Three easy confusions

Remark taught in Bayesian linear regression, part of Bayesian statistics

Three easy confusions

Type	Remark
Taught in	Bayesian linear regression
Part	Bayesian statistics

1. \widehat{\boldsymbol{\beta}} has an exact normal sampling distribution when \sigma^2 is known. The t distribution applies after studentisation by s.

Remark

2. The marginal variance posterior has inverse-Gamma shape (n-p)/2, not (n+p)/2. The p/2 belongs to the coefficient normalising constant and must be subtracted from the original power.

3. Drawing first from p(\sigma^2\mid{\boldsymbol{y}}) and then from p({\boldsymbol{\beta}}\mid\sigma^2,{\boldsymbol{y}}) produces iid posterior draws. A genuine Gibbs sampler instead uses p(\sigma^2\mid{\boldsymbol{\beta}},{\boldsymbol{y}}) and therefore produces a Markov chain.

Used in

• Taught in Bayesian linear regression, where the derivation and examples are.
• All 8 questions on Bayesian linear regression

See also

Inverse-Gamma distribution · Systematic-scan Gibbs sampler · Markov chain · Gaussian linear model · Error and residual · Ordinary least squares · Sampling distribution of the OLS estimator · Studentised coefficient pivot

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