Monte Carlo Inference - STAT - EPFL

Galaxy dataSpeed10 15 20 25 30 35Normal Q−Q Plot−2 −1 0 1 2Theoretical QuantilesMonte Carlo Inference Spring 2009 – slide 196Mixture density□ Natural model for such data is a p-component mixture densityf(y;θ) =p∑π r f r (y;θ), 0 ≤ π r ≤ 1,r=1p∑π r = 1,where π r is the probability that Y comes from the rth component and f r (y;θ) is its densityconditional on this event.□ Widely used class of models, often with number of components p unknown.□ Aside: such models are non-regular for some likelihood inferences:– non-identifiable under permutation of components;– setting π r = 0 eliminates parameters of f r ;– maximum of likelihood can be +∞, achieved for several θMonte Carlo Inference Spring 2009 – slide 197r=1197