L5B Appendix: Composite Hypothesis and Conflicting Priors
In this note we compare the consequences of the priors for hypotheses being induced by the priors on the
parameter(s) and the priors for hypotheses being specified independent of the parameters.
Suppose for a single parameter, θ, which has parameter space Θ, that the following two hypotheses are
posed.
H0 : θ ≤ θ0
H1 : θ > θ0
and Θ0 is the subspace of Θ where θ ≤ θ0, and vice versa for Θ1.
Two scenarios.
Induced: A prior for θ is specified, pi(θ), and the prior probabilities for each hypothesis are thus induced :
p0 = Pr(H0) = Pr(θ ∈ Θ0) =
∫ θ0
−∞
pi(θ)dθ
p1 = Pr(H1) = Pr(θ ∈ Θ1) =
∫ ∞
θ0
pi(θ)dθ
Specified: Priors for H0 and H1 are simply specified, independent of pi(θ).
Given data from the sampling distribution f(y|θ), the posterior probability for H0 is:
Pr(H0|y) = Pr(H0,y)
m(y)
=
f(y|H0)p0
m(y)
=
p0
∫
f(y|θ,H0)pi(θ|H0)dθ
m(y)
=
p0
∫
θ∈Θ0 f(y|θ)
pi(θ)
p0
dθ
m(y)
=
∫
θ∈Θ0 f(y|θ)pi(θ)dθ
m(y)
= Pr(θ ∈ Θ0|y)
Comments:
1. pi(θ|H0) is the prior restricted to Θ0, which is then pi(θ)/p0.
2. The posterior for H0 is thus the integral over Θ0 of the posterior for θ.
3. The prior for H0, p0, disappears from Pr(H0|y) in either scenario.
4. The Bayes Factor, however, retains p0 and p1, i.e., there is no cancellation as in the case of the simple
vs simple and simple vs composite hypotheses cases.
5. There is a potential for strange situations where a specified prior for H0 is extremely different than
what the induced prior for H0 would have been. For example, let θ be a Poisson parameter with two
hypotheses: H0: θ ≤ 3 and H1: θ > 3. Specify probabilities for the hypotheses of p0=0.1 and p1=0.9,
but let the prior for θ be Gamma(7,3.5). Thus the induced probabilities are p0=0.9 and p1=0.1.
October 16, 2021






















































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