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

The gist ISBO introduces INLA–SPDE Spatio-Temporal Bayesian Optimization, a scalable framework for spatio-temporal data that models intensity with a LogGaussian Cox Process and performs inference via Integrated Nested Laplace Approximation and Stochastic Partial Differential Equations to stably locate high-intensity regions with minimal evaluations.

How it works

The method combines a scalable LGCP surrogate model with Bayesian optimization over the latent intensity, which utilizes off-grid SPDE modeling, modern INLA inference, and penalized complexity priors for efficient uncertainty quantification during sequential optimization <ref:2610.12213#pg6>. The LGCP models the latent event intensity as a LogGaussian Cox Process where the log-intensity is modeled as a Gaussian process <ref:2610.12213#pg9>. This intensity, denoted by λ(s) = exp η(s), governs an inhomogeneous Poisson process conditional on its realization <ref:2610.12213#pg8>.

The surrogate model construction involves several key steps:

**((35)) The latent field is assigned a Gaussian prior x θ ∼ N 0, Q(θ)−1, where Q(θ) denotes the precision matrix <ref:2610.12213#pg9>. The linear predictors are obtained through η = Ax, where A is a sparse design matrix linking the latent field to the observations <ref:2610.12213#pg8>. The conditional posterior approximation is N µ(θ), QX(θ)−1, where QX(θ) = Q(θ) + A⊤D(θ)A <ref:2610.12213#pg9>. The marginalization over the INLA support points gives p˜(η(s) y) ≈ X K k=1 pG (η(s) θk, y) ˜p (θk y) δk <ref:2610.12213#pg9>. The posterior summaries for the latent predictor are obtained directly by transforming the posterior of η(x), yielding sd [λ(x) At] ≈ exp µη,t(x) + 1/2 σ squared η,t(x) qexp σ squared η,t(x) − 1 <ref:2610.12213#pg10>. The posterior surrogate used by the acquisition rule is therefore defined on the latent Gaussian scale as at(x) = µη,t(x) + κt sη,t(x). The time-varying UCB acquisition with masking avoids revisits. The next centre is selected by xt+1 = arg max x∈Ct at(x). This process ensures that the sequential update can be summarized as At −→ p(η At) −→ xt+1 −→ W(xt+1; r) −→ At+1. The constant refitting time is maintained because the dominant INLA refitting cost remains constant across ISBO iterations as the revealed dataset grows. The spatial LGCP setup uses a triangular SPDE mesh and PC priors for Mat´ern hyperparameters to achieve computational efficiency. The PC priors regularize the latent field when relatively few events have been revealed, which is particularly relevant during early BO iterations. The low-rank variational Bayes correction improves the accuracy of posterior means without reverting to a full nested Laplace approximation. The corrected mean is defined as µ∗(θ) = µ(θ) + Mλ, where λ ∈ R p, and M is a propagation matrix constructed from selected columns of QX(θ)−1. This improved mean µ∗(θ) can then be used in the Gaussian approximation for the linear predictors. The marginal posteriors of the linear predictors are obtained by numerical integration over the hyperparameters, yielding p˜(ηj y) ≈ X K k=1 pG(ηj θk, y) ˜p(θk y) δk. The proof of Theorem 1 shows that arg max x∈G η(x) = arg max x∈G λ(x), where λ(x) = exp η(x). The posterior standard deviation on the intensity scale is approximated by sd [λ(x) At] ≈ exp µη,t(x) + 1/2 σ squared η,t(x) qexp σ squared η,t(x) − 1. The proof of Theorem 2 establishes that the dominant INLA refitting cost remains constant across ISBO iterations as the revealed dataset grows. The computational complexity is therefore independent of the increasing number of revealed events. The method enables direct two-dimensional spatial BO by fitting and optimizing a spatial Cox-process surrogate rather than just temporal intensity. The final procedure is summarized in Algorithm 3 for 2D Spatial ISBO Implementation. The evaluation on real-world data shows substantial improvements over existing baselines in optimization performance and computational efficiency. The results demonstrate that ISBO consistently improves over existing baselines in optimization performance and computational efficiency, while ablation studies quantify the contribution of its main design components. The framework enables efficient optimization of latent intensity functions across temporal, spatial, and spatio-temporal settings. The framework could be extended to multi-agent reinforcement learning where scalable Bayesian inference may support hyperparameter tuning, adaptive exploration, and machine decision making. The code and experimental materials for this paper are available in the anonymous repository. The data and figures used to generate the results reported in the paper are also available in the anonymous repository. The experiments were executed locally on a MacBook Air with an Apple M2 processor and 16 GB RAM, without HPC resources. The choice of local hardware is motivated by the intended use of ISBO in resource-constrained settings. The method demonstrates that ISBO is sufficiently fast and memory-efficient for repeated Bayesian optimization without relying on specialized computing infrastructure. The paper uses generative AI tools, including ChatGPT, only for limited writing support, such as improving sentence structure, clarity, fluency, and consistency of presentation. All core algorithms and scientific contributions were developed by the authors. The paper uses generative AI tools to improve sentence structure, clarity, fluency, and consistency of presentation. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code. All core algorithms, experimental pipelines, implementation logic, and scientific contributions were developed by the authors. The paper uses generative AI tools for routine coding support including code comments and code completion in Visual Studio Code.

1. Spatial Intensity Estimation via ISBO: The improved system can perform direct two-dimensional spatial BO by optimizing a two-dimensional spatial intensity surface, which is crucial for tasks like analyzing crime incidents or taxi trajectories directly in their native spatio-temporal domain.

2. Scalable Inference for Large Point Processes: The system achieves constant refitting time across BO iterations as the number of revealed events increases, enabling it to handle large datasets (up to 55,000 events) efficiently without increasing computational cost during sequential optimization.

3. Robust Peak Localization in Sparse Data: By using a time-varying UCB acquisition with masking and leveraging PC priors, the system can achieve accurate peak discovery, intensity recovery, and substantial speedups over an RKHS-based baseline, even when data is sparse or noisy, as shown by ISBO reaching zero simple regret in synthetic benchmarks.

4. Uncertainty Quantification via Latent Scale Acquisition: The acquisition strategy utilizes the transformation that maps Gaussian uncertainty to a log-normal posterior for λ(x) At, motivating the use of UCB-style acquisition directly using the posterior mean and standard deviation of the latent predictor (Equation 25).

5. Real-Time Adaptive Data Acquisition: The framework enables adaptive data acquisition under structured event observations by sequentially updating a probabilistic surrogate, allowing it to determine which regions to query next based on maximizing the acquisition function over masked candidate sets.

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.