stat.MLOct 8, 2026

ISBO: Scalable Spatio-Temporal Bayesian Optimization with Log Gaussian Cox Process Models via the INLA-SPDE Approach

Authors: Kaichuang Yang, Håvard Rue, Jakob Zeitler

Organizations: KAUST · University of Oxford

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

Bayesian Optimization (BO) is a popular method for efficiently optimizing expensive black-box objectives. However, BO utilizing standard Gaussian Processes is ill-suited for doubly stochastic Cox Processes that are often used in spatio-temporal problem spaces. We introduce INLA-SPDE Spatio-Temporal Bayesian Optimization (ISBO): the first scalable BO framework for spatio-temporal data, that models the log-intensity with a Log-Gaussian Cox Process(LGCP) and performs inference via Integrated Nested Laplace Approximation and Stochastic Partial Differential Equations (INLA-SPDE) approach. Using a Matern field on meshes yields a sparse Gaussian Markov Random Field, where INLA provides fast and accurate posterior inference throughout sequential optimization. ISBO stably locates high-intensity regions and the peak of the latent intensity with minimal evaluations. A time-varying Upper Confidence Bound acquisition with masking avoids revisits, while penalized-complexity priors regularize early rounds. Experiments on synthetic and real-world spatio-temporal datasets show accurate peak discovery, intensity recovery, and substantial speedups over an RKHS-based baseline, positioning ISBO as a practical choice for BO with point-process data.

