternatives. This point is especially important to take into account if the analysis aims to predict how choices would change if one alternative were to disappear (for instance if one political candidate withdraws from a three candidate race). Other models like the [[nested logit]] or the [[multinomial probit]] may be used in such cases as they allow for violation of the IIA.<ref>{{cite journal |
==Model==
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