Robust Inference of Trees

TL;DR

This paper introduces a reliable method for inferring the structure of dependency trees using imprecise Dirichlet models, providing robust results even with scarce data and offering credible bounds for mutual information.

Contribution

It develops an exact algorithm for inferring common substructures of plausible trees under the imprecise Dirichlet model, enhancing reliability over traditional methods.

Findings

The method reliably infers tree edges with limited data.

The algorithm operates in O(m^4) time, where m is the number of variables.

Credibility limits for mutual information are provided under the model.

Abstract

This paper is concerned with the reliable inference of optimal tree-approximations to the dependency structure of an unknown distribution generating data. The traditional approach to the problem measures the dependency strength between random variables by the index called mutual information. In this paper reliability is achieved by Walley's imprecise Dirichlet model, which generalizes Bayesian learning with Dirichlet priors. Adopting the imprecise Dirichlet model results in posterior interval expectation for mutual information, and in a set of plausible trees consistent with the data. Reliable inference about the actual tree is achieved by focusing on the substructure common to all the plausible trees. We develop an exact algorithm that infers the substructure in time O(m^4), m being the number of random variables. The new algorithm is applied to a set of data sampled from a known distribution. The method is shown to reliably infer edges of the actual tree even when the data are very scarce, unlike the traditional approach. Finally, we provide lower and upper credibility limits for mutual information under the imprecise Dirichlet model. These enable the previous developments to be extended to a full inferential method for trees.