Flexible Intensities Matter: A Comprehensive Re-evaluation of Hawkes Processes and Neural Temporal Point Processes

Hendrik Mehrtens; Florian Buettner; Oliver Stegle

cs.LGNov 4, 2026

Flexible Intensities Matter: A Comprehensive Re-evaluation of Hawkes Processes and Neural Temporal Point Processes

Authors: Hendrik Mehrtens, Florian Buettner, Oliver Stegle

Organizations: German Cancer Research Center (DKFZ), Heidelberg, Germany · Goethe-Universität Frankfurt, Frankfurt, Germany · European Molecular Biology Laboratory (EMBL), Heidelberg, Germany · Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

Abstract

Temporal point processes (TPPs) are the canonical framework for event sequence data, with applications spanning seismology, social media, and electronic health records. Recently, neural TPPs have been reported to substantially outperform classical TPPs, such as Hawkes processes, but these comparisons pitted flexible neural models against Hawkes processes in their most restricted form, and did not consider large-scale medical data. Here, we revisit this comparison through a systematic evaluation of Poisson and Hawkes processes of increasing flexibility against neural TPPs, on five established benchmark datasets and on two large-scale medical evaluation tasks derived from UK Biobank disease histories and MIMIC-IV ICU records, under autoregressive next-event prediction. First, the previously reported gap between Hawkes processes and neural TPPs largely disappears once Hawkes processes are equipped with flexible, learnable baselines and kernels. Second, the flexibility of the intensity parametrization also limits prior neural TPPs: a Transformer with a spline-based intensity head outperforms previous intensity-based neural TPPs and is on par with the intensity-free model. Decomposing the log-likelihood into a time term $\log p(t)$ and a type term $\log p(d \mid t)$ , we find that this flexibility advantage lies almost entirely in the time density: more flexibility does not improve the type term. On these benchmarks, the gains of neural TPPs thus lie mostly in modeling when events occur, not which. Third, our results underscore the value of complex datasets for evaluating TPPs: only on the MIMIC-IV ICU records do neural TPPs retain an advantage in the type term. Our results inform both future method development and model choice. We release our framework, TorchPP, as an open-source package with consistent implementations of classical and neural TPPs.

Figures & tables

Panel of Figure 1 — Figure 1: Conceptual differences between Hawkes processes (1) and neural TPPs (2) on an event sequence ( $\bullet$ type A, $\blacktriangle$ type B). Top (Hawkes process). Each past event interacts via a kernel response $\phi_{d_{i}\to d}(t-t_{i})$ (colored curves) on top of a (possibly time-varying) baseline $\mu_{d}(t)$ (thin solid line). The total intensity (upper sub-panel) is the sum of baseline and active kernels. At a query time $t$ , the intensity thus decomposes into the sum of the baseline and the kernel values. Bottom (Neural TPP). The same event history is consumed by a neural encoder, which produces a hidden state at each event location. The head of the neural TPP projects this encoding to an intensity function. Conceptually, this allows the neural TPP to model higher-order non-additive effects between events, in exchange for losing the interpretable decomposition of the Hawkes process.

Dataset	$D$	Term.	Train / Val / Test	Total Events	Mean Len
Taxi	10	No	1,400 / 200 / 400	74K	37.0
StackOverflow	22	No	1,401 / 401 / 401	143K	64.6
Retweet	3	No	9,000 / 1,535 / 1,520	494K	41.1
Amazon	16	No	6,454 / 922 / 1,851	413K	44.7
Taobao	17	No	1,300 / 200 / 500	113K	56.7
UKB	1,257	Yes	351,656 / 100,473 / 50,238	6.2M	12.4

Table 1: Dataset statistics. Small- to medium-sized, low-dimensional standard benchmarks are shown in the top rows, and the medical evaluation tasks, with orders of magnitude more instances and event types, in the bottom rows. “

D

”: number of event types. “Term.”: whether each type occurs at most once. “Mean Len”: mean number of events per sequence. Sequences are ICU-stay-level (with each patient only in one split) for MIMIC-IV and patient-level for UKB.

Poisson processes
Model	Taxi	StackOv.	Retweet	Amazon	Taobao
H. Poisson	$0.51/{-}1.14$	${-}0.86/{-}1.90$	${-}3.58/{-}0.82$	${-}0.36/{-}2.24$	$2.07/{-}2.18$
S. Poisson	$0.51/{-}1.13$	${-}0.85/{-}1.88$	${-}3.43/{-}0.82$	${-}0.35/{-}2.24$	$2.11/{-}2.19$
Hawkes processes
C/E Hawkes	$0.45/{-}0.74$	${-}0.76/{-}1.62$	${-}3.42/{-}0.79$	${-}0.10/{-}2.03$	$2.61/{-}1.42$
S/E Hawkes	$0.45/{-}0.74$	${-}0.75/{-}1.60$	${-}3.38/\mathbf{{-}0.78}$	${-}0.11/{-}2.04$	$2.61/{-}1.43$

Table 2: Main results: per-event test log-likelihood on the standard benchmark datasets (nats, higher is better; mean over 5 seeds; EasyTPP convention, i.e. conditioned on the first event and without the final survival term, Section 3.3 ). Each cell is

\log p(t)\,/\,\log p(d\mid t)

(time / type), and the joint

\log p(t,d)=\log p(t)+\log p(d\mid t)

is their sum (Table 8 ;

\pm

std in Appendix H ). Models differ chiefly in the time term

\log p(t)

, the axis that flexibility improves, while the type term

\log p(d\mid t)

is comparatively tight across history-aware models (near-flat on StackOverflow and Retweet). Row labels: baseline/kernel (C: constant, S: spline, E: exponential), I.: inhibitive, N.: neural. Bold : best per dataset and component; underline : best joint log-likelihood; ties at the displayed precision are all marked.

