Given a series of data, we want to make inferences about two kinds of latent variables: cluster labels and cluster parameters. The Fearnhead approach only uses particles to represent the cluster assignments, and relies on conjugate priors for the cluster parameters to get the posteriors of the cluster parameters conditional on the assignments. There are three kinds of operations you might want to do:
-
filter one observation (get the posterior distribution of the cluster
assignment after absorbing one observation).
$p(z_T | x_{1:T})$ . This uses only past and present data to infer the state for present data. -
smooth a series of observations (get the posterior distribution of the
sequence of cluster assignments after a series of observation).
$p(z_{1:T} | x_{1:T})$ . This uses "future" data, in the sense that inferences about the state$z_k$ at$k<T$ incorporates information from future data at times$k+1:T$ . -
predict future data or states:
$p(x_{T+1} | x_{1:T})$ or$p(z_{T+1} | x_{1:T})$ .
The Fearnhead approach can be used to do non-parametric clustering, where the number of clusters is unknown. This makes the filtering/smoothing a little tricky, since the cluster labels might not be consistent across particles. So just marginalizing across the particles doesn't make sense. But that's a bridge to cross when we come to it. One possibility is to return a similarity matrix sort of thing, which is the probability that any pair of observations belongs to the same cluster. That can be marginalized over (and compared to the "correct" clustering solution).
There's no need to keep anything other than the putatives for the old particles. And in fact you might be able to represent the whole population more efficiently if you keep a vector of all the components and then just index into that...that might be too fiddly and not buy you too much though.
I guess the question always is, what's the problem this optimization is trying to solve? The run-away copying of vectors that causes some out of control memory use. But that's not the major bottleneck except for the really easy 1-D cases.
The filters should be re-factored into the filter itself and the propogation/resampling method.