An Efficient Algorithm for Bayesian Nearest Neighbours

TL;DR

This paper introduces a fast, exact Bayesian algorithm for determining the optimal number of neighbors in k-NN classification and regression, eliminating the need for simulation and significantly reducing computation time.

Contribution

It presents a novel Bayesian method that computes the posterior distribution of k efficiently without MCMC, using change-point detection in the data.

Findings

Favorable results on UCI datasets

Exact posterior computation without simulation

Significant reduction in computational time

Abstract

K-Nearest Neighbours (k-NN) is a popular classification and regression algorithm, yet one of its main limitations is the difficulty in choosing the number of neighbours. We present a Bayesian algorithm to compute the posterior probability distribution for k given a target point within a data-set, efficiently and without the use of Markov Chain Monte Carlo (MCMC) methods or simulation - alongside an exact solution for distributions within the exponential family. The central idea is that data points around our target are generated by the same probability distribution, extending outwards over the appropriate, though unknown, number of neighbours. Once the data is projected onto a distance metric of choice, we can transform the choice of k into a change-point detection problem, for which there is an efficient solution: we recursively compute the probability of the last change-point as we move towards our target, and thus de facto compute the posterior probability distribution over k. Applying this approach to both a classification and a regression UCI data-sets, we compare favourably and, most importantly, by removing the need for simulation, we are able to compute the posterior probability of k exactly and rapidly. As an example, the computational time for the Ripley data-set is a few milliseconds compared to a few hours when using a MCMC approach.