Figure 2: Flexibility acts on the time density. Change in per-event test log-likelihood (nats) vs. spline knot count

K

(relative to

K{=}20

), for NeuralSplineTPP (intensity-head knots) and Spline/Spline Hawkes (kernel knots), on three datasets (all datasets in Tables 22 and 19 ). Left: the continuous time term

\log p(t)

increases with

K

on Retweet and Amazon. Middle: scored by the probability of the recorded time bin (Retweet: 1, Amazon: 0.01, MIMIC-IV: 1 min), it increases only on Amazon. Right: the type-conditional term

\log p(d\mid t)

does not improve with

K

Poisson processes
Model	UKB	MIMIC-IV
H. Poisson	${-}8.08/{-}5.59$	$2.60/{-}3.78$
S. Poisson	${-}7.51/{-}5.36$	$2.64/{-}3.71$
Hawkes processes
C/E Hawkes	${-}7.36/{-}5.15$	$3.38/{-}2.71$
S/E Hawkes	${-}7.26/{-}5.07$	$3.39/{-}2.70$

Table 3: Large-scale medical results (nats, same convention as Table 2 ): cell =

\log p(t)/\log p(d\mid t)

(time/type). On UKB, inhibitive Hawkes were not evaluated and NHP did not converge. Joint,

\pm

std and full metrics in Appendix H . Bold : best per column and component; underline : best joint log-likelihood (ties all marked).

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Model	LL	Acc $\|t$	RMSE	LL	Acc $\|t$	RMSE	Acc $\|t$	RMSE
		TorchPP			EasyTPP (rerun)			EasyTPP (paper)
Taxi	SAHP	0.25	0.918	0.290	$-0.39$	0.897	0.313	0.903	0.372
Taxi	THP	0.25	0.913	0.285	0.24	0.909	0.305	0.913	0.370
StackOverflow	SAHP	$-2.36$	0.462	1.040	$-8.23$	0.425	1.307	0.439	1.375
StackOverflow	THP	$-2.40$	0.452	1.029	$-2.54$	0.443	1.130	0.450	1.374
Retweet	SAHP	$-4.14$	0.596	15.95	$-5.14$	0.540	19.69	0.584	22.40

Appendix

Table 4: Code equivalence validation: TorchPP re-implementations vs. EasyTPP framework on four benchmark datasets, with reference numbers from ( Xue et al., 2024 ) . LL: joint log-likelihood (nats/event),

\mathrm{Acc}|t

: accuracy given the true next event time, as used by EasyTPP (as a fraction), RMSE: next-event time RMSE. TorchPP values are means over 5 seeds. EasyTPP (rerun) values are single-framework runs we performed locally. EasyTPP (paper) values are taken from the original EasyTPP benchmark; the reported type error rates are converted to accuracies. The TorchPP values use the EasyTPP-matched training configuration of Appendix D.1 (20,000 steps, 100 integration points) and therefore differ from Appendix H . EasyTPP’s reported Hawkes baseline is analyzed separately in Appendix A.6 .

Dataset	Majority	tick LS	MLE	MLE, $\mu{=}0.2$	EasyTPP (reported)	Ours: C/E / best Hawkes
		tick MHP (our fit)
Taxi	44.3	79.6	84.2	84.6	90.5	85.0 / 92.0
StackOverflow	42.5	44.5	43.6	44.2	35.0	46.2 / 46.7
Retweet	54.0	53.8	54.1	42.2	44.3	57.5 / 58.6
Amazon	30.0	29.9	28.9	32.8	24.1	35.0 / 39.1
Taobao	43.6	54.1	53.7	54.5	31.9	61.0 / 61.5

Appendix

Table 5: Reproducing EasyTPP’s Hawkes (MHP) baseline. Time-conditional next-type accuracy

\mathrm{Acc}|t

(%). The tick fits, the majority rate and EasyTPP use the full test sets, our models the first 1,000 test sequences (Section 3.3 ). Majority : always predicting the most frequent type. tick LS : tick HawkesExpKern with fixed decay 1.0 and its default least-squares fit; MLE : the same exponential Hawkes fitted by positivity-constrained maximum likelihood; MLE,

\mu{=}0.2

: additionally fixing the base intensity to 0.2, the literal recipe of EasyTPP’s Appendix B. Ours: our learned Const./Exp Hawkes and the best of our (inhibitive) Hawkes variants, from Table 11 .

