Martingale Posterior Cox Processes

Joshua Rooijakkers; Edwin Fong

stat.MLOct 17, 2026

Martingale Posterior Cox Processes

Authors: Joshua Rooijakkers, Edwin Fong

Organizations: Imperial College London · University of Hong Kong

Abstract

Cox processes provide a principled Bayesian nonparametric framework for modeling event-based data through an underlying intensity function. However, existing inference methods for estimating the intensity typically rely on computationally expensive Markov chain Monte Carlo sampling or approximate inference techniques. In addition, to ensure tractability, these methods often define priors over indirect latent structures, making them challenging to interpret. In this paper, we introduce the Martingale Posterior Cox Process (MPCP), which recasts Bayesian intensity estimation as inference over unobserved events given the observed event history. This perspective avoids the need to define complex priors while retaining the flexibility of nonparametric modeling. We propose a scalable, GPU-friendly copula-based predictive update, and provide strong theoretical guarantees for the convergence of the predictive resampling algorithm. Empirically, we show that MPCP provides excellent performance in both simulations and real data at a fraction of the computational cost of competing Bayesian approaches.

Figures & tables

Figure 1: Predictive view of Cox processes, where darker shades of blue indicate greater uncertainty. The posterior intensity given the observed data (bottom) decomposes into a density (left) and count (right) component. While uncertainty in the density decreases with the number of observed events, uncertainty in the count decreases with the number of repeated realizations of the process.

Figure 2: Frank copula predictive update. Left: copula kernel

k_{i}(x;x_{i})=c_{\phi}(u_{i-1},v_{i-1})\,p_{i-1}(x)

with

u_{i-1}=P_{i-1}(x)

and

v_{i-1}=P_{i-1}(x_{i})

. Right: updated predictive

p_{i}(x)

for

x_{i}=0.75

\alpha_{i}=0.5

and

\phi=1,5,10

( , , ).

Figure 3: Posterior mean intensity and corresponding 90% credible interval, together with the observed events (vertical ticks) generated from the true intensity function

\lambda_{1}

(dashed).

	$\lambda_{1}$ $(d=1,n\approx 46)$			$\lambda_{2}$ $(d=2,n\approx 118)$
	$L_{2}$	$L_{\textup{test}}$	Runtime (s)	$L_{2}$	$L_{\textup{test}}$	Runtime (s)
LGCP	$\bm{0.10\pm 0.01}$	$-34.3\pm 1.06$	$92.3\pm 0.84$	$\bm{0.11\pm 0.01}$	$-54.1\pm 1.82$	$138\pm 0.26$
DPMM	$0.28\pm 0.03$	$-46.7\pm 0.51$	$62.2\pm 1.91$	–	–	–
VLGCP	$0.12\pm 0.01$	$-22.5\pm 0.33$	$0.95\pm 0.03$	$0.14\pm 0.01$	$-38.2\pm 0.59$	$14.4\pm 0.59$
VBPP	$0.11\pm 0.01$	$-23.9\pm 0.36$	$2.36\pm 0.37$	$0.12\pm 0.01$	$-39.4\pm 0.54$	$19.0\pm 0.64$
MPCP	$0.13\pm 0.01$	$\bm{-22.1\pm 0.25}$	$\bm{0.49\pm 0.07}$	$0.17\pm 0.02$	$\bm{-38.0\pm 1.15}$	$\bm{4.01\pm 0.05}$

Table 1: Mean and standard error of squared error

L_{2}

, test log-likelihood

L_{\textup{test}}

and runtime across 50 seeds for synthetic intensity functions

\lambda_{1}

and

\lambda_{2}

. Methods are grouped into MCMC-based methods, approximate inference methods, and our proposed method, from top to bottom.

Figure 4: Coverage plots of the

q

-level credible intervals for

\gamma=\int_{\mathcal{X}}\lambda(x)\textup{d}x

Figure 5: Posterior mean intensity over time for earthquake data for MPCP, VBPP, and VLGCP (top to bottom) at four time points (left to right), with black points indicating the observed earthquake events. Lighter colors indicate higher mean intensity.

Figure 6: Posterior density of

\gamma=\int\lambda(x)\,\textup{d}x

, together with the true observed count

n

(dashed).

	bei		earthquake
	$L_{\textup{test}}$	Runtime (s)	$L_{\textup{test}}$	Runtime (s)
VLGCP	$-10958\pm 8.68$	$44.5\pm 2.40$	$-51.4\pm 26.4$	$300\pm 8.55$
VBPP	$-10929\pm 12.0$	$77.0\pm 4.63$	$-196\pm 13.7$	$911\pm 45.2$
MPCP	$\bm{-10778\pm 10.3}$	$\bm{21.9\pm 0.97}$	$\bm{1094\pm 64.3}$	$\bm{48.6\pm 8.76}$

Table 2: Mean and standard error of the test log-likelihood

L_{\textup{test}}

and runtime across 10 seeds for bei and earthquake datasets.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Comparison of predictive updates under the Frank and Gaussian copulas. For each copula, the left panel shows the copula kernel $k_{i}(x;x_{i}):=c(P_{i-1}(x),P_{i-1}(x_{i}))p_{i-1}(x)$ , while the right panel shows the resulting predictive density $p_{i}(x)$ , with $x_{i}=0.75$ and $\alpha_{i}=0.5$ .

Figure 8: Posterior mean intensity and corresponding 90% credible interval for MPCP with the Frank (left) and Gaussian (right) copula. The observed events (vertical ticks) are generated from the true intensity function

\lambda_{1}

(dashed).

