Amortized Variational Inference of Latent Intensities from Replicated Point Patterns

Wenqing Liu; Xenia Miscouridou; Déborah Sulem

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

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.

Method	Intensity inference	Intensity UQ	Replicated realizations	Amortized 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.

	Recovery	Uncertainty		Efficiency
Method	RMSE $\downarrow$	C95 (%)	IW95	Time (ms) $\downarrow$
FIM-PP	14.19	—	—	5.81
PSTH	2.45	—	—	0.04
KDE	1.91	—	—	0.75
LGCP	$1.86_{\pm 0.01}$	$92.70_{\pm 0.10}$	$6.67_{\pm 0.01}$	$6{,}311.76_{\pm 36.06}$
Ours ( $K_{\mathrm{train}}=4$ )	$\mathbf{1.55}_{\pm 0.01}$	${93.28}_{\pm 0.18}$	${5.73}_{\pm 0.00}$	${7.53_{\pm 0.44}}$

Table 2: Synthetic results on 500 test tasks with

K_{\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.

$K_{\mathrm{test}}$	RMSE $\downarrow$	C95 (%)	IW95
1	$2.68_{\pm 0.01}$	$92.21_{\pm 0.05}$	$9.36_{\pm 0.02}$
2	$1.99_{\pm 0.00}$	$93.18_{\pm 0.09}$	$7.20_{\pm 0.02}$
4	$1.55_{\pm 0.01}$	$93.28_{\pm 0.18}$	$5.73_{\pm 0.00}$
8	$1.23_{\pm 0.01}$	$93.78_{\pm 0.25}$	$4.78_{\pm 0.01}$
16	$1.00_{\pm 0.01}$	$94.32_{\pm 0.28}$	$4.23_{\pm 0.01}$
32	$0.88_{\pm 0.01}$	$93.84_{\pm 0.21}$	$3.91_{\pm 0.02}$

Table 3: Evidence accumulation for the model trained with

K_{\mathrm{train}}=4

. Results are averaged over four seeds.

Figure 2: Adaptation to additional realizations for the model trained with $K_{\mathrm{train}}=8$ . Four test tasks are shown as $K_{\mathrm{test}}\in\{1,4,8,16\}$ realizations become available. Across these examples, additional realizations generally improve intensity recovery and yield narrower CIs.

Method	HLL $\uparrow$	Time (ms) $\downarrow$
FIM-PP	59.53	76.03
PSTH	102.73	0.19
KDE	107.10	10.41
LGCP	$108.80_{\pm 0.01}$	$24{,}407.47_{\pm 76.83}$
Ours	$\mathbf{110.99}_{\pm 0.03}$	$26.82_{\pm 0.13}$

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

Module	Input	Transformation	Output
Event encoder $f_{e}$	$\bar{t}_{\tau,k,i}\in\mathbb{R}$	$\operatorname{MLP}_{\mathrm{SiLU}}(1,512,512,512,256)$	${\bm{h}}_{\tau,k,i}\in\mathbb{R}^{256}$
Realization representation	$\{{\bm{h}}_{\tau,k,i}\}_{i=1}^{N_{\tau,k}}$	${\bm{u}}_{\tau,k}=N_{\tau,k}^{-1}\sum_{i}{\bm{h}}_{\tau,k,i}$ ; ${\bm{x}}_{\tau,k}=[{\bm{u}}_{\tau,k}^{\top},c_{\tau,k}]^{\top}$	${\bm{x}}_{\tau,k}\in\mathbb{R}^{257}$
Task statistics	$\{{\bm{x}}_{\tau,k}\}_{k=1}^{K_{\tau}}$	${\bm{x}}_{\mathrm{stat},\tau}=[\bar{{\bm{x}}}_{\tau}^{\top},\bm{v}_{\tau}^{\top},\kappa_{\tau}]^{\top}$	${\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}$
Mean head $f_{\mu}$	${\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}$	$\operatorname{Linear}(515,256)$	${\bm{\mu}}_{{\mathbb{D}}_{\tau}}\in\mathbb{R}^{256}$
Log-variance head $f_{\sigma}$	${\bm{x}}_{\mathrm{stat},\tau}\in\mathbb{R}^{515}$	$\operatorname{MLP}_{\mathrm{SiLU}}(515,128,256)$	$\bm{\ell}_{{\mathbb{D}}_{\tau}}\in\mathbb{R}^{256}$

Appendix

Table 5: Context encoder architecture.

Split	Tasks	Sequences	Events	Events/sequence	Range
Train	4,500	72,000	11,625,135	$161.46\pm 67.80$	15–484
Validation	500	8,000	1,265,973	$158.25\pm 68.72$	7–441
Test	500	8,000	1,275,955	$159.49\pm 64.91$	13–444

Appendix

Table 6: Synthetic dataset statistics.