Panel of Figure 3 — Figure 3: Relationships between the key quantities of a multivariate temporal point process. The top row constructs the survival objects from the per-type conditional intensities $\lambda_{d}^{*}(t)$ : summation across types yields the ground intensity $\lambda^{*}(t)$ , integration over time yields the cumulative intensity $\Lambda^{*}(t_{i-1},t)$ , and the survival $S^{*}(\tau)$ and CDF $F^{*}(\tau)$ of the next-event time follow from $\Lambda^{*}$ by $\exp(-\,\cdot\,)$ and $1-\cdot$ . The middle row shows how the next-event densities are constructed at merge nodes ( $\times$ , $/$ ): each density is formed by combining two top-row quantities, and the resulting expression is the label on the outgoing arrow. In addition, the time marginal is recovered as $p^{*}(t)=\sum_{d}p^{*}(t,d)$ , the type marginal as $p^{*}(d)=\int_{t_{i-1}}^{\infty}p^{*}(t,d)\,dt$ , and the time conditional as $p^{*}(t\mid d)=p^{*}(t,d)/p^{*}(d)$ . Quantities shown in red ( $p^{*}(d)$ and $p^{*}(t\mid d)$ ) are expensive to evaluate when starting from intensities: both require the unbounded integral defining $p^{*}(d)$ , which has no closed form except for trivial cases such as the homogeneous Poisson process and otherwise demands numerical quadrature with $\Lambda^{*}$ evaluated at every quadrature point. Stars denote conditioning on the history $\mathcal{H}_{t_{i-1}}$ ; $\tau=t-t_{i-1}$ .

Family	Configs / dataset	Swept hyperparameters	Integration	Params (MIMIC-IV)
Poisson, Const./Exp Hawkes	2	–	closed form	117 – 27,495
Spline Poisson, flexible Hawkes	$\approx 16$	$K_{b}$ , $K_{k}\times T_{f}$ ; $K$ validation-selected	closed form	2,340 – 285,481
Inhibitive Hawkes	2	inherited from analytical twin, clamping function	numerical (grid)	27,495 – 285,481
NeuralConstTPP, NeuralSplineTPP	$\approx 20$	$L\times H\times d$ ; $K_{k}\times T_{f}$ ; $K$ validation-selected	closed form	216,554 – 286,052
THP, SAHP	–	architecture from EasyTPP, validated	numerical (per interval)	216,905 / 231,764
NHP, IntensityFree	7	encoder (GRU/Transformer), $L$ , $d$ , learning rate	numerical / closed form	73,066 / 64,746

Appendix

Table 6: Tuning budget per model family on the standard benchmarks (search configurations per dataset of the systematic grid), integration method, and trainable parameters on MIMIC-IV (

D=117

) at the final configuration. Early stopping (patience 4 on the standard benchmarks, 6 on UKB/MIMIC-IV, best validation checkpoint restored) is identical for all families.

K_{b}

K_{k}

: baseline and kernel knots;

T_{f}

: kernel support in mean inter-event times;

L,H,d

: Transformer layers, heads, embedding dimension.

	Hyperparameter tuning		Replica runs
Dataset	Runs	GPU-h	Runs	GPU-h
Standard benchmarks	396	144	370	73
MIMIC-IV ICU	28	4	75	156
UK Biobank	205	289	133	98
Total	629	437	578	327

Appendix

Table 7: Compute used for the experiments in this paper. Run counts and wall-clock GPU-hours are aggregated from WandB logs of the tuning sweeps and of the replica runs, which include the runs that produced the numbers in the tables. “Runs” counts only runs that reached the final evaluation step, and runtime is wall-clock time on a single GPU. Some tuning sweeps logged only metrics and not runtime. These are included in the run counts but not in the GPU-hour totals, which therefore slightly underestimate the true cost.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$-0.63_{\pm.00}$	$-2.76_{\pm.00}$	$-4.41_{\pm.00}$	$-2.60_{\pm.00}$	$-0.11_{\pm.00}$
Spline Poisson	$-0.62_{\pm.00}$	$-2.73_{\pm.00}$	$-4.25_{\pm.00}$	$-2.59_{\pm.00}$	$-0.07_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$-0.29_{\pm.00}$	$-2.38_{\pm.00}$	$-4.20_{\pm.00}$	$-2.13_{\pm.02}$	$1.19_{\pm.00}$
Spline/Exp Hawkes	$-0.29_{\pm.00}$	$-2.35_{\pm.01}$	$-4.16_{\pm.00}$	$-2.15_{\pm.01}$	$1.19_{\pm.00}$

Appendix

Table 8: Joint test log-likelihood per event

\log p(t,d\mid\mathcal{H})

(nats, higher is better) on the standard benchmarks. It is the sum of Tables 9 and 10 . Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$0.51_{\pm.00}$	$-0.86_{\pm.00}$	$-3.58_{\pm.00}$	$-0.36_{\pm.00}$	$2.07_{\pm.00}$
Spline Poisson	$0.51_{\pm.00}$	$-0.85_{\pm.00}$	$-3.43_{\pm.00}$	$-0.35_{\pm.00}$	$2.11_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$0.45_{\pm.00}$	$-0.76_{\pm.00}$	$-3.42_{\pm.00}$	$-0.10_{\pm.00}$	$2.61_{\pm.00}$
Spline/Exp Hawkes	$0.45_{\pm.00}$	$-0.75_{\pm.00}$	$-3.38_{\pm.00}$	$-0.11_{\pm.00}$	$2.61_{\pm.00}$

Appendix

Table 9: Time log-likelihood per event

\log p(t\mid\mathcal{H})

