(cvalidate the model bayesian statistics bayesian statisticsin this approach we average over all possible values of the parameter th using user-specified weight function (thand obtain the model (thg(thdth for practical computationsthis means that we can treat th as random variable with pdf (thbayesformula (th tg( thg(this used to learn th based on the observed data example (iid samplethe most fundamental statistical model is where the data xn is such that the random variables xn are assumed to be independent and identically distributediid xn distrandom sample according to some known or unknown distribution dist an iid sample is often called random sample in the statistics literature note that the word "samplecan refer to both collection of random variables and to single random variable it should be clear from the context which meaning is being used often our guess or model for the true distribution is specified up to an unknown parameter thwith th th the most common model isiid xn (us )in which case th (us and th rc estimator estimate bias mean squared error efficient estimation suppose the model (thfor the data is completely specified up to an unknown parameter vector th the aim is to estimate th on the basis of the observed data only (an alternative goal could be to estimate ps(thfor some vector-valued function psspecificallythe goal is to find an estimator ( that is close to the unknown th the corresponding outcome (tis the estimate of th the bias of an estimator of th is defined as et th an estimator of th is said to be unbiased if eth th we often write th for both an estimator and estimate of th the mean squared error (mseof real-valued estimator is defined as mse eth ( th) an estimator is said to be more efficient than an estimator if the mse of is smaller than the mse of the mse can be written as the sum mse (eth th) varth relative time variance products the first term measures the unbiasedness and the second is the variance of the estimator in particularfor an unbiased estimator the mse of an estimator is simply equal to its variance for simulation purposes it is often important to include the running time of the estimator in efficiency comparisons one way to compare two unbiased estimators and is to compare their relative time variance productsri var ( ) (