A ranked linear assignment approach to Bayesian classification

Abstract

We show how a ranked linear assignment algorithm can be used to solve the problem of accurately matching the n components of a random data sample vector to n data classes. It is assumed that each component of the random data sample vector corresponds to exactly one data class and, conversely, that each data class corresponds to exactly one data sample vector component, and that the sample data vector components are always arranged in the same but unknown order. That is, we assume that the mth sample random vector, X m =[ x 1 m , x 2 m , x 3 m ,…, x n m ] T, of sample-to-sample independent normally distributed random vector components has the property: x i m∈N(μ ϕ(i),σ ϕ(i) 2), μ i≠μ j if i≠j, i,j=1,2,…,n for some fixed but unknown permutation ϕ in the symmetric group S n on the letters [1,2,3,…, n], m=1,2,3,…, M,… We derive the averaged permutation matrix representation over S n , given the mean of M data vector samples. This matrix turns out to be a doubly stochastic matrix, P M , whose ( i, j) element is the probability that data stream component i corresponds to class j, given the sample mean vector X M . We prove that P M converges to ϕ in probability. Moreover, it is shown that P M can be accurately approximated by extending from the solution of the classical linear assignment problem of finding the permutation with maximum likelihood to the solution of the ranked linear assignment problem in which a small number of permutations with rapidly decreasing likelihoods are identified.