(nats, higher is better) on the standard benchmarks (Equation 12 ). Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$-1.14_{\pm.00}$	$-1.90_{\pm.00}$	$-0.82_{\pm.00}$	$-2.24_{\pm.00}$	$-2.18_{\pm.00}$
Spline Poisson	$-1.13_{\pm.00}$	$-1.88_{\pm.00}$	$-0.82_{\pm.00}$	$-2.24_{\pm.00}$	$-2.19_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$-0.74_{\pm.00}$	$-1.62_{\pm.00}$	$-0.79_{\pm.00}$	$-2.03_{\pm.03}$	$-1.42_{\pm.00}$
Spline/Exp Hawkes	$-0.74_{\pm.00}$	$-1.60_{\pm.00}$	$\mathbf{-0.78}_{\pm.00}$	$-2.04_{\pm.00}$	$-1.43_{\pm.00}$

Appendix

Table 10: Time-conditional type log-likelihood per event

\log p(d\mid t,\mathcal{H})

at the true next-event time (nats, higher is better; Equation 13 ). Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$44.29_{\pm.00}$	$42.50_{\pm.00}$	$53.92_{\pm.00}$	$29.87_{\pm.00}$	$43.57_{\pm.00}$
Spline Poisson	$44.26_{\pm.10}$	$42.50_{\pm.00}$	$54.66_{\pm.06}$	$29.87_{\pm.00}$	$43.57_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$85.00_{\pm.08}$	$46.16_{\pm.04}$	$57.49_{\pm.16}$	$35.03_{\pm 2.50}$	$60.96_{\pm.05}$
Spline/Exp Hawkes	$84.74_{\pm.12}$	$46.16_{\pm.08}$	$58.65_{\pm.10}$	$32.48_{\pm.01}$	$60.91_{\pm.06}$

Appendix

Table 11: Time-conditional top-1 type accuracy

\mathrm{Acc}|t

(%, higher is better):

\arg\max_{d}p(d\mid t,\mathcal{H})

at the true next-event time, the next-type metric reported by EasyTPP ( Xue et al., 2024 ) . Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$44.29_{\pm.00}$	$42.50_{\pm.00}$	$53.92_{\pm.00}$	$29.87_{\pm.00}$	$43.57_{\pm.00}$
Spline Poisson	$44.25_{\pm.28}$	$42.50_{\pm.00}$	$54.71_{\pm.01}$	$29.87_{\pm.00}$	$43.57_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$85.29_{\pm.40}$	$44.39_{\pm.06}$	$57.62_{\pm.12}$	$34.06_{\pm 1.22}$	$60.91_{\pm.05}$
Spline/Exp Hawkes	$85.50_{\pm.28}$	$44.45_{\pm.03}$	$58.71_{\pm.11}$	$33.21_{\pm.02}$	$60.79_{\pm.06}$

Appendix

Table 12: Marginal top-1 type accuracy

\mathrm{Acc}

(%, higher is better):

\arg\max_{d}p(d\mid\mathcal{H})

from the marginal type distribution (Equation 14 ), computed by numerical integration. Grid-based metrics are evaluated on the first 1,000 test sequences. ∗ : mean over 3–4 seeds, as the evaluation of a seed is missing or its RMSE diverged. Spline/Spline Hawkes on Amazon was not evaluated with the grid-based metrics. Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$-1.14_{\pm.00}$	$-1.90_{\pm.00}$	$-0.82_{\pm.00}$	$-2.24_{\pm.00}$	$-2.18_{\pm.00}$
Spline Poisson	$-1.14_{\pm.00}$	$-1.88_{\pm.00}$	$-0.82_{\pm.00}$	$-2.24_{\pm.00}$	$-2.19_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$-0.81_{\pm.00}$	$-1.64_{\pm.00}$	$-0.79_{\pm.00}$	$-2.03_{\pm.02}$	$-1.42_{\pm.00}$
Spline/Exp Hawkes	$-0.80_{\pm.00}$	$-1.63_{\pm.00}$	$\mathbf{-0.78}_{\pm.00}$	$-2.03_{\pm.00}$	$-1.42_{\pm.00}$

Appendix

Table 13: Marginal type log-likelihood per event

\log p(d\mid\mathcal{H})

(nats, higher is better; Equation 14 ), computed by numerical integration. Grid-based metrics are evaluated on the first 1,000 test sequences. ∗ : mean over 3–4 seeds, as the evaluation of a seed is missing or its RMSE diverged. Spline/Spline Hawkes on Amazon was not evaluated with the grid-based metrics. Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao
Homogeneous Poisson	$0.30_{\pm.00}$	$1.07_{\pm.00}$	$17.82_{\pm.01}$	$0.34_{\pm.00}$	$\mathbf{0.13}_{\pm.00}$
Spline Poisson	$0.30_{\pm.00}$	$1.07_{\pm.00}$	$16.22_{\pm.01}$	$0.34_{\pm.00}$	$\mathbf{0.13}_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$0.31_{\pm.00}$	$1.03_{\pm.00}$	$16.65_{\pm.01}$	$0.36_{\pm.01}$	$\mathbf{0.13}_{\pm.00}$
Spline/Exp Hawkes	$0.31_{\pm.00}$	$1.03_{\pm.00}$	$\mathbf{15.83}_{\pm.01}$	$0.34_{\pm.00}$	$\mathbf{0.13}_{\pm.00}$

