Priors, Normal Models, Computing Posteriors

1. Priors, Normal Models, Computing Posteriors st5219: Bayesian hierarchical modelling lecture 2.2

2. The normal distribution Stupid name

3. 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

4. A normal model Board work

5. Conjugate priors for a normal model • The normal-scaled inverse χ²(NSIχ²) distribution is conjugate for the normal distribution • If (μ,σ²)~NSIχ²(μ0, κ0, ν0, σ0²)and xi~N(μ,σ²) then(μ,σ²)|x ~ NSIχ²(μn, κn, νn, σn²) • Use geoR’sdinvchisq, rinvchisq for the inverse χ² bit • To sample NSIχ², • first draw σ² from Iχ²(μk, κk, νk, σk²) and • then μ |σ² from N(μk, σ²/κk) • See Gelman et al (2003) Bayesian Data Analysis Chapman & Hall

6. In practice See computing posteriors (next section)