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Beyond Point Prediction: Score Matching-based Pseudolikelihood Estimation of Neural Marked Spatio-Temporal Point Process
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Proposes SMASH, a normalization-free score-matching pseudolikelihood estimator for neural marked spatio-temporal point processes that also yields confidence intervals/regions via Langevin-based sampling.
Problem
Learning neural marked spatio-temporal point processes requires an intractable normalization integral in the likelihood, and existing methods provide only point predictions without uncertainty quantification (confidence intervals/regions).
Approach
Decompose the joint intensity into time-mark and conditional spatial parts, apply score matching (with denoising) to the conditional time and location densities and a conditional log-likelihood for marks, giving a normalization-free pseudolikelihood; then generate samples with Langevin dynamics and Tweedie denoising to build confidence intervals/regions.
Key results
- On Earthquake, SMASH reduces time calibration score to 3.53% vs 14.80% for DSTPP (next best), and time MAE to 0.29 vs 0.35.
- On Crime, SMASH achieves space calibration score 1.08% vs 1.92% for DSTPP and time CS 6.93% vs 14.50%.
- On StackOverflow, SMASH achieves time CS 0.29% vs 0.34% for SMURF-THP and mark ECE 3.89% vs 5.41%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SMASH instead of the strongest reported baseline | SMASH · Earthquake | CStime (%): 14.80±0.67 -> 3.53±0.55 | Helped |
| Use SMASH instead of DSTPP | SMASH · Crime | CSspace (%): 1.92±1.48 -> 1.08±0.12 | Helped |
| Use SMASH instead of DSTPP | SMASH · Football | MAEtime (%): 3.76±0.24 -> 3.40±0.08 | Helped |
| Use SMASH instead of SMURF-THP | SMASH · StackOverflow | CS (%): 0.34±0.04 -> 0.29±0.12 | Helped |
| Use SMASH instead of SMURF-THP | SMASH · Retweet | CS (%): 0.35±0.04 -> 0.27±0.09 | Helped |
| Use SMASH instead of SMURF-THP | SMASH · MIMIC | CS (%): 1.14±0.23 -> 0.85±0.38 | Helped |
| Add Gaussian noise and use denoising score matching (ablation) | SMASH · Earthquake | CS-time | Helped |
| Increase the Gaussian noise scale (ablation) | SMASH · Earthquake | CS-time | Mixed |
| Increase the Gaussian noise scale (ablation) | SMASH · Earthquake | CS-space | Hurt |
| Increase the Gaussian noise scale (ablation) | SMASH · Earthquake | MAE | Helped |
| Increase the mark-loss weight α (ablation) | SMASH · Earthquake | CS-time | Hurt |
| Increase the mark-loss weight α (ablation) | SMASH · Earthquake | CS-space | Hurt |
Surprising findings
- Complex diffusion processes appear unnecessary for STPP modeling, per the authors, possibly because spatio-temporal data dimension is much lower than images.
- Adding noise (denoising) improves performance only at a suitable noise scale; calibration score for time first decreases then increases with noise.
Limitations
- Sampling uses two cascaded denoising steps, potentially less efficient than parallel sampling in DSTPP.
- Score-based sampling and denoising hyperparameters (noise scale, alpha) require tuning; performance sensitive to noise scale.
- Consistency result (Theorem 3.1) is local and relies on strong assumptions such as global minimum finding.
Experimental setup
Datasets: Earthquake (USGS Japan 1990-2020) Crime (Atlanta Police Department 2015-2020) Football (WyScout Open Access) StackOverflow Retweet MIMIC-II Financial Transactions (NYSE)
Benchmarks: Earthquake Crime Football StackOverflow Retweet MIMIC-II Financial Transactions
Metrics: Calibration Score MAE Accuracy Expected Calibration Error
Baselines: Poisson process Hawkes process NJSDE NSTPP NSMPP DeepSTPP DSTPP NHP NCE-TPP SAHP THP SMURF-THP
Builds on: score matching (Hyvärinen 2005) denoising score matching (Vincent 2011) Transformer Hawkes process (THP) SMURF-THP DSTPP
Closest papers by meaning
- Seahorse: A Unified Benchmarking Framework for Spatiotemporal Event Modeling (NeurIPS 2026 · similarity 0.88)
- EasyTPP: Towards Open Benchmarking Temporal Point Processes (ICLR 2024 · similarity 0.85)
- Arbitrarily Conditioned Hierarchical Flows for Spatiotemporal Events (NeurIPS 2026 · similarity 0.85)
- Interacting Diffusion Processes for Event Sequence Forecasting (ICML 2024 · similarity 0.84)
- Deep Continuous-Time State-Space Models for Marked Event Sequences (NeurIPS 2025 · similarity 0.84)
- Unlocking Point Processes through Point Set Diffusion (ICLR 2025 · similarity 0.84)
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