Appendix

Table 14: Next-event time RMSE (lower is better) using the predictive mean

\mathbb{E}[t\mid\mathcal{H}]

, in each dataset’s native EasyTPP time unit (not comparable across datasets). ‡ IntensityFree: the closed-form mean of its log-normal mixture is dominated by components with large variance and overflows for individual seeds, so we report the median predictor instead and exclude it from the bold marking. Grid-based metrics are evaluated on the first 1,000 test sequences. ∗ : mean over 3–4 seeds, as the evaluation of a seed is missing or its RMSE diverged. Spline/Spline Hawkes on Amazon was not evaluated with the grid-based metrics. Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	$\log p(t,d)$	$\log p(t)$	$\log p(d\|t)$	$\log p(d)$	Acc $\|t$ (%)	Acc (%)	RMSE $\times 100$
Homogeneous Poisson	$-1.18_{\pm.02}$	$2.60_{\pm.01}$	$-3.78_{\pm.02}$	$-3.81_{\pm.01}$	$13.6_{\pm.0}$	$13.1_{\pm.0}$	$5.79_{\pm.01}$
Spline Poisson	$-1.07_{\pm.01}$	$2.64_{\pm.00}$	$-3.71_{\pm.01}$	$-3.73_{\pm.01}$	$13.6_{\pm.0}$	$13.1_{\pm.0}$	$5.75_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$0.68_{\pm.00}$	$3.38_{\pm.00}$	$-2.71_{\pm.00}$	$-2.81_{\pm.00}$	$30.9_{\pm.1}$	$30.0_{\pm.1}$	$5.48_{\pm.01}$
Spline/Exp Hawkes	$0.68_{\pm.00}$	$3.39_{\pm.00}$	$-2.70_{\pm.00}$	$-2.81_{\pm.00}$	$30.9_{\pm.0}$	$30.1_{\pm.1}$	$5.39_{\pm.00}$

Appendix

Table 15: Full MIMIC-IV results (

D=117

, including the OTHER class).

\log p(t,d)=\log p(t)+\log p(d|t)

;

\log p(d)

, Acc and RMSE are computed by numerical integration on the first 500 test stays, all other metrics on the first 1,000. RMSE in days, multiplied by 100. ‡ Median predictor (see Table 14 ). § NeuralConstTPP uses a Softplus head on MIMIC-IV, as its exponential head diverged for individual seeds. Mean

\pm

std over 5 seeds. Bold : best per column; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Poisson processes
Model	$\log p(t,d)$	$\log p(t)$	$\log p(d\|t)$	$\log p(d)$	Acc $\|t$ (%)	Acc (%)	RMSE $/100$
Homogeneous Poisson	$-13.67_{\pm.01}$	$-8.08_{\pm.02}$	$-5.59_{\pm.00}$	$-5.59_{\pm.00}$	$4.23_{\pm.04}$	$4.26_{\pm.02}$	$23.57_{\pm.29}$
Spline Poisson	$-12.87_{\pm.00}$	$-7.51_{\pm.00}$	$-5.36_{\pm.00}$	$-5.40_{\pm.00}$	$4.56_{\pm.07}$	$4.37_{\pm.06}$	$17.06_{\pm.15}$
Hawkes processes
Const./Exp Hawkes	$-12.51_{\pm.00}$	$-7.36_{\pm.00}$	$-5.15_{\pm.00}$	$-5.03_{\pm.00}$	$6.45_{\pm.06}$	$6.91_{\pm.04}$	$17.93_{\pm.06}$
Spline/Exp Hawkes	$-12.33_{\pm.00}$	$-7.26_{\pm.00}$	$-5.07_{\pm.00}$	$-5.00_{\pm.00}$	$6.77_{\pm.10}$	$7.09_{\pm.04}$	$15.26_{\pm.01}$

Appendix

Table 16: Full UKB results (

D=1{,}257

, terminating). UKB models use the configuration of Appendix D.2 , and IntensityFree uses its Transformer encoder. Inhibitive Hawkes processes were not evaluated on UKB (cost of numerical integration at

D=1{,}257

), and NHP did not converge. RMSE in days, divided by 100, using the predictive mean for all models. Mean

\pm

std over seeds. Bold : best per column; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

Dataset	Encoder	$\log p(t,d)$	$\log p(t)$	$\log p(d\|t)$
Taxi	Transformer	$0.44_{\pm.00}$	$0.73_{\pm.00}$	$-0.29_{\pm.00}$
	GRU	$0.44_{\pm.00}$	$0.73_{\pm.00}$	$-0.29_{\pm.00}$
StackOverflow	Transformer	$-2.33_{\pm.02}$	$-0.73_{\pm.01}$	$-1.60_{\pm.01}$
	GRU	$-2.31_{\pm.01}$	$-0.70_{\pm.01}$	$-1.61_{\pm.00}$
Retweet	Transformer	$-4.45_{\pm.26}$	$-3.66_{\pm.26}$	$-0.79_{\pm.01}$
	GRU	$-4.49_{\pm.27}$	$-3.71_{\pm.27}$	$-0.78_{\pm.02}$

Appendix

Table 17: IntensityFree with a GRU encoder (as in Shchur et al. (2020a) ) vs. the Transformer encoder used in the main tables. Per-event test log-likelihood, mean

\pm

std over 5 seeds (higher is better). The Transformer is clearly better on Amazon, the GRU on MIMIC-IV and UKB and slightly on Taobao, and the two are close on Taxi, StackOverflow, and Retweet.

