Читать книгу Computational Statistics in Data Science - Группа авторов - Страница 118

The most common quantity of interest in Monte Carlo simulations is the expectation of a function of the target distribution. Let denote the Euclidean norm, and let , so that interest is in estimating

where we assume . If is identity, then the mean of the target is of interest. Alternatively, can be chosen so that moments or other quantities are of interest. A Monte Carlo estimator of is

For IID and MCMC sampling, the ergodic theorem implies that as . The Monte Carlo average is naturally unbiased as long as the samples are either IID or the Markov chain is stationary.