title: ## Contents
style: nestedOrderedList # TOC style (nestedList|nestedOrderedList|inlineFirstLevel)
minLevel: 0 # Include headings from the specified level
maxLevel: 0 # Include headings up to the specified level
includeLinks: true # Make headings clickable
hideWhenEmpty: false # Hide TOC if no headings are found
debugInConsole: false # Print debug info in Obsidian console1 Monte-Carlo Variance Reduction
In the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results.
Suppose one wants to compute \(z:=\mathbb{E}(Z)\) with the random variable \(Z\) defined on the probability space \((Ω,\mathcal{F},\mathbb{P})\). Monte Carlo does this by sampling i.i.d. copies \(Z_1,...,Z_n\) of \(Z\) and then to estimate \(z\) via the sample-mean estimator \[ \bar{ z }=\frac{1}{n}\sum_{i=1}^{n}Z_{i}. \] Under further mild conditions such as \(\text{Var}(Z)< \infty\), a central limit theorem will apply such that for large \(n→∞\), the distribution of \(\bar{ z}\) converges to a normal distribution with mean \(z\) and standard error \(\frac{σ}{n}\). Because the standard deviation only converges towards 0 at the rate \(\sqrt{ n }\), implying one needs to increase the number of simulations \(n\) by a factor of 4 to halve the standard deviation of \(\bar{z}\), variance reduction methods are often useful for obtaining more precise estimates for \(z\) without needing very large numbers of simulations.
In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main variance reduction methods are:
- Common Random Numbers
- [[antithetic-variates]]
- Control Variates
- Importance Sampling
- Stratified Sampling
- Moment Matching
- Conditional Monte Carlo
- Quasi-Random-Variables (in Quasi-Monte Carlo method)
2 Antithetic Variates
[[antithetic-variates]]