Bayesian GLMs

Summary so far

We have seen two models that share the same general structure:

• “Bernoulli regression” for binary observations
• “Normal regression” for continuous observations.

Generalization of these two special cases

• Look at the data type of the output variable, this will guide the choice of likelihood model
• Are you trying to predict..
  • a real number? us a normal likelihood (or better: fat tail distribution such as t distribution, for robustness to outliers)
  • a non-negative integer? Replace Bernoulli by Poisson (or Negative Binomial, or other integer valued distribution)
  • etc
• General pattern:

\[\text{output} | \text{inputs}, \text{parameters} \sim \text{SuitableDistribution}(f(\text{parameters}, \text{inputs}))\]

• \(f\) should map to the range of parameters permitted by your “SuitableDistribution”
• known in the GLM literature as the inverse of the link function
• often a composition of a linear function with a non-linear “squashing” function if the output is constrained.