## Monte Carlo Simulation

When the probability density function (PDF) is completely known, the statistical characteristics (mean, variance, etc) of distribution can be obtained in the form of integration

Monte Carlo integration. Monte Carlo simulation is used to approximate the integration by drawing a large number of random variables from .

Sampling via probability inverse transform. Let be a cumulative distribution function (cdf) on the real line . Then we can define the quantile function by

Algorithm.

- Generate an uniform random variable on .
- Return the value .

Rejection sampling methods. Sample first from a different distribution which satisfies

Algorithm.

- Generate a random variable from .
- Generate an uniform random variable on .
- Accept if ; otherwise, reject it.

Emergence of Markov chain Monte Carlo simulation. In reality the ``state'' space for is not . Either it is a subset of (in Bayesian applications), or it has a complex discrete structure (e.g., Ising model). For such models both algorithms via inverse probability transform and resampling method are not applicable in general. By way of Markov chain convergence theorem one can construct a Markov chain whose stationary distribution is .

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