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and an exponential decay as a function of time. The data is available in Bates and Watts [40], Section A4.1]. Let
,
be the time points, and let
be the BOD at time
. Assume for
and
. Newton and Raftery [41] assume a default prior on
,
, and a transformation invariant design‐dependent prior for
such that
, where
is an
where the
th element of
. The resulting posterior distribution of
is intractable and up to normalization and can be written as
The goal is to estimate the posterior mean of . We implement an MCMC algorithm to estimate the posterior mean and implement the relative‐standard deviation sequential stopping rule via effective sample size.
We sample from the posterior distribution via a componentwise random walk Metropolis–Hastings algorithm updating first and then , with step size for both components chosen so that the acceptance probability is around 30%. Since the posterior distribution is three‐dimensional, the minimum ESS required for and in Equation (7) is 8123. Thus, we first run the sampler for and obtain early estimates of and the corresponding effective sample size. We then proceed to run the sampler until ESSn using and with in Equation (4) is more than 8123.
At , ESSn was 237, and the estimated density plot is presented in Figure 3 by the dashed line. We verify the termination criteria in Equation (7) incrementally, and simulation terminates at iterations. The final estimated density is presented in Figure 3 by the solid line.
Figure 3 Estimated density of the marginal posterior for
from an initial run of
(dashed) and at termination (solid).
At termination, the estimated posterior mean is , and 80% credible intervals are , , and for , , and , respectively.
It is possible to run a more efficient linchpin sampler [42] by integrating out from the posterior. That is, , where
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