stat.MLOct 12, 2026

Amortized Variational Inference of Latent Intensities from Replicated Point Patterns

Authors: Wenqing Liu, Xenia Miscouridou, Déborah Sulem

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

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Abstract

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.

Figures & tables

Figure 1: Architecture overview. Replicated point patterns parameterize a variational distribution whose samples are decoded into shared latent intensity functions.
MethodIntensity inferenceIntensity UQReplicated realizationsAmortized inference
LGCP✓\checkmark✓\checkmark✓\checkmark×\times
Meta-TPP×\times×\times×\times✓\checkmark
FIM-PP✓\checkmark×\times✓\checkmark✓\checkmark
LGCP-BayesFlow×\times×\times✓\checkmark✓\checkmark
DRS✓\checkmark✓\checkmark×\times✓\checkmark
NDIM✓\checkmark✓\checkmark×\times✓\checkmark
Table 1: Comparisons of related point process methods.
 RecoveryUncertaintyEfficiency
MethodRMSE ↓\downarrowC95 (%)IW95Time (ms) ↓\downarrow
FIM-PP14.19——5.81
PSTH2.45——0.04
KDE1.91——0.75
LGCP1.86±0.011.86_{\pm 0.01}92.70±0.1092.70_{\pm 0.10}6.67±0.016.67_{\pm 0.01}6,311.76±36.066{,}311.76_{\pm 36.06}
Ours ( Ktrain=4K_{\mathrm{train}}=4 )1.55±0.01\mathbf{1.55}_{\pm 0.01}93.28±0.18{93.28}_{\pm 0.18}5.73±0.00{5.73}_{\pm 0.00}7.53±0.44{7.53_{\pm 0.44}}
Table 2: Synthetic results on 500 test tasks with Ktest=4K_{\mathrm{test}}=4 . C95 is reported as a percentage (%). For learned models, ±\pm denotes standard deviation across four training seeds on the same test set. Classical baseline metrics are deterministic.
KtestK_{\mathrm{test}}RMSE ↓\downarrowC95 (%)IW95
12.68±0.012.68_{\pm 0.01}92.21±0.0592.21_{\pm 0.05}9.36±0.029.36_{\pm 0.02}
21.99±0.001.99_{\pm 0.00}93.18±0.0993.18_{\pm 0.09}7.20±0.027.20_{\pm 0.02}
41.55±0.011.55_{\pm 0.01}93.28±0.1893.28_{\pm 0.18}5.73±0.005.73_{\pm 0.00}
81.23±0.011.23_{\pm 0.01}93.78±0.2593.78_{\pm 0.25}4.78±0.014.78_{\pm 0.01}
161.00±0.011.00_{\pm 0.01}94.32±0.2894.32_{\pm 0.28}4.23±0.014.23_{\pm 0.01}
320.88±0.010.88_{\pm 0.01}93.84±0.2193.84_{\pm 0.21}3.91±0.023.91_{\pm 0.02}
Table 3: Evidence accumulation for the model trained with Ktrain=4K_{\mathrm{train}}=4 . Results are averaged over four seeds.
Figure 2: Adaptation to additional realizations for the model trained with Ktrain=8K_{\mathrm{train}}=8 . Four test tasks are shown as Ktest∈{1,4,8,16}K_{\mathrm{test}}\in\{1,4,8,16\} realizations become available. Across these examples, additional realizations generally improve intensity recovery and yield narrower CIs.
MethodHLL ↑\uparrowTime (ms) ↓\downarrow
FIM-PP59.5376.03
PSTH102.730.19
KDE107.1010.41
LGCP108.80±0.01108.80_{\pm 0.01}24,407.47±76.8324{,}407.47_{\pm 76.83}
Ours110.99±0.03\mathbf{110.99}_{\pm 0.03}26.82±0.1326.82_{\pm 0.13}
Table 4: Divvy results (held-out log likelihood and time) on 674 test tasks. ±\pm denotes standard deviation across four seeds for Ours and LGCP.
Figure 3: Held-out model fit for a test task. All estimates use only the three context realizations shown in the lower, light rows. The upper, dark row shows the held-out realization and is used only for evaluation. Solid curves show the inferred mean intensities for Ours and LGCP, with 95% CIs; the dashed and step curves show the KDE and PSTH estimates, respectively.
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