Dataset	Recorded resolution	Bins	Tied (%)
Taxi	1 s	1 s, 10 s	0.0
StackOverflow	float32 ( $2^{-13}$ )	$2^{-13}$ , $10\cdot 2^{-13}$	0.5
Retweet	integer time stamps	1, 10	7.4
Amazon	none (minimum gap 0.01)	0.01, 0.1	0.0
Taobao	gaps of $10^{-4}$	$10^{-4}$ , $10^{-3}$	0.0
MIMIC-IV	1 min (dequantized)	1 min, 5 min	0.0

Appendix

Table 18: Time resolution of the evaluated event times (first 1,000 test sequences, MIMIC-IV truncated to 316 events) and the bin widths used for the binned time likelihood, in the time unit of each dataset. Tied: fraction of evaluated events with a zero gap to the previous event.

		$\log p(t)$ , continuous		$\Delta$ binned
Dataset	Model	$K=20$	$\Delta$ to $K=400$	recorded	coarser	centered
Taxi	NeuralSplineTPP	0.73	-0.05	-0.05	-0.04	-0.05
	Spline/Spline Hawkes	0.49	+0.01	+0.01	+0.01	+0.01
StackOverflow	NeuralSplineTPP	-0.72	-0.04	-0.05	-0.06	-0.09
	Spline/Spline Hawkes	-0.73	-0.03	-0.04	-0.04	-0.04
Retweet	NeuralSplineTPP	-3.32	+1.47	-0.06	+0.00	-0.28

Appendix

Table 19: Change of

\log p(t)

from

K=20

K=400

knots (nats, mean over 5 seeds), continuous and binned at the recorded resolution, at a ten-fold (MIMIC-IV: five-fold) coarser bin, and with bins centered on the grid points instead of starting at them (Table 18 ). Same checkpoints as Table 22 .

Poisson processes
Model	Taxi	StackOverflow	Retweet	Amazon	Taobao	MIMIC-IV
Homogeneous Poisson	$-7.68_{\pm.00}$	$-9.86_{\pm.00}$	$-3.62_{\pm.00}$	$-4.96_{\pm.00}$	$-7.14_{\pm.00}$	$-4.90_{\pm.01}$
Spline Poisson	$-7.68_{\pm.00}$	$-9.85_{\pm.00}$	$-3.48_{\pm.00}$	$-4.96_{\pm.00}$	$-7.10_{\pm.00}$	$-4.87_{\pm.00}$
Hawkes processes
Const./Exp Hawkes	$-7.74_{\pm.00}$	$-9.76_{\pm.00}$	$-3.47_{\pm.00}$	$-4.70_{\pm.00}$	$-6.60_{\pm.00}$	$-4.13_{\pm.00}$
Spline/Exp Hawkes	$-7.74_{\pm.00}$	$-9.75_{\pm.00}$	$\mathbf{-3.43}_{\pm.00}$	$-4.71_{\pm.00}$	$-6.60_{\pm.00}$	$-4.12_{\pm.00}$

Appendix

Table 20: Binned time log-likelihood

\log\Pr(t\in\text{bin})

at the recorded resolution (nats, higher is better; bins in Table 18 ) for the models of the main tables. Only differences within a column are meaningful, as the level depends on the bin width. On Retweet the history-aware models are within a few tenths of a nat, while the continuous

\log p(t)

(Table 9 ) spreads over more than two nats. Mean

\pm

std over 5 seeds. Bold : best per dataset; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

	$\log p(t)$		Binned $\log p(t)$			$\log p(d\|t)$			Acc $\|t$ (%)
Model	all	other	all	other	same	all	other	same	all	other	same
Homogeneous Poisson	$2.60_{\pm.01}$	$1.71_{\pm.04}$	$-4.90_{\pm.01}$	$-5.57_{\pm.04}$	$-3.97_{\pm.04}$	$-3.78_{\pm.02}$	$-3.91_{\pm.02}$	$-3.60_{\pm.02}$	$13.6_{\pm.0}$	$12.7_{\pm.0}$	$14.7_{\pm.0}$
Spline Poisson	$2.64_{\pm.00}$	$1.79_{\pm.03}$	$-4.87_{\pm.00}$	$-5.49_{\pm.03}$	$-3.99_{\pm.03}$	$-3.71_{\pm.01}$	$-3.84_{\pm.01}$	$-3.53_{\pm.01}$	$13.6_{\pm.0}$	$12.7_{\pm.0}$	$14.7_{\pm.0}$
Const./Exp Hawkes	$3.38_{\pm.00}$	$2.05_{\pm.00}$	$-4.13_{\pm.00}$	$-5.24_{\pm.00}$	$-2.57_{\pm.00}$	$-2.71_{\pm.00}$	$-3.12_{\pm.00}$	$-2.13_{\pm.00}$	$30.9_{\pm.1}$	$20.1_{\pm.1}$	$46.2_{\pm.2}$
Spline/Exp Hawkes	$3.39_{\pm.00}$	$2.05_{\pm.00}$	$-4.12_{\pm.00}$	$-5.23_{\pm.00}$	$-2.57_{\pm.00}$	$-2.70_{\pm.00}$	$-3.12_{\pm.00}$	$-2.12_{\pm.00}$	$30.9_{\pm.0}$	$20.1_{\pm.1}$	$46.2_{\pm.1}$
Const./Spline Hawkes	$3.78_{\pm.00}$	$2.11_{\pm.00}$	$-3.73_{\pm.00}$	$-5.17_{\pm.00}$	$-1.71_{\pm.01}$	$-2.62_{\pm.00}$	$-2.95_{\pm.00}$	$-2.15_{\pm.00}$	$32.7_{\pm.0}$	$22.4_{\pm.0}$	$47.3_{\pm.0}$

