Page 387 - Special Topic Session (STS) - Volume 3

STS551 Zamira Hasanah Zamzuri et al.
measures the proportion of reported accident whereas τ is the true accident
rate. Then the specification of this model is
| , , π~( )

= πτ

= exp( + log( 1 ) + ⋯ + (−1) log ( (−1) ))

0
1
~ (0, )

The posterior distribution of this proposed model is proportional to

{∏ ( | , 0 −1 )} ( −1 | , ) (π| , ) ∏ ( |) ∏ ( | , , π)

0

0
1
0

2
=1 =1 =1
in which is the k-variate normal distribution, is the Wishart density, is

the Beta density and is the Poisson density. Parameter estimation in this
model is performed in two phases; first by estimating the proportion of
reported accident rate π; then secondly, the estimation of other parameters.
Hence, four sampling stages are identified:
1) Sampling π
2) Sampling

3) Sampling

4) Sampling −1
In Bayesian framework, if the posterior distribution is in the same family as
the prior, it is called as conjugate distributions; in which the prior is called as
conjugate prior to the likelihood. In cases that the conditional posterior
distribution is identified, the Gibbs sampling algorithm can be used. In
contrast, when the conditional posterior distribution is unidentified, there is a
need for Metropolis Hastings algorithm, in which a sampling density will be
proposed.
We present the details of these four sampling stages in the APL model.
1) Sampling π
We want to sample from a density proportional
to (π| , ) ∏ =1 ( | , , π). Since this is not a recognized

1

2

distribution, the Metropolis-Hastings algorithm is needed. A technique as
suggested in Chib & Winkelmann will be applied here in which we will
maximize the log of this density using Newton-Raphson algorithm. Then,
the parameter estimates will be sampled from a proposed density.
2) Sampling

We want to sample from a density proportional to
∏ ( |) ∏ =1 ( | , , π). Since this is also not a recognized
=1

distribution, the Metropolis-Hastings algorithm is needed. The same
technique as (1) is used for this stage. Then, we sample from the proposal
density, multivariate-t.
Let
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