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of multiplicities m ∗ indicates how many times each unique history appears in the original dataset, where history y ∗ i occurs in y a total of m ∗ i times, for i = 1 , . . . , n $^{∗}$. 

Using this reduced representation, we can express (3) such that the likelihood of each unique observation history is calculated only once. This computational approach makes use of Bayes law in the form: 

<img src='content_image/304238.jpg'>

Computing according to (7) requires only n ∗ applications of the filtering likelihood calcula- tion, rather than n applications when using the filtering MCMC approach on the full dataset. We expect to this provide an approximate factor of n/n ∗ improvement in computational ef- ficiency relative to the filtering MCMC on the original dataset. 

## Filtering MCMC with block sampling 

As a final approach, we consider joint (a.k.a. block) MCMC sampling of model parameters (Roberts and Sahu, 1997). In the case of correlated posteriors, it is well known that block sampling of highly-correlated parameter dimensions can result in improved MCMC mixing ( e.g. , Liu, Wong, and Kong, 1994). The general problem of determining posterior dimensions for block sampling is difficult, as a practitioner cannot reliably guess what blocking arrange- ment will result in efficient MCMC sampling. Further, existing literature on the efficiency of block sampling generally only considers the mixing properties of univariate versus block sam- pling, and fails to consider computational demands (Mengersen and Tweedie, 1996; Roberts and Tweedie, 1996; Roberts, Gelman, and Gilks, 1997, among others). 

We make use of NIMBLE’s automated procedure for determining an efficient problem- specific block sampling MCMC algorithm, which exemplifies how the flexibility and pro- grammability of NIMBLE facilitates a higher level of algorithmic control than other statis- tical software packages. This procedure dynamically determines a partition of the model 

