# 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.