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I have the following distribution. $$\operatorname{logpdf}(x) = -\sum_{i = 0}^n \max(v_i \cdot x + b_i, 0) -1/2 (x - \mu)^T M (x - \mu) + \text{const}$$ Where $M$ is symmetric positive definite, $x \in \mathbb{R}^m$, with $m$ around $500$ and $n$ around $100,000$. I want to calculate the mean and the variance in less than 1 second. I understand this is quite difficult, so I would like to find the best approximation.

I have tried distributed MCMC methods

  1. Metropolis Hestings
  2. Metropolis-adjusted Langevin algorithm (MALA)
  3. Gibbs sampling
  4. Importance sampling (using multivariate normal distribution suggested by the quadratic term).

MALA seems to perform the best, but it is still not good enough. I also tried to approximate the distribution with a normal distribution using variational bayesian methods, but it seems to have too much bias.

I have run out of things to try. Can I get suggestions on different sampling methods or approximations?

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  • $\begingroup$ Also, is operations research the right forum for this question? $\endgroup$
    – JEK
    Aug 16, 2022 at 0:57
  • $\begingroup$ stats.stackexchange.com is a site which might be better. $\endgroup$ Aug 16, 2022 at 17:30

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