I have a serious model fitting issue, and wanted to pick the brains of the forum members here.

My model involves multivariate (d \ge 3) latent variables which contribute to the likelihood. Each of the components have identical marginal distributions, but have a *very* complicated dependence structure. I have figured the proper multivariate *_lpdf functions for d=3, but higher values are intractable.

However, it turns out I can easily (d \le\ \sim 20) find the expressions for the telescoping product of conditional densities, but of the *order statistics* of my original variables. That is, these functions:

f(x_{(1)})\_\mathrm{lpdf},

f(x_{(2)}|x_{(1)})\_\mathrm{lpdf},

f(x_{(3)}|x_{(1)},x_{(2)})\_\mathrm{lpdf},

and so on. I then wrote these into the likelihood, then performed a random permutation of the resulting order statistics to propagate the latent variable into my model calculations. Here however, the convergence of my model failed dramatically (n_{\mathit{eff}} < 10) and I could not improve it.

I suppose this is due to the discontinuity of the gradient of the permuting process, though am not sure. Is there any workaround available for multivariate situations like this, where the raw unordered variables are not tractable but the order statistics are, and which can successfully exploit them?

Thanks in advance,

JZ