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
- Metropolis Hestings
- Metropolis-adjusted Langevin algorithm (MALA)
- Gibbs sampling
- 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?
