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Register a new userConditional formulae for Gibbs-type exchangeable random partitions
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Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.
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Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.
Suppose $Pi$ is an exchangeable random partition of the positive integers and $Pi_n$ is its restriction to ${1, ..., n}$. Let $K_n$ denote the number of blocks of $Pi_n$, and let $K_{n,r}$ denote the number of blocks of $Pi_n$ containing $r$ integers. We show that if $0 < alpha < 1$ and $K_n/(n^{alpha} ell(n))$ converges in probability to $Gamma(1-alpha)$, where $ell$ is a slowly varying function, then $K_{n,r}/(n^{alpha} ell(n))$ converges in probability to $alpha Gamma(r - alpha)/r!$. This result was previously known when the convergence of $K_n/(n^{alpha} ell(n))$ holds almost surely, but the result under the hypothesis of convergence in probability has significant implications for coalescent theory. We also show that a related conjecture for the case when $K_n$ grows only slightly slower than $n$ fails to be true.
We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as $n$ tends to infinity while $m$ is fixed or tends to infinity. In particular, if $m$ goes to infinity not fast enough, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We derive the asymptotic distributions of the parts jointly and that of the largest part by taking limit of $n$ and $m$ in the same manner as that in the first probability measure.
Kallenberg (2005) provided a necessary and sufficient condition for the local finiteness of a jointly exchangeable random measure on $R_+^2$. Here we note an additional condition that was missing in Kallenbergs theorem, but was implicitly used in the proof. We also provide a counter-example when the additional condition does not hold.
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