Theory of Statistics - George Mason University

360 4 Bayesian Inference
For a 0-1 loss function, the Bayes estimator of IΘ0(θ) is the function that
minimizes the posterior risk, EΘ|x(L(Θ, s)). The risk is just the posterior
probability, so the Bayesian solution using this loss is
1 if Pr(θ ∈ Θ0|x) > Pr(θ /∈ Θ0|x)
S(x) =
0 otherwise,
where Pr(·) is evaluated with respect to the posterior distribution PΘ|x.
4.5.2 Loss Functions
Due to the discrete nature of the decision regarding a test of an hypothesis,
discrete loss functions are often more appropriate.
The 0-1-γ Loss Function
In a Bayesian approach to hypothesis testing using the test δ(X) ∈ {0, 1}, we
often formulate a loss function of the form
cd for θ ∈ Θ0
L(θ, d) =
bd for θ ∈ Θ1
where c1 > c0 and b0 > b1, with c0 = b1 = 0, b0 = 1, and c1 = γ > 0. (This is
a 0-1-γ loss function; see page 257.)
A Bayesian action for hypothesis testing with a 0-1-γ loss function is fairly
easy to determine. The posterior risk for choosing δ(X) = 1, that is, for
rejecting the hypothesis, is
cPr(Θ ∈ ΘH0|X = x),
and the posterior risk for choosing δ(X) = 0 is
Pr(Θ ∈ ΘH1|X = x),
hence the optimal decision is to choose δ(X) = 1 if
which is the same as
cPr(Θ ∈ ΘH0|X = x) < Pr(Θ ∈ ΘH1|X = x),
Pr(Θ ∈ ΘH0|X = x) < 1
1 + c .
In other words, the Bayesian approach says to reject the hypothesis if its
posterior probability is small. The Bayesian approach has a simpler interpretation
than the frequentist approach. It also makes more sense for other loss
functions.
Theory of Statistics c○2000–2013 James E. Gentle