Notes 18 : Chinese Restaurant Process
نویسنده
چکیده
We begin with a proof of the ESF. This is essentially Kingman’s proof (understanding ESF was the main motivation for his introduction of the coalescent). Proof: Let Π be the partition generated by the infinite-alleles model on n sample. We call each set in Π a cluster. Assume there are k clusters. Looking backwards in time, to obtain Π it must be that each cluster undergoes a sequence of coalescences followed by a single mutation: more precisely, a cluster of size λi goes through λi − 1 coalescences then a single mutation. Once this happens, we say that the cluster has merged. Moreover, no coalescence event can occur between two unmerged clusters—we call such an event invalid. We now compute the probability that these events occur. The probability that a coalescence occurs within a cluster with j lineages remaining when the total number of lineages remaining is i is given by
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