the left panel of figure shows the (us points generated by the gibbs sampler also shownvia the white circleis the point (xs )where is the sample mean and the sample variance this posterior point cloud visualizes the considerable uncertainty in the estimates by projecting the (us points onto the -axis -that isby ignoring the values -one obtains (approximatesamples from the posterior pdf of uthat isf ( xthe right panel of figure shows kernel density estimate (see section of this pdf the corresponding and sample quantiles were found to be - and respectivelygiving the credible interval (- for uwhich contains the true expectation similarlyan estimated credible interval for is ( )which contains the true variance monte carlo estimation in this section we describe how monte carlo simulation can be used to estimate complicated integralsprobabilitiesand expectations number of variance reduction techniques are introduced as wellincluding the recent cross-entropy method crude monte carlo the most common setting for monte carlo estimation is the followingsuppose we wish to compute the expectation ey of some (say continuousrandom variable with pdf but the integral ey (ydy is difficult to evaluate for exampleif is complicated function of other random variablesit would be difficult to obtain an exact expression for (ythe idea of crude monte carlo -sometimes abbreviated as cmc -is to approximate by simulating many independent copies yn of and then take their sample mean as an estimator of all that is needed is an algorithm to simulate such copies by the law of large numbersy converges to as provided the expectation of exists moreoverby the central limit theoremy approximately has (us /ndistribution for large nprovided that the variance vary this enables the construction of an approximate ( aconfidence interval for us - / - / crude monte carlo confidence interval ( where is the sample standard deviation of the {yi and zg denotes the -quantile of the ( distributionsee also section instead of specifying the confidence intervalone often reports only the sample mean and the estimated standard errors nor the estimated relative errors /( nthe basic estimation procedure for independent data is summarized in algorithm below it is often the case that the output is function of some underlying random vector or stochastic processthat isy ( )where is real-valued function and is random vector or process the beauty of monte carlo for estimation is that ( holds regardless of the dimension of estimated standard error estimated relative error