Theory of Statistics - George Mason University

330 4 Bayesian Inference
4.2.2 Regularity Conditions for Bayesian Analyses
Many interesting properties of statistical procedures depend on common sets
of assumptions, for example, the Fisher information regularity conditions. For
some properties of a Bayesian procedure, or of the posterior distribution itself,
there is a standard set of regularity conditions, often called the Walker regularity
conditions, after Walker (1969), who assumed them in the proofs of various
asymptotic properties of the posterior distribution. The regularity conditions
apply to the parameter space Θ, the prior PDF fΘ(θ), the conditional PDF
of the observables fX|θ(x), and to the support X = {x : fX|θ(x) > 0}. All
elements are real, and µ is Lebesgue measure.
The Walker regularity conditions are grouped into three sets:
A1. Θ is closed.
When a general family of distributions is assumed for the observables,
this condition may allow for a distribution that is not in that family (for
example, in a Bernoulli(π), the parameter is not allowed to take the values
0 and 1), but the convenience of this condition in certain situations more
than pays for this anomaly, which occurs with 0 probability anyway.
A2. X does not depend on θ.
This is also one of the FI regularity conditions.
A3. For θ1 = θ2 ∈ Θ, µ({x : fX|θ(x) = fX|θ(x)}) > 0.
This is identifiability; see equation (1.24). Without it parametric inference
does not make much sense.
A4. Given x ∈ X and θ1 ∈ Θ and δ a sufficiently small real positive number,
then ∀θ ∋ θ − θ1 < δ,
| log(fX|θ(x)) − log(fX|θ1 (x))| < Hδ(x, θ1)
where Hδ(x, θ1) is a measurable function of x and θ1 such that
lim Hδ(x, θ1) = 0
δ→0+
and, ∀˜ θ ∈ Θ,
lim
δ→0+
Hδ(x, θ1)fX| θ ˜(x)dx X
= 0.
This is a continuity condition.
A5. If Θ is not bounded, then for any ˜ θ ∈ Θ and a sufficiently large real
number ∆,
θ > ∆ =⇒ log(fX|θ(x)) − log(f X| ˜ θ (x)) < K∆(x, ˜ θ),
where K∆(x, ˜ θ) is a measurable function of x and ˜ θ such that
lim
δ→0+
K∆(x, ˜
θ)fX| θ ˜(x)dx < 0.
Theory of Statistics c○2000–2013 James E. Gentle
X