	Validation HLL $\uparrow$
Update	$K_{\mathrm{train}}=4$	$K_{\mathrm{train}}=8$	$K_{\mathrm{train}}=16$
1,000	308.32	308.69	309.38
2,000	310.11	309.21	310.71
3,000	310.76	309.61	310.85
4,000	310.62	309.89	310.90
5,000	310.81	310.11	311.02

Appendix

Table 7: Validation HLL across saved checkpoints for seed 42. Values are shown to two decimal places.

	RMSE $\downarrow$		C95 (%)		IW95
$K_{\mathrm{test}}$	$K_{\mathrm{train}}=8$	$K_{\mathrm{train}}=16$	$K_{\mathrm{train}}=8$	$K_{\mathrm{train}}=16$	$K_{\mathrm{train}}=8$	$K_{\mathrm{train}}=16$
1	$2.75_{\pm 0.01}$	$2.81_{\pm 0.01}$	$92.23_{\pm 0.16}$	$89.81_{\pm 2.62}$	$9.60_{\pm 0.03}$	$9.14_{\pm 0.61}$
2	$2.03_{\pm 0.01}$	$2.06_{\pm 0.01}$	$92.66_{\pm 0.10}$	$90.64_{\pm 2.35}$	$7.12_{\pm 0.01}$	$6.77_{\pm 0.44}$
4	$1.56_{\pm 0.01}$	$1.56_{\pm 0.01}$	$91.90_{\pm 0.16}$	$89.73_{\pm 1.77}$	$5.37_{\pm 0.01}$	$5.04_{\pm 0.24}$
8	$1.19_{\pm 0.01}$	$1.18_{\pm 0.01}$	$92.78_{\pm 0.26}$	$90.94_{\pm 0.57}$	$4.23_{\pm 0.01}$	$3.89_{\pm 0.06}$
16	$0.93_{\pm 0.01}$	$0.91_{\pm 0.01}$	$93.74_{\pm 0.31}$	$92.40_{\pm 0.74}$	$3.54_{\pm 0.01}$	$3.18_{\pm 0.07}$

Appendix

Table 8: Evidence accumulation for models trained with

K_{\mathrm{train}}\in\{8,16\}

. Results are averaged over four seeds.

PSTH bin width	Validation RMSE	KDE bandwidth	Validation RMSE
0.05	8.67	0.05	4.65
0.10	6.13	0.10	3.32
0.20	4.33	0.20	2.42
0.30	4.93	0.30	2.08
0.50	2.80	0.50	1.87
0.75	5.78	0.75	2.03

Appendix

Table 9: Validation selection for deterministic baselines on the synthetic dataset.

Method	Hardware	Training time	Checkpoint / fit budget
Ours, $K_{\mathrm{train}}=4$	NVIDIA GH200	2,969 s	20,000 updates
Ours, $K_{\mathrm{train}}=8$	NVIDIA GH200	3,497 s	20,000 updates
Ours, $K_{\mathrm{train}}=16$	NVIDIA GH200	4,305 s	20,000 updates
LGCP	NVIDIA GH200	—	10 inducing points; 10,000 steps per task
FIM-PP	NVIDIA GH200	Pretrained	Released checkpoint

Appendix

Table 10: Synthetic compute and implementation record.

Test set	RMSE	C95 (%)	IW95
OOD1	$0.89_{\pm 0.01}$	$95.72_{\pm 0.10}$	$3.77_{\pm 0.01}$
OOD2	$1.23_{\pm 0.02}$	$91.99_{\pm 0.17}$	$4.25_{\pm 0.01}$

Appendix

Table 11: OOD results. Results are averaged over four training seeds.

Panel of Figure 4 — Figure 4: OOD intensity recovery at $K_{\mathrm{test}}=8$ for OOD1 (left) and OOD2 (right).

Split	Tasks	Sequences	Stations	Events	Events/task
Train	5,387	21,548	182	1,437,806	163–1,534
Validation	674	2,696	21	170,078	163–1,018
Test	674	2,696	21	184,032	163–1,154

Appendix

Table 12: Divvy dataset after preprocessing.

Update	Seed 42	Seed 43	Seed 44	Seed 45
	Validation HLL $\uparrow$
1,000	82.87	82.63	81.67	81.95
2,000	83.62	83.61	83.30	82.95
3,000	83.55	83.75	83.72	83.59
4,000	83.45	83.71	83.83	83.47
5,000	83.99	84.00	83.87	83.99

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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