Figures & tables

 xt+1=arg⁡max⁡x∈Ct{μη,t(x)+κt+1ση,t(x)};x_{t+1}=\arg\max_{x\in\mathcal{C}_{t}}\left\{\mu_{\eta,t}(x)+\kappa_{t+1}\sigma_{\eta,t}(x)\right\}; 
Algorithm 1 ISBO for point-process data
Figure 1: Temporal intensity λ(t)\lambda(t) (ours)
Figure 2: Average simple regret for BO over the synthetic 1D intensity, with 95% confidence intervals.
Figure 3: Mean fitting time for LGCP inference on local commodity hardware. Panel (a) compares INLA–SPDE, the Nyström-based inference procedure of Mei et al. (2024a) , and the Laplace and variational approximations of Dovers et al. (2023) over 200–2,500 events. Panel (b) extends the comparison between INLA–SPDE and the two SCAMPR approximations over 5,000 to 55,000 events.
Figure 4: Temporal Bayesian optimization on the Washington, D.C. crime data, comparing ISBO with Mei et al. (2024a) . Results are averaged across repeated runs, with shaded regions denoting 95% confidence intervals.
Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Observed firearm-related crime incidents time at Washington, D.C. between May and June 2022
Figure 6: 1D time mesh, integration points, and events
Figure 7: Posteriors for the hyperparameters
Figure 8: Temporal intensity λ(t)\lambda(t) (baseline)
Figure 9: Stepwise visualization of ISBO over the synthetic intensity. The red line is the posterior mean intensity, the pink band is the 95% credible interval, the green dashed line is the acquisition function, the black line is the ground-truth intensity, the black vertical ticks denote all incident times, and the red vertical ticks denote the revealed incident times.
Algorithm 2 1D Temporal ISBO Implementation
InputEvent times A={ti}\mathcal{A}=\{t_{i}\} ; candidate grid G\mathcal{G} ; initial centres C0\mathcal{C}_{0} ; query radius rr ; budget TT ; κstart,κfinal\kappa_{\mathrm{start}},\kappa_{\mathrm{final}} .
InitialiseConstruct the 1D SPDE domain and integration meshes, define the Matérn LGCP surrogate, reveal events around C0\mathcal{C}_{0} , and fit the initial posterior.
for t=1,…,Tt=1,\ldots,T do
1Predict μη,t−1(x)\mu_{\eta,t-1}(x) and ση,t−1(x)\sigma_{\eta,t-1}(x) on G\mathcal{G} .
2Compute κt=κstart−t−1T−1(κstart−κfinal)\kappa_{t}=\kappa_{\mathrm{start}}-\frac{t-1}{T-1}(\kappa_{\mathrm{start}}-\kappa_{\mathrm{final}}) and at(x)=μη,t−1(x)+κtση,t−1(x).a_{t}(x)=\mu_{\eta,t-1}(x)+\kappa_{t}\sigma_{\eta,t-1}(x).
3Mask previously revealed regions and select xt∈arg⁡max⁡x∈Ct−1at(x).x_{t}\in\arg\max_{x\in\mathcal{C}_{t-1}}a_{t}(x).
Appendix
Table 1: Detailed one-dimensional implementation of ISBO.
Figure 10: Starting locations of the full Porto taxi dataset.
Figure 11: Spatial intensity estimated by the Nyström baseline using 200 taxi trips. Black points denote the observed starting locations.
Figure 12: Posterior spatial intensity for the Porto taxi data using 200 events (top) and 66,666 events (bottom). Left panels show the response scale and right panels the latent scale.
Figure 13: Direct two-dimensional spatial Bayesian optimization on the Washington, D.C. crime data over 10 seeds. Shaded regions denote 95% confidence intervals.
Figure 14: Visualization of BO step-wise spatial intensity, red dots: used crime locations for current step
Algorithm 3 2D Spatial ISBO Implementation
InputSpatial events A={si}\mathcal{A}=\{\mathbf{s}_{i}\} ; spatial domain SS ; candidate grid Gs\mathcal{G}_{s} ; initial centres C0\mathcal{C}_{0} ; query radius rr ; budget TT ; κstart,κfinal\kappa_{\mathrm{start}},\kappa_{\mathrm{final}} .
InitialiseProject event locations to a metric coordinate system and retain events inside SS . Construct the triangular SPDE mesh, define the Matérn LGCP surrogate, and generate the common spatial candidate grid Gs\mathcal{G}_{s} from the prediction pixels.
 Sample the initial centres C0\mathcal{C}_{0} , reveal all events within radius rr of these centres, and fit the initial spatial LGCP posterior. Use the full-data fitted intensity λ^ref(s)\widehat{\lambda}_{\mathrm{ref}}(\mathbf{s}) as the reference surface for regret evaluation.
for t=1,…,Tt=1,\ldots,T do
1Fit and predict: fit the spatial LGCP to the currently revealed events and obtain posterior mean and uncertainty over Gs\mathcal{G}_{s} .
2Update exploration: compute κt=κstart−t−1T−1(κstart−κfinal)\kappa_{t}=\kappa_{\mathrm{start}}-\frac{t-1}{T-1}(\kappa_{\mathrm{start}}-\kappa_{\mathrm{final}}) and construct the spatial UCB acquisition.
Appendix
Table 2: Detailed two-dimensional spatial implementation of ISBO.

Explore similar work

CardsList

  1. Isotropic Gaussian Processes Improve Vanilla Bayesian Optimization in High Dimensions

    Oct 5, 2026Wei-Ting Tang, Madhav Muthyala, Joel A. PaulsonBayesian OptimizationGaussian Processes

    Department of Chemical and Biological Engineering University of Wisconsin–Madison Madison, WI 53706, USA

  2. Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

    Jun 30, 2026Rui-Yang Zhang, Lachlan Astfalck, Edward Cripps +2Efficient InferenceGaussian Processes

    School of Mathematical Sciences, Lancaster University. · School of Mathematics and Statistics, The University of New South Wales. · School of Physics, Mathematics and Computing, the University of Western Australia.

  3. Local Preferential Bayesian Optimization

    Jun 1, 2026Johanna Menn, Miriam Kober, Paul Brunzema +2Pairwise Preference LearningPairwise Preference Evaluation

    Institute for Data Science in Mechanical Engineering, RWTH Aachen University, Aachen, Germany · Department of Clinical Research, University of Bern, Bern, Switzerland · Center for Reproducible Science and Research Synthesis, University of Zurich, Zurich, Switzerland +1