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12.3 Bernoulli-logit generalized linear model (Logistic Regression)

Stan also supplies a single function for a generalized linear model with Bernoulli likelihood and logit link function, i.e. a function for a logistic regression. This provides a more efficient implementation of logistic regression than a manually written regression in terms of a Bernoulli likelihood and matrix multiplication.

12.3.1 Probability mass function

If xRnm,αRn,βRm, then for y{0,1}n, BernoulliLogitGLM(y | x,α,β)=1inBernoulli(yi | logit1(αi+xiβ))=1in{logit1(αi+1jmxijβj)if yi=1, and1logit1(αi+1jmxijβj)if yi=0.

12.3.2 Sampling statement

y ~ bernoulli_logit_glm(x, alpha, beta)

Increment target log probability density with bernoulli_logit_glm_lupmf(y | x, alpha, beta).

12.3.3 Stan Functions

real bernoulli_logit_glm_lpmf(int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

real bernoulli_logit_glm_lpmf(int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

real bernoulli_logit_glm_lpmf(int[] y | row_vector x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int[] y | row_vector x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

real bernoulli_logit_glm_lpmf(int[] y | row_vector x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int[] y | row_vector x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

real bernoulli_logit_glm_lpmf(int[] y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int[] y | matrix x, real alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.

real bernoulli_logit_glm_lpmf(int[] y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta).

real bernoulli_logit_glm_lupmf(int[] y | matrix x, vector alpha, vector beta)
The log Bernoulli probability mass of y given chance of success inv_logit(alpha + x * beta) dropping constant additive terms.