Chapter 4 Bayesian Computation (Monte Carlo Methods)
Let be the data observed and
,
be of interest. Sometimes, it might be very difficult to find the explicit form the above integral. In such cases, Monte Carlo method might be an alternative choice.
-
Direct sampling
The direct sampling is to generate from and then use
to estimate r. Note that N is usually large. The variance of is
.
Thus, the standard error of is
.
An approximate 95% confidence interval for r is
.
Further, the estimate of
is
.
The standard error of is
.
-
Indirect sampling
As can not be generated from the posterior directly, the following sampling methods can be used:
-
important sampling
-
rejection sampling
-
the weighted boostrap
(a) Important sampling
Since
where
and is a density which the data can be generated from easily and be in generally chosen to approximate the posterior density, i.e.,
.
is called importance function. can be estimated by
,
where
The accuracy of the important sampling can be estimated by plugging in and .
(b) Rejection sampling
Let be a density which the data can be generated from easily and be in generally chosen to approximate the posterior density. In addition, there is a finite known constant such that
,
for every . The steps for the rejection sampling are:
1. generate from .
2. generate independent of from .
3. If
accept ; otherwise reject .
4. Repeat steps 1~3 until the desired sample (accepted )
,
are obtained. Note that will be the data generated from the posterior density. Then,
.
Note:
Differentiation with respect to yield , the posterior density function evaluated at .
(c) Weighted boostrap
It is very similar to the important sampling method. The steps are as follows:
1. generate from .
2. draw from the discrete distribution over which put mass
,
at . Then,
.
-
Markov chain Monte Carlo method
There are several Markov chain Monte Carlo methods. One of the commonly used methods is Metropolis-Hastings algorithm. Let be generated from which is needed only up to proportionality constant. Given an auxiliary function such that is a probability density function and , the Metropolis algorithm is as follows:
-
Draw from the p.d.f. , where is the current state of the Markov chain.
-
Compute the odds ratio .
-
If , then .
If , then
4. Repeat steps 1~3 until the desired sample (accepted )
,
are obtained. Note that will be the data generated from the posterior density. Then,
.
Note:
For the Metropolis algorithm, under mild conditions, converge in distribution to the posterior density as .
Note:
is called the candidate or proposal density. The most commonly used is the multivariate normal distribution.
Note:
Hastings (1970) redefine
,
where is not necessarily symmetric.
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