# Traceplots and gelman.rubin statistics for analysing convergence of mixtures of discrete and continuous distributions

I have a Bayesian hierarchical model which contains a number of distributions which are mixtures of a point-mass at zero and a continuous random variable. The model is fitted using a gibbs sampler. For the complete conditionals of the point-mass and continuous mixtures I use a Metropolis Hastings (MH) step to pick the next sample value in the simulation. The MH proposal is a mixture of a point-mass at zero and normal distribution centred on the current value, with equal weighting (0.5) given to each part of the proposal. The model contains hundreds of parameters and I can't look at each of them individually so I use parallel chains and Gelman-Rubin (GR) statistics to give me some idea of approximate convergence. I discard a number of initial simulations as burnin and I've chosen 1.2 as the cut-off for GR statistic before I look at the traceplots to see what is happening. Although most of the examples on the GR statistic I've seen are for distributions which are absolutely continuous w.r.t Lebesgue measure I'm using it for the point-mass mixture because it could be considered a measure of similarity between the chains when they start from over-dispersed initial values.

However, this approach has raised a number of issues for me for which I haven't been able to locate any papers or other texts for guidance for these types of distributions.

I've really got two questions:

1) Mostly the GR statistics are less than 1.1 for all model parameters but occassionally for the point-mass + continuous distribution they will exceed 1.1 (or occasionally 1.2). If the model is run again this may occur for different parameters. Given the large number (sometimes 1000+) of parameters in the model is this type of "random" departure from "close to 1" to be expected or an indication that there may be something not quite right?

2) For point-mass + continuous distributions with larger GR statistics the traceplot will sometimes show that one or more of the chains appears to be in particular region for a short while, and this of course has an effect on the GR statistic. Even a relatively small number of iterations like this can cause it to increase.

For example: With 5 parallel chains one particular parameter reports a GR statistic with value 1.12. The traceplot for this data using coda is: Looking at the first chain I can that there are sections made up completely of zeros - for example between iterations 10000 and 20000 (slightly difficult to see): Looking at the second chain between iteration 30000 and 40000 there is a section where most of the values are non-zero, without ever visiting zero, similarly between 40000 and 50000: Looking at the GR stat and the traceplots I think I can say that because we are looking at a mixture with a discrete term that the sample values remaining at zero for a number of iterations, or remaining above zero for a number of iterations is not unexpected behaviour, particularly given the weighting of the proposal distribution (approximately every second proposal value will be 0 leading to the possibility of many rejections in certain regions) and for chain 2, it has reached an area (between 45000 and 50000) that the other chains starting from different initial values haven't (yet). It remains longer in this area because proposed zeros are rejected. My reasoning for this is that if, for example, if it turned out that in "reality" the point-mass mixture was actually degenerate at zero then then I'd expect the sample to be mostly zeros, in which case iterations of consecutive zeros would be common, so this type of occurrence can't be ruled out just because the traceplot does not show the type of behaviour we might expect for a distribution which is absolutely continuous w.r.t Lebesgue measure.

Is this a realistic assessment of what is happening, the GR statistic suggests looking at the traceplot but there is nothing really sinister happening, or do the traceplots indicate that further tuning of the parameters may be needed, for example re-weighting the proposal, and I should in fact try to get rid of either of these periods in the traceplots - or in fact is there any real general way of approaching this type of analysis which I should be considering?

Any input is appreciated and apologies if this or similar has been covered elsewhere (I looked but it is always possible to miss something).