A proper approximation to the flat prior
Remark taught in Conjugate analysis for normal models, part of Bayesian statistics
A proper approximation to the flat prior
| Type | Remark |
| Taught in | Conjugate analysis for normal models |
| Part | Bayesian statistics |
The honest bounded version of the improper kernel p(\mu)\propto1 is
p_C(\mu)=\frac1{2C}\mathbf{1}\{-C\leq\mu\leq C\}.
Taking, for example, C=10^4 makes this density nearly constant over an ordinary data range while keeping it proper. It also exposes what “very diffuse” means. For any fixed 0<M<C,
\mathbb{P}_C(|\mu|\leq M)=\int_{-M}^{M}\frac1{2C}\,\mathrm{d}\mu=\frac MC \longrightarrow0 \qquad(C\longrightarrow\infty).
Thus the limiting sequence puts essentially all of its probability beyond every fixed finite range; it does not converge to a uniform probability distribution on \mathbb{R}. Human height makes the point concrete: no observed person is three metres tall, yet a sufficiently broad uniform prior assigns almost no mass to the entire range of heights that have actually been encountered.
Used in
- Taught in Conjugate analysis for normal models, where the derivation and examples are.
- All 19 questions on Conjugate analysis for normal models
See also
Normal–Normal model · Normal–Normal conjugacy · Precision weighting · Inverse-Gamma distribution · Normal–inverse-Gamma variance update · Full conditionals under independent priors