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I'm using an adaptive MCMC (metropolis-hastings) scheme to infer some parameters. I've run 7 chains, each starting from a random point.

6 of the chains vaguely converge to the same area, but one of the chains converges to a different value, and with a much smaller variance.

The data I generated was synthetic, so I know that the other 6 chains were converging to the right area, and this other chain is converging to the wrong area. But what is this indiciative of? Or how can I determine what this is indicative of? Have I just got stuck in a local maxima somewhere? If so, how do I overcome this? Or is that the whole point of using multiple chains?

Another potential issue is that I'm currently using a diagonal covariance matrix. The parameters are biological, and so are probably correlated in some way? I'm aware that this is an issue I should probably correct, but could this specifically be causing this wrong convergence issue?

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    $\begingroup$ Yes, finding convergence problems is the whole point of running multiple chains. If chain #7 found a lokal maximum, that exists but was not also found by the other 6 chains you are in trouble, because you do not know, whether there are even more local maxima to be found. Running much longer chains is the first thing to try. Hopefully eventually they will all converge. Next step is to find out, which settings are available to play with in your software. $\endgroup$
    – Bernhard
    Commented May 12, 2020 at 10:13
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    $\begingroup$ The likelihood and prior values at the endpoints should be considered to determine if the #7 chain got stuck in a terrible part of the space. $\endgroup$
    – Xi'an
    Commented May 12, 2020 at 10:46

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You basically answered your question. We use multiple chains to diagnose problems with convergence (see e.g. Roy, 2020; Vehtari et al, 2019). If one of the chains explores different area of the posterior then the others, this is a clear sign of problems with convergence. This would often mean that you cannot trust the results. Try running the simulation longer, maybe after more iterations all the chains would converge? Otherwise, by the folk theorem of statistical computing, computational problems often suggest problems with the model itself.

Here you can find a good set advice & links by George Ho on Bayesian models, MCMC, diagnostics checks etc.

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  • $\begingroup$ I'm not using a mixture model. I'm fitting parameters to a mechanistic model with a likelihood given by the solution to a forward equation. I'm guessing it will vary a lot from model to model, but is there any rough guess as to how long the chain should be before you decide the issue is probably elsewhere? Up to now, I've been running it until I have 100,000 accepted samples, which feels like a lot (though as I said, it's probably all relative - just to what extent?) I don't the model is the issue, as I simulate data from the markov chain I'm using to calculate my likelihood. $\endgroup$
    – user112495
    Commented May 12, 2020 at 10:19
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    $\begingroup$ @user112495 how many samples is "enough" depends on the model and the data, 100K may, or may not be enough. For complicated model, it might be the case you need to run the model for hours, or days. Still, worth checking if there is no issues that may lead to local maximas. Label switching was given as one of the examples of where this commonly happens because of local maximas. $\endgroup$
    – Tim
    Commented May 12, 2020 at 10:22
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    $\begingroup$ @user112495 if you are using diagonal cov matrix, but the data is correlated, then it might be the case that model has hard time fitting it & it leads to bad results. Worth checking if swiching to unrestricted cov matrix fixes it. $\endgroup$
    – Tim
    Commented May 12, 2020 at 10:23
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    $\begingroup$ The diagonal matrix is a fixture of the model, not of the sampler. Once the model is chosen, the posterior distribution is a given and whether or not it fits the data well, it is still conceivable to analyse the convergence properties of a MCMC sampler for that target. I thus tend to abstain from mixing modelling with MCMC design, even though I acknowledge poorly fitting models make the target structure mode challenging. $\endgroup$
    – Xi'an
    Commented May 12, 2020 at 10:49
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    $\begingroup$ @Xi'an that was by the folk theorem of statistical computing $\endgroup$
    – Tim
    Commented May 12, 2020 at 15:31

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