Fast and scalable non-parametric Bayesian inference for Poisson point processes

Gugushvili, Shota; van der Meulen, Frank; Schauer, Moritz; Spreij, Peter

doi:10.48550/arxiv.1804.03616

Cited by 5 publications

(10 citation statements)

References 33 publications

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“…The posterior µ and φ are estimated using the resulting cluster processes. Our framework is close to the stochastic Expectation-Maximization (EM) algorithm [Celeux and Diebolt, 1985] where posterior µ and φ are estimated [Lloyd et al, 2015;Walder and Bishop, 2017;Gugushvili et al, 2018] in the Mstep and random samples of µ and φ are drawn. We adapt the approach of the recent non-parametric Bayesian estimation for Poisson process intensities, termed Laplace Bayesian Poisson process (LBPP) [Walder and Bishop, 2017], to estimate the posterior φ given the sampled branching structure.…”

Section: Hawkes Processmentioning

confidence: 99%

Efficient Non-parametric Bayesian Hawkes Processes

Zhang

Walder

Rizoiu

et al. 2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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In this paper, we develop an efficient nonparametric Bayesian estimation of the kernel function of Hawkes processes. The non-parametric Bayesian approach is important because it provides flexible Hawkes kernels and quantifies their uncertainty. Our method is based on the cluster representation of Hawkes processes. Utilizing the stationarity of the Hawkes process, we efficiently sample random branching structures and thus, we split the Hawkes process into clusters of Poisson processes. We derive two algorithms -a block Gibbs sampler and a maximum a posteriori estimator based on expectation maximization -and we show that our methods have a linear time complexity, both theoretically and empirically. On synthetic data, we show our methods to be able to infer flexible Hawkes triggering kernels. On two largescale Twitter diffusion datasets, we show that our methods outperform the current state-of-the-art in goodness-of-fit and that the time complexity is linear in the size of the dataset. We also observe that on diffusions related to online videos, the learned kernels reflect the perceived longevity for different content types such as music or pets videos.

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Section: Hawkes Processmentioning

confidence: 99%

Efficient Non-parametric Bayesian Hawkes Processes

Zhang

Walder

Rizoiu

et al. 2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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“…Crucially, by following methods proposed in Ref. [17] we avoid the computationally intensive task of computing the explicit posterior distribution, and instead use a Gibbs sampler [18] to efficiently obtain sample trajectories. This allows not only to find the most likely trajectory, but also to estimate uncertainty (e.g.…”

Section: Heart Rate (Inferred)mentioning

confidence: 99%

“…Here we specify a generative model for heart rate dynamics, which follows the approach introduced in Ref. [17] for general point processes.…”

Section: Heart Rate Dynamics Via Gamma Markov Chainsmentioning

confidence: 99%

Bayesian at heart: Towards autonomic outflow estimation via generative state-space modelling of heart rate dynamics

Rosas¹,

Candia-Rivera²,

Luppi³

et al. 2023

Preprint

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Recent research is revealing how cognitive processes are supported by a complex interplay between the brain and the rest of the body, which can be investigated by the analysis of physiological features such as breathing rhythms, heart rate, and skin conductance. Heart rate dynamics are of particular interest as they provide a way to track the sympathetic and parasympathetic outflow from the autonomic nervous system, which is known to play a key role in modulating attention, memory, decision-making, and emotional processing. However, extracting useful information from heartbeats about the autonomic outflow is still challenging due to the noisy estimates that result from standard signal-processing methods. To advance this state of affairs, we propose a paradigm shift in how we conceptualise and model heart rate: instead of being a mere summary of the observed inter-beat intervals, we introduce a modelling framework that views heart rate as a hidden stochastic process that drives the observed heartbeats. Moreover, by leveraging the rich literature of state-space modelling and Bayesian inference, our proposed framework delivers a description of heart rate dynamics that is not a point estimate but a posterior distribution of a generative model. We illustrate the capabilities of our method by showing that it recapitulates linear properties of conventional heart rate estimators, while exhibiting a better discriminative power for metrics of dynamical complexity compared across different physiological states.

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“…We note that the case of i.i.d. data as considered in (Kirichenko and Van Zanten, 2015) and (Gugushvili et al, 2018) is a special case of this model, where γ j (x) = 1, ∀x ∈ [0, 1] d , ∀j = 1, . .…”

Section: Likelihoodmentioning

confidence: 99%

Posterior Contraction Rates for Gaussian Cox Processes with Non-identically Distributed Data

Grant,

Leslie

2019

Preprint

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This paper considers the posterior contraction of non-parametric Bayesian inference on nonhomogeneous Poisson processes. We consider the quality of inference on a rate function λ, given non-identically distributed realisations, whose rates are transformations of λ. Such data arises frequently in practice due, for instance, to the challenges of making observations with limited resources or the effects of weather on detectability of events. We derive contraction rates for the posterior estimates arising from the Sigmoidal Gaussian Cox Process and Quadratic Gaussian Cox Process models. These are popular models where λ is modelled as a logistic and quadratic transformation of a Gaussian Process respectively. Our work extends beyond existing analyses in several regards. Firstly, we consider non-identically distributed data, previously unstudied in the Poisson process setting. Secondly, we consider the Quadratic Gaussian Cox Process model, of which there was previously little theoretical understanding. Thirdly, we provide rates on the shrinkage of both the width of balls around the true λ in which the posterior mass is concentrated and on the shrinkage of posterior mass outside these balls -usually only the former is explicitly given. Finally, our results hold for certain finite numbers of observations, rather than only asymptotically, and we relate particular choices of hyperparameter/prior to these results.

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Fast and scalable non-parametric Bayesian inference for Poisson point processes

Cited by 5 publications

References 33 publications

Efficient Non-parametric Bayesian Hawkes Processes

Efficient Non-parametric Bayesian Hawkes Processes

Bayesian at heart: Towards autonomic outflow estimation via generative state-space modelling of heart rate dynamics

Posterior Contraction Rates for Gaussian Cox Processes with Non-identically Distributed Data

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