27 lines
609 B
Markdown
27 lines
609 B
Markdown
# Variance and Convergence
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## Definition of Variance
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$$\text{Var}(X) = E\{(x - \mu)^2\}$$
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## Variance of the Monte Carlo Estimator
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$$\text{Var}(\hat{I}) = \frac{(b-a)^2}{n} \text{Var}(f(x))$$
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### Convergence Rate
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Standard deviation scales as:
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$$\sigma \sim \frac{1}{\sqrt{n}}$$
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This is good for a small number of dimensions.
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**However:** With higher dimensionality, variance gets exponentially worse (curse of dimensionality).
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---
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## Multi-Dimensional Case
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Combine with Monte Carlo Integration techniques (importance sampling, stratified sampling, etc.) to manage variance in high dimensions.
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