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I'm fitting a Bayesian HLM in JAGS using k-fold cross-validation (k=5). I'd like to know whether estimates of parameter $\beta$ are stable across all folds. What's the best way to do this?

One idea is to find the differences of the posteriors of $\beta$ and to see if 0 is in the 95% CI of the difference. In other words, is 0 in the 95% interval of $\beta_{k=1}-\beta_{k=2}$ (and then repeat for all pairs of folds).

Another idea is to treat the posteriors from each fold as different MCMC chains, and to compute Gelman's $\hat{R}$ (Potential Scale Reduction Factor) across these pseudo-chains.

Is one of these preferable, and are there alternatives?

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    $\begingroup$ Feels strange to see if zero is among the credible differences as surely you expect there to be some difference between the folds. One suggestion would be to calculate point estimates of $\beta$ for each fold and look at the spread of these. $\endgroup$ – Rasmus Bååth Feb 24 '14 at 16:28
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    $\begingroup$ Just a general comment about cross-validation and Bayesian stuff: Why not just calculate WAIC? It's asymptotically equivalent to LOOCV, and you can still use all of your data. $\endgroup$ – Brash Equilibrium May 30 '14 at 5:34
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    $\begingroup$ How would you generate posterior simulations of $\beta_{k=1}-\beta_{k=2}$ ? $\endgroup$ – Stéphane Laurent Aug 7 '14 at 11:03
  • $\begingroup$ In our tests in my former factory we had to prove that 0% yield loss was in the 95% CI. Questions of adequate, independent samples, and the nature of the binomial test dominated. Can you give an idea of what your sample sizes are? $\endgroup$ – EngrStudent Jan 13 '15 at 13:41
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I don't know if this qualifies as a comment or as an answer. I'm putting here because it feels like an answer.

In k-fold cross-validation you are partitioning your data into k groups. If you are covering even the "basics" then you are uniformly randomly selecting members for each of the k bins.

When I speak of data, I think of each row as a sample, and each column as a dimension. I'm used to using various methods to determine variable importance, column importance.

What if you, as a thought exercise, departed from the "textbook" uniform random, and determined which rows were important? Maybe they inform a single variable at a time, but maybe they inform more. Are there some rows that are less important than others? Maybe many of the points are informative, maybe few are.

Knowing the importance of the variable, perhaps you could bin them by importance. Maybe you could make a single bin with the most important samples. This could define the size of your "k". In this way, you would be determining the "most informative" kth bucket and comparing it against others, and against the least informative bucket.

This could give you an idea of the maximal variation of your model parameters. It is only one form.

A second way of splitting the kth buckets is by the magnitude and the direction of the influence. So you could put samples that sway a parameter or parameters in one direction into one bucket and put samples that sway the same parameter or parameters in the opposite direction into a different bucket.

The parameter variation in this form might give a wider sweep to the variables, based not on information density, but on information breed.

Best of luck.

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It may not be a complete answer, but if 0 is NOT in the 95% CI for several differences it is quite safe to say that they are not identical at a 0.05 level.

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