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In binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value).
The simplest direct probabilistic model is the logit model, which models the log-odds as a linear function of the explanatory variable or variables. The logit model is "simplest" in the sense of generalized linear models (GLIM): the log-odds are the natural parameter for the exponential family of the Bernoulli distribution, and thus it is the simplest to use for computations.
The formulation of binary logistic regression as a log-linear model can be directly extended to multi-way regression. That is, we model the logarithm of the probability of seeing a given output using the linear predictor as well as an additional normalization factor , the logarithm of the partition function :
The resulting model is known as logistic regression (or multinomial logistic regression in the case that K-way rather than binary values are being predicted). For the Bernoulli and binomial distributions, the parameter is a single probability, indicating the likelihood of occurrence of a single event.
As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. [2] It is most often estimated using the maximum likelihood procedure, [3] such an estimation being called a probit regression.
This is also known as the log loss (or logarithmic loss [4] or logistic loss); [5] the terms "log loss" and "cross-entropy loss" are used interchangeably. [6] More specifically, consider a binary regression model which can be used to classify observations into two possible classes (often simply labelled and ).
Binomial regression is closely connected with binary regression. If the response is a binary variable (two possible outcomes), then these alternatives can be coded as 0 or 1 by considering one of the outcomes as "success" and the other as "failure" and considering these as count data : "success" is 1 success out of 1 trial, while "failure" is 0 ...
For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model .