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Importance Sampling and Variance Reduction

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Importance Sampling and Variance Reduction

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Now · Importance Sampling

Importance Sampling and Variance Reduction

Importance Sampling

Estimate

E_f[h(X)]

by sampling from

g(x)

instead of

f(x)

\hat{\theta} = \frac{1}{n} \sum_{i=1}^n h(X_i) \frac{f(X_i)}{g(X_i)}

Control Variates

E[U] = \mu_U

known, use

Y_{CV} = Y - \beta(U - \mu_U)

. Optimal

\beta = \text{Cov}(Y, U) / \text{Var}(U)


python
import numpy as np
# Control variate: use known mean of U
u = np.random.uniform(0, 1, 1000)
y = np.exp(u)  # want to estimate integral of e^x
u_mean = 0.5
beta = np.cov(y, u)[0,1] / np.var(u)
y_cv = y - beta * (u - u_mean)
print(f"Var(Y): {np.var(y):.4f}, Var(Y_CV): {np.var(y_cv):.4f}")

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