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Bayesian Optimization through Gaussian Cox Process Models for Spatio-temporal Data
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Proposes a MAP inference method for Gaussian Cox processes via Laplace approximation and a kernel change into a new RKHS (with Nyström approximation), yielding posterior mean and covariance that enable Bayesian optimization with GCP-based acquisition functions.
Problem
Bayesian optimization cannot be applied to doubly-stochastic Gaussian Cox processes because existing inference gives only posterior means for specific link functions, not the full posterior mean and covariance needed for acquisition functions.
Approach
Apply Laplace approximation and a change-of-kernel technique (merging the L2 integral term into the RKHS norm via Mercer expansions) so the GCP MAP problem admits a representer-theorem solution, giving a functional posterior mean and covariance for any smooth link; then use these in BO with UCB, idle-time, cumulative-arrival and change-point acquisition functions.
Key results
- On synthetic intensity estimation with the quadratic link, the proposed method achieves the best IQL_0.50 on lambda1 (11.44 vs. 12.50 for PIF) and lambda2 (12.56), and beats baselines on 7 of 9 settings per the authors.
- On lambda2, l2-norm error is 28.58 versus 30.01 for STVB, a 1.43 improvement.
- BO with UCB on 2022 DC crime data (343 events) identifies the main spatial intensity pattern within 10 steps and completes 25 steps in 91.56 seconds.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed Gaussian Cox process inference instead of STVB | Gaussian Cox process intensity estimation · Synthetic λ2 | l2-norm: 30.01 -> 28.58 | Helped |
| Use the proposed inference instead of MFVB | Gaussian Cox process intensity estimation · Synthetic λ3 | IQL.50: 32.77 -> 30.02 | Helped |
| Use the proposed inference instead of MFVB | Gaussian Cox process intensity estimation · Synthetic λ3 | IQL.85: 17.89 -> 24.32 | Hurt |
| Replace the quadratic link with the softplus link (ablation) | Ours Gaussian Cox process inference · Synthetic λ1 | l2-norm: 2.98 -> 2.91 | Helped |
| Replace the quadratic link with the exponential link (ablation) | Ours Gaussian Cox process inference · Synthetic λ1 | l2-norm: 2.98 -> 3.11 | Hurt |
| Replace the quadratic link with the softplus link (ablation) | Ours Gaussian Cox process inference · Synthetic λ1 | IQL.85: 7.38 -> 7.21 | Helped |
| Replace the quadratic link with the exponential link (ablation) | Ours Gaussian Cox process inference · Synthetic λ2 | l2-norm: 28.58 -> 31.22 | Hurt |
| Replace the quadratic link with the exponential link (ablation) | Ours Gaussian Cox process inference · Synthetic λ3 | l2-norm: 2.94 -> 4.00 | Hurt |
| Use UCB rather than EI (ablation) | Gaussian Cox process BO for synthetic peak prediction · Synthetic intensity peak 1 | l2-norm: 13.68 -> 10.79 | Helped |
| Use PI rather than UCB (ablation) | Gaussian Cox process BO for synthetic peak prediction · Synthetic intensity peak 2 | l2-norm: 8.91 -> 14.50 | Helped |
Surprising findings
- On lambda3 the 0.85-quantile IQL is worse than MFVB (24.32 vs. 17.89), so the method does not win every setting.
Limitations
- Relies on a BIC-style approximation that assumes a large number of events, so performance with sparse data is weaker.
- Requires discretization of the observation space to ensure tractability.
- Scalability is limited: runtime grows with number of events (USA tornado BO takes 820 s total).
Experimental setup
Datasets: synthetic intensity functions (lambda1, lambda2, lambda3) DC crime incidents 2022 2D neuronal data Porto taxi data coal mining disaster data 2022 USA tornado data
Benchmarks: synthetic intensity estimation (l2-norm, IQL) 2D neuronal data Porto taxi data DC crime incidents coal mining disaster data
Metrics: l2-norm IQL_0.50 IQL_0.85 runtime
Baselines: RKHS MFVB STVB PIF
Builds on: Gaussian Cox process Laplace approximation representer theorem Nyström approximation Bayesian online change point detection
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