Appendix

Table 21: MIMIC-IV results split by whether the evaluated event was recorded in the same minute as the previous event (same,

41.6\%

of the evaluated events) or not (other). Same checkpoints and test sequences as Table 15 . Binned:

\log\Pr(t\in\text{bin})

at one minute (Appendix I ). Mean

\pm

std over 5 seeds. Bold : best per column; underline : best classical (Poisson or Hawkes) model; ties at the displayed precision are all marked.

NeuralSplineTPP
$K$	Taxi	StackOverflow	Retweet	Amazon	Taobao	MIMIC-IV
10	$0.72/{-}0.21$	${-}0.73/{-}1.58$	${-}3.32/{-}0.78$	$1.63/{-}1.90$	$2.66/{-}1.43$	$3.99/{-}2.49$
20	$0.73/{-}0.22$	${-}0.72/{-}1.59$	${-}3.32/{-}0.78$	$1.41/{-}1.89$	$2.70/{-}1.45$	$3.99/{-}2.61$
50	$0.71/{-}0.23$	${-}0.73/{-}1.64$	${-}3.13/{-}0.79$	$2.22/{-}1.92$	$2.72/{-}1.47$	$3.97/{-}2.82$
100	$0.70/{-}0.24$	${-}0.74/{-}1.69$	${-}2.87/{-}0.78$	$2.43/{-}1.90$	$2.73/{-}1.49$	$3.97/{-}2.89$
200	$0.67/{-}0.26$	${-}0.75/{-}1.74$	${-}2.44/{-}0.79$	$2.53/{-}1.91$	$2.72/{-}1.54$	$3.95/{-}3.20$

Appendix

Table 22: Knot-count sweep on all datasets:

\log p(t)\,/\,\log p(d\mid t)

(nats, mean over 5 seeds) for NeuralSplineTPP (head knots

K

) and Spline/Spline Hawkes (kernel knots

K

, baseline fixed at 100 knots). The sweep runs are separate from the reported runs (20,000 steps on the standard benchmarks, learning rates without per-model adjustment), so values differ from the main tables.

$K$	#params	$\log p(t)$	$\log p(d\|t)$	Acc $\|t$ (%)	#params	$\log p(t)$	$\log p(d\|t)$	Acc $\|t$ (%)
	Spline/Spline Hawkes				NeuralSplineTPP
10	148,591	$3.69_{\pm.00}$	$-2.60_{\pm.00}$	$32.5_{\pm.0}$	77,220	$3.99_{\pm.02}$	$-2.49_{\pm.11}$	$36.9_{\pm 1.3}$
20	285,481	$3.83_{\pm.00}$	$-2.61_{\pm.00}$	$32.9_{\pm.1}$	154,440	$3.99_{\pm.04}$	$-2.61_{\pm.21}$	$35.4_{\pm 2.7}$
50	696,151	$3.83_{\pm.00}$	$-2.62_{\pm.00}$	$32.8_{\pm.0}$	386,100	$3.97_{\pm.02}$	$-2.82_{\pm.11}$	$32.7_{\pm 1.4}$
100	1,380,601	$3.82_{\pm.00}$	$-2.64_{\pm.00}$	$32.8_{\pm.0}$	772,200	$3.97_{\pm.05}$	$-2.89_{\pm.37}$	$30.9_{\pm 7.4}$
200	2,749,501	$3.81_{\pm.00}$	$-2.65_{\pm.00}$	$32.7_{\pm.0}$	1,544,400	$3.95_{\pm.08}$	$-3.20_{\pm.50}$	$23.5_{\pm 10.0}$

Appendix

Table 23: Knot-count sweep on MIMIC-IV: Spline/Spline Hawkes (kernel knots

K

; baseline fixed at 100 knots) vs. NeuralSplineTPP (head knots

K

). Mean

\pm

std over 5 seeds.

\log p(t)

is flat in

K

for both families, and

\log p(d|t)

does not improve. #params: trainable parameters of the Hawkes model and of the NeuralSplineTPP head (its Transformer encoder adds 208,832).

