Читать книгу Computational Statistics in Data Science - Группа авторов - Страница 135

Consider the biomedical oxygen demand (BOD) data collected by Marske [39] where BOD levels were measured periodically from cultured bottles of stream water. Bates and Watts [40] and Newton and Raftery [41] study a Bayesian nonlinear model with a fixed rate constant 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 ESS_n using and with in Equation (4) is more than 8123.

At , ESS_n 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

and

The sampler then proceeds to implement a random walk Metropolis–Hastings step to update , and a draw from yields a joint MCMC draw from the posterior. We empirically note that this linchpin variable sampler yields lower marginal autocorrelation in as illustrated by Figure 4.

Repeating the previous procedure with the linchpin sampler, we have an estimated ESS at of 652, and the sequential stopping rule terminates at . The resulting estimates of posterior mean and quantiles are similar. Thus, using a more efficient sampler requires substantially fewer iterations to obtain estimates of similar quality.

Figure 4 Estimated autocorrelations for nonlinchpin sampler (a) and linchpin sampler (b).