-step given the current estimate tht :[ut ]compute the expectations nt :ez and zt : ( ut ) where (ut )conditional on cthat isphs ( ut nt :ut ph(( ut )/st phs ( ut zt :st ( ut ph(( ut )/st -step update the estimate to tht+ :[ut+ + ]via the formulasp nc nt :xi < xi ut+ np :xi < (xi ut+ + empirical distribution and density estimation bn obtained from an iid training set in section we saw how the empirical cdf { xn from an unknown distribution on rgives an estimate of the unknown cdf bn is genuine cdfas it is right-continuousf of this sampling distribution the function increasingand lies between and the corresponding discrete probability distribution is called the empirical distribution of the data random variable distributed according to this empirical distribution takes the values xn with equal probability / the concept of empirical distribution naturally generalizes to higher dimensionsa random vector that is distributed according to the empirical distribution of xn has discrete pdf [ xi /ni sampling from such distribution -in other words resampling the original data -was discussed in section the preeminent usage of such sampling is the bootstrap methoddiscussed in section in waythe empirical distribution is the natural answer to the unsupervised learning questionwhat is the underlying probability distribution of the datahoweverthe empirical distribution isby definitiona discrete distributionwhereas the true sampling distribution might be continuous for continuous data it makes sense to also consider estimation of the pdf of the data common approach is to estimate the density via kernel density estimate (kde)the most prevalent learner to carry this out is given next empirical distribution definition gaussian kde let xn rd be the outcomes of an iid sample from continuous pdf gaussian kernel density estimate of is mixture of normal pdfsof the form kx-xi gtn ( se = ( ) / sd where is called the bandwidth rd ( gaussian kernel density estimate