Generalized linear models extend linear regression to accommodate non-normal outcome distributions by combining a likelihood family with a link function. Below are the standard likelihoods commonly used.

1. Continuous Data

Linear regression is a special case of the generalized linear model with the identity link function, $g(\mu) = \mu$.

For strictly positive continuous responses, one may instead model $\log(y)$ as normally distributed. If the log-normal model is inadequate, gamma or Weibull likelihoods are often considered as alternatives because they naturally constrain outcomes to be positive and can flexibly model skewness.

2. Poisson Likelihood for Count Data

Count data are often modeled using the Poisson generalized linear model. The model assumes:

  • $y_i \sim \text{Poisson}(\mu_i)$
  • $\operatorname{Var}(y_i) = \mu_i$

The canonical link is the log link, so that:

$\log(\mu_i) = X_i \beta,$
where $\eta_i = X_i \beta$ is the linear predictor.

The likelihood for the full dataset $y = (y_1, \ldots, y_n)$ is:

$\displaystyle p(y \mid \beta) = \prod_{i=1}^n \frac{1}{y_i!} \exp\big(-\exp(\eta_i)\big)\big(\exp(\eta_i)\big)^{y_i}.$

When forming the Bayesian posterior distribution, $\prod_i 1/y_i!$ is constant with respect to $\beta$ and can be ignored.

3. Binomial Likelihood

For binary or binomial outcomes, suppose:

$y_i \sim \text{Bin}(n_i, \mu_i)$
with known $n_i$.

The model is usually expressed in terms of the success probability $\mu_i$, where the link is the logit function:

$g(\mu_i) = \log \left( \frac{\mu_i}{1 – \mu_i} \right).$

This produces the logistic regression model. The likelihood is:

$\displaystyle p(y \mid \beta) = \prod_{i=1}^n \binom{n_i}{y_i} \left(\frac{e^{\eta_i}}{1 + e^{\eta_i}}\right)^{y_i} \left(\frac{1}{1 + e^{\eta_i}}\right)^{n_i – y_i}.$

Because $\binom{n_i}{y_i}$ depends only on observed data, these factors can be absorbed into a proportionality constant in the posterior.

4. Probit Likelihood

Another common link for binomial outcomes is the probit link, defined as:

$g(\mu_i) = \Phi^{-1}(\mu_i),$

where $\Phi$ is the standard normal cumulative distribution function.

The corresponding likelihood is:

$\displaystyle p(y \mid \beta) = \prod_{i=1}^n \binom{n_i}{y_i} \left(\Phi(\eta_i)\right)^{y_i} \left(1 – \Phi(\eta_i)\right)^{n_i – y_i}.$

The probit model can be viewed as a normal latent-variable system in which an underlying continuous response is dichotomized. In practice, the probit and logit models behave similarly; differences appear mainly in the tails.

Both logistic and probit models can be extended to multivariate outcomes.

5. Other Link Functions

A robust alternative to logit and probit uses the $t$ distribution for latent variation.

Another commonly used link is the complementary log-log:

$g(\mu) = \log(-\log(1 – \mu)),$

which is asymmetric and appropriate for modeling hazard-type processes.

6. Overdispersed Models

In applied settings, the variance of the data often exceeds the variance implied by the assumed likelihood—a phenomenon known as overdispersion.

For example, if the unit of observation is a litter of mice and $y_i/n_i$ is modeled as binomial, real data may show more variation than a binomial model allows because litters differ systematically (e.g., by mother). A hierarchical model naturally accounts for this by including mother-specific effects.

These effects may follow a normal distribution (often interpretable) or a beta distribution (conjugate to the binomial).

A similar idea applies in Poisson regression: extra variation can be modeled by adding an individual-level normal error term to the linear predictor. This produces a Poisson model with extra-Poisson variation.