Figure 9: Trajectories of

L_{N}

, as defined in ( 9 ), over

N-n

predictive resampling steps for different posterior samples in the synthetic data examples.

Figure 10: Density component of MPCP for

n=50

(left),

n=500

(middle), and

n=5,000

(right), together with the true density (dashed).

	$(x,x^{\prime})$	$(x^{\prime},x)$
$L_{2}$	$0.17\pm 0.02$	$0.16\pm 0.02$
$L_{\textup{test}}$	$-38.0\pm 1.15$	$-37.4\pm 1.06$
90% coverage	0.90	0.90
90% interval length	$35.4\pm 0.27$	$35.4\pm 0.27$

Appendix

Table 3: Mean and standard error across 50 seeds for

L_{2}

L_{\textup{test}}

, 90% coverage, and the corresponding 90% interval length, under different orderings of the covariate dimensions.

Figure 11: Posterior mean (top) and variance (bottom) inferred from the observed events (black dots), with the true intensity function

\lambda_{2}

shown in the top-right panel. Lighter colors indicate higher mean and variance.

Figure 12: Lengths of the posterior credible intervals for

\gamma=\int_{\mathcal{X}}\lambda(x)\textup{d}x

corresponding to level

q

Figure 13: Posterior mean (top) and variance (bottom) of MPCP, VBPP and VLGCP (left to right) on the bei dataset. Lighter colors indicate higher mean and variance.

Figure 14: Posterior variance of the intensity over time for earthquake data for MPCP, VBPP, and VLGCP (top to bottom) at four time points (left to right), with black points indicating the observed earthquake events. Lighter colors indicate larger variance.

Figure 15: Mean and standard error of

L_{\textup{test}}

for VBPP and VLGCP as the number of inducing points/bins increases, compared with MPCP, for the bei (left) and two-dimensional earthquake (right) datasets.

Temporal Point Processes

Temporal point processes model the timing, types, and dependencies of events occurring irregularly over time.

See all 33 papers

Feb 27, 2026

Neural Diffusion Intensity Models for Point Process Data

Cox processes model overdispersed point process data via a latent stochastic intensity, but both nonparametric estimation of the intensity model and posterior inference over intensity paths are typically intractable, relying on expensive MCMC methods. We introduce Neural Diffusion Intensity Models, a variational framework for Cox processes driven by neural SDEs. Our key theoretical result, based on enlargement of filtrations, shows that conditioning on point process observations preserves the diffusion structure of the latent intensity with an explicit drift correction. This guarantees the variational family contains the true posterior, so that ELBO maximization coincides with maximum likelihood estimation under sufficient model capacity. We design an amortized encoder architecture that maps variable-length event sequences to posterior intensity paths by simulating the drift-corrected SDE, replacing repeated MCMC runs with a single forward pass. Experiments on synthetic and real-world data demonstrate accurate recovery of latent intensity dynamics and posterior paths, with orders-of-magnitude speedups over MCMC-based methods.

Xinlong Du, Harsha Honnappa, Vinayak Rao

Edwardson School of Industrial Engineering, Purdue University · Department of Statistics, Purdue University

Temporal Point Processes Variational Inference Stochastic Differential Equations

Ask PDF

Oct 12, 2026

Amortized Variational Inference of Latent Intensities from Replicated Point Patterns

Bayesian inference for latent point process intensities often relies on per-dataset sampling or optimization, making repeated inference computationally expensive. We propose a neural-network-based amortized variational framework that learns a reusable inference rule mapping a variable number of replicated point patterns to a distribution over their shared latent intensity. Trained from event times using the point process likelihood, the model learns an amortized inference rule without ground-truth intensity supervision. At test time, it performs inference of the latent distribution in a forward pass. On synthetic data, additional realizations improve intensity recovery and yield narrower credible intervals with broadly stable coverage. Our method is tested on bicycle-sharing data and achieves the highest mean held-out log likelihood among the evaluated methods while requiring substantially less inference time than the Log-Gaussian Cox Process baseline.

Wenqing Liu, Xenia Miscouridou, Déborah Sulem

Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland · Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus

Amortized Inference Variational Inference Temporal Point Processes

Ask PDF

Oct 6, 2026

Generalized Matheron Variational Implicit Processes

Implicit-process priors specify distributions over functions through sample-forward mechanisms such as Bayesian neural networks and stochastic simulators, but their function-space densities are typically unavailable. We introduce Generalized Matheron Variational Implicit Processes (GMVIP), a pathwise variational family for posterior inference with such priors. For Gaussian-process priors, GMVIP recovers the standard inducing-variable variational GP construction; for general implicit priors, its empirical covariance construction preserves the prior mean and covariance in the population limit. GMVIP constructs posterior samples by drawing a function from the prior and applying a correction anchored at a set of inducing inputs. The effect of this correction away from the inducing inputs is determined directly from prior samples, allowing the posterior to retain the structure and variability of the original implicit process. The (surrogate) prior and variational posterior use the same pathwise construction and differ only in the distribution of whitened inducing coefficients, yielding a tractable coefficient-space Kullback-Leibler divergence. Experiments on regression, classification, and forecasting with simulator-defined and retrieval-conditioned empirical trajectory priors show that GMVIP is broadly competitive with existing methods.

Luis A. Ortega, Andrés R. Masegosa, Thomas D. Nielsen

Aalborg University

Bayesian Inference Gaussian Processes Variational Inference

Ask PDF

Abstract

Temporal Point Processes

Explore similar work

Neural Diffusion Intensity Models for Point Process Data

Amortized Variational Inference of Latent Intensities from Replicated Point Patterns

Generalized Matheron Variational Implicit Processes