AIS

One Simple Way

1. start with a trial density, say $g_0(x) = t_\alpha (x; \mu_0,\Sigma_0)$
2. with weighted Monte Carlo samples, estimate the parameters, mean and covariate matrix, and construct new trial density, say $g_1(x) = t_\alpha (x; \mu_1,\Sigma_1)$
3. construct a certain measure of discrepancy between the trial distribution and the target distribution, such as the coefficient of variation of importance weights, does not improve any more.

One Example

Implement an Adaptive Importance Sampling algorithm to evaluate mean and variance of a density

$$\pi(\mathbf{x})\propto N(\mathbf{x;0}, 2I_4) + 2N(\mathbf{x; 3e}, I_4) + 1.5 N(\mathbf{x; -3e}, D_4)$$

where $\mathbf{e} = (1,1,1,1), I_4 = diag(1,1,1,1), D_4 = diag(2,1,1,.5)$.

A possible procedure is as follows:

1. start with a trial density $g_0 =t_{\nu}(0, \Sigma)$
2. Recursively, build

   $$g_k(\mathbf{x})=(1-\epsilon)g_{k-1}(\mathbf {x}) + \epsilon t_{\nu}(\mu, \Sigma)$$

   in which one chooses $(\epsilon, \mu, \Sigma)$ to minimize the variation of coefficient of the importance weights.