Through a round-about series of links I ended up reading a recent Xian’s Og post about a paper by one of my former QUT colleagues, Chris Drovandi, and my former supervisor, Tony Pettitt, on a simulation-based methodology for posterior exploration and model choice tailored to the case of low count time series data. In particular, the models considered have a Markovian structure in discrete time steps, and the specification of low count data is so that we have a non-negligible chance of generating mock data at each time step that matches exactly (or near to, i.e., ABC) the observed data. One example considered is for a compartmental model of disease outbreak (which piqued my interest in regards to my current malaria modelling work) and another is for an Integer Auto Regressive Moving Average model (IARMA): this latter falling within the class of models now being explored for modelling astronomical time series (e.g. this talk, or this other talk). The power of the simulation-based approach arises in this context when the process is only partially observed (e.g. some observations are just lower/upper limits or there are gaps in the time series) such exact computation of the likelihood becomes an intractable summation problem. Instead, the method presented by Drovandi et al. uses the “alive particle filter” strategy to produce an unbiased estimate of the likelihood, which is suitable for both posterior sampling and model choice (e.g. with an RJMCMC setup). I’m sure there must be some great applications for this method in astronomy … ideas welcome!?
[A side note: the trick of the alive particle filter is to use negative binomial sampling of N+1, rather than N, draws to recover an unbiased likelihood estimate from the acceptance ratio … a trick which I was also playing with (but abandoned) in the context of our Importance Nested Sampling algorithm; rather than stopping with one L > L_target particle I thought we could go for two such particles to produce an unbiased estimate of the ratio between the proposal ellipse and L > L_target volume, but it didn’t really pay off with an improved accuracy in practice.]