ST5219: BAYESIAN HIERARCHICAL MODELLING LECTURE 2.2 Priors, Normal Models, Computing Posteriors

The normal distribution Stupid name

The normal distribution Although data are normally not normal, the normal distribution is a popular model for data Assume normal distributions in:  paired t tests  two sample t tests  ANOVAs  regression  multiple regression ++? and use it as a limiting distribution for other models We’ll look at how to deal with a single sample now Next week: multiple normal data sets

A normal model Board work

Conjugate priors for a normal model The normal-scaled inverse χ² (NSIχ²) distribution is conjugate for the normal distribution If (μ,σ²)~ NSIχ²(μ 0, κ 0, ν 0, σ 0 ²) and x i ~N(μ,σ²) then (μ,σ²)|x ~ NSI χ²(μ n, κ n, ν n, σ n ²) Use geoR’s dinvchisq, rinvchisq for the inverse χ² bit To sample NSIχ²,  first draw σ² from Iχ²(μ k, κ k, ν k, σ k ²) and  then μ | σ² from N(μ k, σ²/κ k ) See Gelman et al (2003) Bayesian Data Analysis Chapman & Hall

In practice See computing posteriors (next section)