Generalized linear models Generalized Linear model (GLM)

This paragraph is mainly about the definition and assumption of the generalized linear model, in order to see the logical regression, we have to read the patience.


1.The Exponential family exponential distribution family

Because the generalized linear model is around the exponential distribution family, it needs to be introduced first, in the words of ng great God, "although not all, most of the distributions we have seen are of exponential distribution, such as: Bernoulli Bernoulli distribution, Gaussian Gaussian distribution, multinomial distribution, Poisson poisson distribution, gamma distribution, exponential distribution, Dirichlet distribution ... "The condition that obeys the exponential distribution family is that the probability distribution can be written as follows:

η is called the natural parameter, which is the only parameter of the exponential distribution family.
T (y) is called sufficient statistic, in many cases T (y) =y A (η) is called log partition function
t function, a function, B function together to determine a distribution
Now let's see why the normal distribution (Gaussian distribution) belongs to the exponential distribution family:
Normal distribution (normal distribution has two parameters μ mean and σ standard deviation, when doing linear regression, we are concerned about the mean value and the standard deviation does not affect the model of learning and parameter θ choice, so here σ is set to 1 easy to calculate)

2. Three hypotheses constituting a generalized linear model

P (y | x;θ) ∼exponentialfamily (η). The conditional probability distribution of the output variable based on the input variable obeys the exponential distribution family

Our goal are to predict the expected value of T (Y) given x. For a given input variable x, the goal of learning is to predict the expected value of T (y), which is often the Y natural parameterηa nd the inputs x are related linearly:η=θt x.η and input variable X's association is linear: Η=θt x

These three hypotheses actually indicate how to map the input variable to the output variable and the probability model. For example, the conditional probability distribution of linear regression is the exponential distribution family (the likelihood function part of the linear regression in the reference note); Our goal is to predict the expectations of T (y), by the above calculation we know T (y) =y , and the expectation of Y is also the parameter μ of the normal distribution; we know Μ=η by the above calculation, and Η=θt x. Therefore, linear regression is a special case of generalized linear regression, and its model is:

Classical linear regression: The predictive value y is continuous, assuming that given x and the parameters, Y's probability distribution obeys the Gaussian distribution (corresponding to the first hypothesis of the construction GLM). Logical regression: In two categories, for example, the predictive value y is {1,0} with a value of two, assuming that given x and the parameters, the probability distribution of y obeys the Bernoulli distribution (corresponding to the first assumption of the construction GLM).

Through this study to the establishment of the GLM model.

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