AIS

# One Simple Way

1. 1.
start with a trial density, say
$g_0(x) = t_\alpha (x; \mu_0,\Sigma_0)$
2. 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. 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. 1.
start with a trial density
$g_0 =t_{\nu}(0, \Sigma)$
2. 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.
Last modified 3yr ago
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