Theory of Statistics - George Mason University

344 4 Bayesian Inference
Posterior
Posterior
0 1 2 3
0.0 1.0 2.0 3.0
x=2
0.0 0.2 0.4 0.6 0.8 1.0
π
x=6
0.0 0.2 0.4 0.6 0.8 1.0
π
Posterior
0.0 1.0 2.0 3.0
0.0 1.0 2.0 3.0
x=4
0.0 0.2 0.4 0.6 0.8 1.0
π
x=8
0.0 0.2 0.4 0.6 0.8 1.0
Figure 4.3. Posteriors Resulting from a Beta(3,5) Prior after Various Possible
Observations
Of course, in most cases, we must also take into account the loss function.
Recall the effects in this problem of different hyperparameter values on the
point estimation problem (that is, the choice of the Bayes action to minimize
the posterior risk) when the loss is squared error.
We might also consider what might be the effect of different hyperparameters.
There are several possibilities we could consider. Let’s just look at
one possibility, which happens to be bimodal, as shown in the upper right of
Figure 4.1. In this case, we have chosen α = 0.3 and β = 0.5. This would
correspond to a general prior belief that π is probably either close to 0 or
close to 1.
Now, again we might consider the effect of various observations on our
belief about Π. We get the posteriors shown in Figure 4.4 for various possible
values of the observations.
Compare the posteriors in Figure 4.4 with the prior in Figure 4.1. In each
case in this example we see that our posterior belief is unimodal instead of
bimodal, as our prior belief had been. Although in general a posterior may
Theory of Statistics c○2000–2013 James E. Gentle
Posterior
π