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Monte-Carlo
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One Simple Way

  1. 1.
    start with a trial density, say
    g0(x)=tα(x;μ0,Σ0)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
    g1(x)=tα(x;μ1,Σ1)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
π(x)∝N(x;0,2I4)+2N(x;3e,I4)+1.5N(x;−3e,D4)\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
e=(1,1,1,1),I4=diag(1,1,1,1),D4=diag(2,1,1,.5)\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
    g0=tν(0,Σ)g_0 =t_{\nu}(0, \Sigma)
    ​
  2. 2.
    Recursively, build
    gk(x)=(1−ϵ)gk−1(x)+ϵtν(μ,Σ)g_k(\mathbf{x})=(1-\epsilon)g_{k-1}(\mathbf {x}) + \epsilon t_{\nu}(\mu, \Sigma)
    in which one chooses
    (ϵ,μ,Σ)(\epsilon, \mu, \Sigma)
     to minimize the variation of coefficient of the importance weights.
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