Shrinkage Priors for Nonparametric Bayesian Prediction of Nonhomogeneous Poisson Processes

Abstract

We consider nonparametric Bayesian estimation and prediction for nonhomogeneous Poisson process models with unknown intensity functions. We propose a class of improper priors for intensity functions. Nonparametric Bayesian inference with kernel mixture based on the class improper priors is shown to be useful, although improper priors have not been widely used for nonparametric Bayes problems. Several theorems corresponding to those for finite-dimensional independent Poisson models hold for nonhomogeneous Poisson process models with infinite-dimensional parameter spaces. Bayesian estimation and prediction based on the improper priors are shown to be admissible under the Kullback-Leibler loss. Numerical methods for Bayesian inference based on the priors are investigated.

A semiparametric Bayesian model for queueing arrival processes: An application to call centers

“…Gamma process priors in nonparametric Bayesian analysis of NHPPs have been considered by Sinha (1993) and Kuo and Ghosh (2001). Kuo and Ghosh (2001) assume a gamma process for the cumulative intensity function of an NHPP and show that the posterior distribution for the cumulative intensity function is also obtained as a gamma process if all event times of the NHPP are observed.…”

“…Gamma process priors in nonparametric Bayesian analysis of NHPPs have been considered by Sinha (1993) and Kuo and Ghosh (2001). Kuo and Ghosh (2001) assume a gamma process for the cumulative intensity function of an NHPP and show that the posterior distribution for the cumulative intensity function is also obtained as a gamma process if all event times of the NHPP are observed. More specifically, if we observe n arrival times during the time interval (0, t ] then the posterior of $$\Lambda _0(t)$$ will be given by the gamma distribution $$$$\begin{equation*} \nonumber (\Lambda _0(t)|D_{0},n)\backsim G(M(t)+n,c+1), \end{equation*}$$$$using the gamma process prior in ().…”

Determinants of apartment price volatility in Vietnam: a comparison between Hanoi and Ho Chi Minh City

“…The works in Avanzi et al (2021); Jahani et al (2021) also have similar weakness. Other nonparametric estimators of the rate function were introduced in Kuhl and Bhairgond (2000); Komaki (2021). In Kuhl and Bhairgond (2000), the authors proposed a nonparametric estimator using wavelets and adapted the nonnegative wavelet estimation of general density functions proposed in Walter and Shen (1998) to estimate the rate function.…”

“…However, a drawback is that they require more constraints on the rate function than needed to achieve nonnegativity. As a result, there are several limitations of their procedure in constrained convex nonlinear programming problems (Komaki, 2021). Most of parametric estimators proposed in the literature have a fixed number of parameters, so they do not suffer the storage problems when the number of observed realizations (data) increases.…”

“…Stochastic processes allow us to model uncertain quantities that evolve over time. One of the most fundamental and important of such processes is the Poisson process 1 arXiv:2402.12808v1 [cs.LG] 20 Feb 2024 (Komaki, 2021;Jahani et al, 2021;Moore and Davenport, 2019;Alizadeh and Papp, 2013) as it is appropriate for modelling a large number of completely random phenomena. Therefore, the Poisson process has been widely used in various real-world applications such as taxi routing (Son et al, 2022), queueing systems (Brown et al, 2005), and e-mail arrival counting (Alizadeh et al, 2008;Alizadeh and Papp, 2013).…”