Bayesian Poisson regression for crowd counting

TLDR

Poisson regression models the noisy output of a counting function as a Poisson random variable, with a log‑mean parameter that is a linear function of the input vector. The study analyzes Poisson regression in a Bayesian framework by placing a prior on the linear function weights. The authors introduce a prior on the linear weights, derive a closed‑form approximation to the predictive distribution, compute an approximate marginal likelihood for kernel hyperparameter learning, and relate the model to Gaussian processes. The kernelized predictive distribution enables non‑linear log‑mean functions, and experiments demonstrate its effectiveness for crowd counting from low‑level features.

Abstract

Poisson regression models the noisy output of a counting function as a Poisson random variable, with a log-mean parameter that is a linear function of the input vector. In this work, we analyze Poisson regression in a Bayesian setting, by introducing a prior distribution on the weights of the linear function. Since exact inference is analytically unobtainable, we derive a closed-form approximation to the predictive distribution of the model. We show that the predictive distribution can be kernelized, enabling the representation of non-linear log-mean functions. We also derive an approximate marginal likelihood that can be optimized to learn the hyperparameters of the kernel. We then relate the proposed approximate Bayesian Poisson regression to Gaussian processes. Finally, we present experimental results using Bayesian Poisson regression for crowd counting from low-level features.