Model	$K_{\Delta t}$	$K_{ns}$	$\log p(t)$	$\log p(d\|t)$	Acc $\|t$ (%)
Spline/Exp Hawkes	–	1	$3.39_{\pm.00}$	$-2.70_{\pm.00}$	$30.9_{\pm.0}$
+ NS kernel	–	5	$3.36_{\pm.00}$	$-2.69_{\pm.00}$	$31.1_{\pm.0}$
Spline/Spline Hawkes + NS kernel	12	3	$3.72_{\pm.00}$	$-2.63_{\pm.00}$	$32.9_{\pm.0}$
Spline/Spline Hawkes + NS kernel	12	5	$3.71_{\pm.00}$	$-2.63_{\pm.00}$	$33.0_{\pm.0}$
Spline/Spline Hawkes	20	1	$3.83_{\pm.00}$	$-2.61_{\pm.00}$	$32.9_{\pm.1}$
+ NS kernel	20	3	$3.76_{\pm.00}$	$-2.66_{\pm.00}$	$32.9_{\pm.1}$

Appendix

Table 24: Non-stationary Hawkes control on MIMIC-IV. The kernel heights of the NS variants are themselves linear splines over absolute stay time with

K_{ns}

knots (

K_{ns}=1

: the stationary model);

K_{\Delta t}

: knots of the kernel over elapsed time. Mean

\pm

std over 5 seeds. The NS runs use the configuration of Appendix D.4 with learning rate 0.05 for the NS Spline/Spline kernels. The stationary Spline/Spline rows are taken from the knot sweep (Table 23 ). At

K_{\Delta t}=12

, their nearest counterparts are

K=10

and

20

there. No non-stationary cell improves the joint log-likelihood of its nearest stationary counterpart by more than 0.002 nats, and all remain below NeuralSplineTPP (

\log p(t)=4.05

\log p(d|t)=-2.39

Model (KS / ECE)	Taxi	StackOverflow	Retweet	Amazon	Taobao	MIMIC-IV
Const./Exp Hawkes	.151 / .274	.033 / .023	.060 / .008	.267 / .045	.063 / .022	.315 / .040
Spline/Spline Hawkes	.128 / .221	.027 / .025	.111 / .021	.137 / .018	.035 / .020	.176 / .056
NeuralConstTPP	.057 / .007	.066 / .024	.048 / .021	.344 / .012	.343 / .015	.395 / .031
NeuralSplineTPP	.040 / .008	.009 / .027	.083 / .009	.025 / .015	.037 / .016	.021 / .024

Appendix

Table 25: Calibration of four representative models with a closed-form cumulative intensity (seed 0). Each cell is the time-rescaling Kolmogorov–Smirnov distance

\sup|F_{\mathrm{emp}}(u)-u|

of the probability integral transform

u_{i}=1-\exp(-\Lambda^{*}(t_{i-1},t_{i}))

(events with a zero inter-event time excluded) and the top-label expected calibration error of

p(d\mid t,\mathcal{H})

(15 bins); lower is better for both. Up to 10,000 test events per cell (MIMIC-IV: 5,000). For Spline/Spline Hawkes on StackOverflow, the evaluated checkpoint uses learning rate 0.1 instead of the adjusted rate of Table 8 .

Temporal Point Processes

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Yahya Aalaila, Gerrit Großmann, Sebastian Vollmer

German Research Center for Artificial Intelligence (DFKI), Data Science and its Applications Research Group, Kaiserslautern, Germany · Department of Computer Science, Rhineland-Palatinate Technical University of Kaiserslautern-Landau (RPTU), Kaiserslautern, Germany

Temporal Point Processes

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Jul 23, 2026

Smooth Neural Point Processes via B-Splines

Temporal point processes (TPPs) provide a general and flexible framework for modeling sequences of events in continuous time. Neural networks have been successfully employed to model TPPs in a highly expressive and data-driven way. Neural TPPs are typically trained via Maximum Likelihood Estimation (MLE) by minimizing the negative log-likelihood (NLL), which depends on both the conditional intensity function (CIF) and its integral over time, the compensator. Recent neural TPP approaches enable exact evaluation of the NLL without numerical integration. However, these methods typically model the compensator rather than the CIF directly, impose constraints on the neural network architecture, and are computationally expensive during training, as event contributions to the NLL are evaluated sequentially rather than in parallel. In this work, we propose a novel neural TPP model that directly parametrizes the CIF as a non-negative combination of B-spline basis functions, whose coefficients are predicted by a neural network. This formulation enables exact evaluation of the NLL, preserves full flexibility in the neural architecture, allows efficient parallelization during training, and naturally supports CIF smoothness regularization through the integrated squared second derivative. Experiments on both synthetic and real-world datasets show improved computational efficiency and predictive accuracy compared to the reference neural TPP baseline.

Michele Bellomo, Riccardo Ramaschi, Alberto Dolara +1

Politecnico di Milano Milan, Italy · University College London London, UK

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Feb 27, 2026

Multivariate Spatio-Temporal Neural Hawkes Processes

We propose a Multivariate Spatio-Temporal Neural Hawkes Process for modeling complex multivariate event data with spatio-temporal dynamics. The proposed model extends continuous-time neural Hawkes processes by integrating spatial information into latent state evolution through learned temporal and spatial decay dynamics, enabling flexible modeling of excitation and inhibition without predefined triggering kernels. By analyzing fitted intensity functions of deep learning-based temporal Hawkes process models, we identify a modeling gap in how fitted intensity behavior is captured beyond likelihood-based performance, which motivates the proposed spatio-temporal approach. Simulation studies show that the proposed method successfully recovers sensible temporal and spatial intensity structure in multivariate spatio-temporal point patterns, while existing temporal neural Hawkes process approach fails to do so. An application to terrorism data from Pakistan further demonstrates the proposed model's ability to capture complex spatio-temporal interaction across multiple event types.

Christopher Chukwuemeka, Hojun You, Mikyoung Jun

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Temporal Point Processes

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