ModuleInputTransformationOutput
Event encoder fef_{e}tˉτ,k,i∈R\bar{t}_{\tau,k,i}\in\mathbb{R}MLP⁡SiLU(1,512,512,512,256)\operatorname{MLP}_{\mathrm{SiLU}}(1,512,512,512,256)hτ,k,i∈R256{\bm{h}}_{\tau,k,i}\in\mathbb{R}^{256}
Realization representation{hτ,k,i}i=1Nτ,k\{{\bm{h}}_{\tau,k,i}\}_{i=1}^{N_{\tau,k}}uτ,k=Nτ,k−1∑ihτ,k,i{\bm{u}}_{\tau,k}=N_{\tau,k}^{-1}\sum_{i}{\bm{h}}_{\tau,k,i} ; xτ,k=[uτ,k⊤,cτ,k]⊤{\bm{x}}_{\tau,k}=[{\bm{u}}_{\tau,k}^{\top},c_{\tau,k}]^{\top}xτ,k∈R257{\bm{x}}_{\tau,k}\in\mathbb{R}^{257}
Task statistics{xτ,k}k=1Kτ\{{\bm{x}}_{\tau,k}\}_{k=1}^{K_{\tau}}xstat,τ=[xˉτ⊤,vτ⊤,κτ]⊤{\bm{x}}_{\mathrm{stat},\tau}=[\bar{{\bm{x}}}_{\tau}^{\top},\bm{v}_{\tau}^{\top},\kappa_{\tau}]^{\top}xstat,τ∈R515{\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}
Mean head fμf_{\mu}xstat,τ∈R515{\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}Linear⁡(515,256)\operatorname{Linear}(515,256)μDτ∈R256{\bm{\mu}}_{{\mathbb{D}}_{\tau}}\in\mathbb{R}^{256}
Log-variance head fσf_{\sigma}xstat,τ∈R515{\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}MLP⁡SiLU(515,128,256)\operatorname{MLP}_{\mathrm{SiLU}}(515,128,256)ℓDτ∈R256\bm{\ell}_{{\mathbb{D}}_{\tau}}\in\mathbb{R}^{256}
Appendix
Table 5: Context encoder architecture.
SplitTasksSequencesEventsEvents/sequenceRange
Train4,50072,00011,625,135161.46±67.80161.46\pm 67.8015–484
Validation5008,0001,265,973158.25±68.72158.25\pm 68.727–441
Test5008,0001,275,955159.49±64.91159.49\pm 64.9113–444
Appendix
Table 6: Synthetic dataset statistics.
 Validation HLL ↑\uparrow
UpdateKtrain=4K_{\mathrm{train}}=4Ktrain=8K_{\mathrm{train}}=8Ktrain=16K_{\mathrm{train}}=16
1,000308.32308.69309.38
2,000310.11309.21310.71
3,000310.76309.61310.85
4,000310.62309.89310.90
5,000310.81310.11311.02
Appendix
Table 7: Validation HLL across saved checkpoints for seed 42. Values are shown to two decimal places.
 RMSE ↓\downarrowC95 (%)IW95
KtestK_{\mathrm{test}}Ktrain=8K_{\mathrm{train}}=8Ktrain=16K_{\mathrm{train}}=16Ktrain=8K_{\mathrm{train}}=8Ktrain=16K_{\mathrm{train}}=16Ktrain=8K_{\mathrm{train}}=8Ktrain=16K_{\mathrm{train}}=16
12.75±0.012.75_{\pm 0.01}2.81±0.012.81_{\pm 0.01}92.23±0.1692.23_{\pm 0.16}89.81±2.6289.81_{\pm 2.62}9.60±0.039.60_{\pm 0.03}9.14±0.619.14_{\pm 0.61}
22.03±0.012.03_{\pm 0.01}2.06±0.012.06_{\pm 0.01}92.66±0.1092.66_{\pm 0.10}90.64±2.3590.64_{\pm 2.35}7.12±0.017.12_{\pm 0.01}6.77±0.446.77_{\pm 0.44}
41.56±0.011.56_{\pm 0.01}1.56±0.011.56_{\pm 0.01}91.90±0.1691.90_{\pm 0.16}89.73±1.7789.73_{\pm 1.77}5.37±0.015.37_{\pm 0.01}5.04±0.245.04_{\pm 0.24}
81.19±0.011.19_{\pm 0.01}1.18±0.011.18_{\pm 0.01}92.78±0.2692.78_{\pm 0.26}90.94±0.5790.94_{\pm 0.57}4.23±0.014.23_{\pm 0.01}3.89±0.063.89_{\pm 0.06}
160.93±0.010.93_{\pm 0.01}0.91±0.010.91_{\pm 0.01}93.74±0.3193.74_{\pm 0.31}92.40±0.7492.40_{\pm 0.74}3.54±0.013.54_{\pm 0.01}3.18±0.073.18_{\pm 0.07}
Appendix
Table 8: Evidence accumulation for models trained with Ktrain∈{8,16}K_{\mathrm{train}}\in\{8,16\} . Results are averaged over four seeds.
PSTH bin widthValidation RMSEKDE bandwidthValidation RMSE
0.058.670.054.65
0.106.130.103.32
0.204.330.202.42
0.304.930.302.08
0.502.800.501.87
0.755.780.752.03
Appendix
Table 9: Validation selection for deterministic baselines on the synthetic dataset.
MethodHardwareTraining timeCheckpoint / fit budget
Ours, Ktrain=4K_{\mathrm{train}}=4NVIDIA GH2002,969 s20,000 updates
Ours, Ktrain=8K_{\mathrm{train}}=8NVIDIA GH2003,497 s20,000 updates
Ours, Ktrain=16K_{\mathrm{train}}=16NVIDIA GH2004,305 s20,000 updates
LGCPNVIDIA GH200—10 inducing points; 10,000 steps per task
FIM-PPNVIDIA GH200PretrainedReleased checkpoint
Appendix
Table 10: Synthetic compute and implementation record.
Test setRMSEC95 (%)IW95
OOD10.89±0.010.89_{\pm 0.01}95.72±0.1095.72_{\pm 0.10}3.77±0.013.77_{\pm 0.01}
OOD21.23±0.021.23_{\pm 0.02}91.99±0.1791.99_{\pm 0.17}4.25±0.014.25_{\pm 0.01}
Appendix
Table 11: OOD results. Results are averaged over four training seeds.
Figure 4: OOD intensity recovery at Ktest=8K_{\mathrm{test}}=8 for OOD1 (left) and OOD2 (right).
SplitTasksSequencesStationsEventsEvents/task
Train5,38721,5481821,437,806163–1,534
Validation6742,69621170,078163–1,018
Test6742,69621184,032163–1,154
Appendix
Table 12: Divvy dataset after preprocessing.
 Validation HLL ↑\uparrow
UpdateSeed 42Seed 43Seed 44Seed 45
1,00082.8782.6381.6781.95
2,00083.6283.6183.3082.95
3,00083.5583.7583.7283.59
4,00083.4583.7183.8383.47
5,00083.9984.0083.8783.99
Appendix
Table 13: Validation HLL across saved checkpoints on Divvy. Bold indicates the best checkpoint for each seed, determined before rounding.

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