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Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process

This paper proposes a computationally efficient, likelihood-free method using fully connected neural networks trained on simulated data to estimate parameters of discretely observed Hawkes processes, achieving accuracy comparable to existing approximate likelihood estimators while significantly reducing computational time and enabling fast uncertainty quantification.

Original authors: Jason J. Lambe, Feng Chen, Tom Stindl, Tsz-Kit Jeffrey Kwan

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jason J. Lambe, Feng Chen, Tom Stindl, Tsz-Kit Jeffrey Kwan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Blind Count"

Imagine you are trying to understand a very busy coffee shop. You want to know exactly when every single customer walked through the door and how many new customers arrived because of the buzz of the previous ones (self-excitation). This is what a "Hawkes Process" models: events that trigger more events, like earthquakes triggering aftershocks or a viral tweet sparking more shares.

Usually, to understand the rules of this coffee shop, you need a perfect video recording of every arrival time. But in the real world, you often only have a daily logbook. You know that 15 people came between 9:00 and 10:00, and 20 came between 10:00 and 11:00. You don't know exactly when they arrived, just the total count for that hour.

In statistics, this is called discrete observation or "interval censoring." The problem is that when you only have these counts, the math required to figure out the underlying rules (the parameters) becomes impossibly complex. It's like trying to solve a 1,000-piece puzzle when you've only been given the picture on the box, but the pieces are all mixed up in a bag.

The Old Solutions: Slow or Biased

Scientists have tried to fix this before, but they hit a wall:

The "Guess-and-Check" Method: Some tried to guess the missing times and then refine their guess. This is fast but often leads to the wrong answer (biased).
The "Supercomputer" Method: Others built a method that is incredibly accurate but takes so much computing power that it's like using a nuclear reactor to boil a single egg. It works, but it's too slow for real-time use.

The New Solution: The "AI Coach"

The authors of this paper propose a clever workaround. Instead of trying to solve the impossible math equation every time you get new data, they train a Neural Network (NN)—a type of artificial intelligence—to act like a seasoned coach.

Here is how they trained the coach:

Simulate the World: They created a computer simulation of the coffee shop (the Hawkes Process) thousands of times. They knew the "true rules" for each simulation.
Create the Logbook: For each simulation, they pretended they only had the daily logbook (the counts), hiding the exact arrival times.
The "Quick Estimate" Trick: To help the AI learn, they used a simple, fast trick to make a rough guess of the missing times. Think of this as the coach looking at the logbook and saying, "If 15 people came in an hour, let's pretend they arrived evenly spaced." This isn't perfect, but it's a good starting point.
Training: They fed the AI the "rough guess" and the "logbook counts" and asked it to predict the "true rules" they started with. The AI learned the pattern: "When the logbook looks like X and my rough guess is Y, the true rules are Z."

The Magic of Summary Statistics

The paper's key innovation is what they fed the AI. Instead of giving the AI the entire logbook (which could be huge), they gave it a Summary Statistic.

Imagine you are trying to guess the weather. You don't need to know the temperature of every single second of the day; you just need the average temperature, the highest temperature, and the lowest temperature.

The authors created a "summary" of the data that captures the most important clues about the rules of the process.
Because this summary is small and simple, the AI can learn it very quickly.
Once trained, the AI can look at a new logbook, calculate this simple summary, and instantly spit out the answer.

Why This is a Game Changer

The paper claims three main victories:

Speed: The AI is "amortized." This means the hard work (training) happens only once. After that, making a prediction takes a fraction of a second. It's like buying a map once and then being able to navigate any city instantly, rather than drawing a new map every time you leave the house.
Accuracy: Even though the AI was trained on "rough guesses," it performs just as well as the "Supercomputer" method (PMMH) mentioned earlier.
Uncertainty: The AI doesn't just give one answer; it gives a range. It can say, "I'm 95% sure the rule is between X and Y." This is crucial for knowing how much to trust the result.

Real-World Tests

The authors tested this method on two real-world scenarios:

Measles in Tokyo: They analyzed weekly counts of measles cases.
Salmonella in Australia: They analyzed weekly counts of Salmonella cases, accounting for seasonal changes (like how flu spikes in winter).

In both cases, the AI method produced results very similar to the most accurate existing methods but did it much faster.

The Bottom Line

This paper introduces a way to use Artificial Intelligence to solve a difficult math problem that usually requires massive computing power. By training an AI on simulated data and using a clever "summary" of the information, they created a tool that is fast, accurate, and ready to handle real-world data where we only know the totals, not the exact timing of events.

Technical Summary: Neural Networks for Parameter Estimation of the Discretely Observed Hawkes Process

Problem Statement
The Hawkes process is a self-exciting point process widely used to model event sequences in fields such as seismology, finance, and epidemiology. While parameter estimation is well-established for continuously observed data via Maximum Likelihood (ML) or Expectation Maximisation (EM), these methods become infeasible when the process is observed only discretely (interval-censored). In this setting, only the total event counts within disjoint time intervals are known, rendering the likelihood function analytically intractable.

Existing approaches to this problem face a trade-off between accuracy and computational efficiency. Approximate methods, such as those by Shlomovich et al. (2022b) and Schneider and Weber (2023), attempt to reconstruct latent event times but exhibit significant bias. Conversely, highly accurate methods like the Pseudo-Marginal Metropolis–Hastings (PMMH) algorithm (Chen et al., 2025) are computationally prohibitive, particularly for non-Markovian processes or large datasets. There is a need for an estimator that offers accuracy comparable to PMMH but with significantly reduced computational costs.

Methodology
The authors propose a likelihood-free approach utilizing a fully connected, feed-forward Neural Network (NN) to estimate Hawkes process parameters from summary statistics of count data. The methodology consists of three core components:

Summary Statistic Construction:
Instead of feeding raw count sequences into the network, the authors construct a low-dimensional summary statistic based on "naive" imputation estimates.
- Single Imputation Estimate: For the Markovian (exponential kernel) case, a latent sample path is constructed by uniformly distributing events within their observed intervals. The Maximum Likelihood Estimate (MLE) of the Hawkes parameters is then computed from this imputed path. This serves as a highly informative, minimum-dimension summary statistic.
- Negative Binomial Autoregression (NBAR): For non-Markovian (non-exponential) kernels, the imputation estimate is supplemented with NBAR parameters. An NBAR model is fitted to the observed counts to capture the dependence structure and overdispersion, providing additional information regarding the offspring density.
- Adaptability: The summary statistic is designed to handle unequally sized censoring intervals and time-varying background rates by normalizing rates and using piecewise approximations for background volumes.
Neural Network Architecture and Training:
- Input/Output: The NN takes the summary statistic as input and outputs a vector containing the posterior median (point estimate) and the upper and lower $\tau$ -quantiles for each parameter. This allows for simultaneous point estimation and Bayesian uncertainty quantification.
- Loss Function: The network is trained using a quantile loss function (pinball loss) to target specific quantiles of the posterior distribution $p(\theta | s)$ .
- Amortised Inference: The NN is trained once on a large simulated dataset (parameter-observation pairs generated from a prior). Once trained, the network performs inference on new data via a single forward pass, making the cost of inference negligible regardless of the sample path length.
Prior Distribution:
The authors utilize an "inverse softplus normal" (ISN) prior for positive parameters and a logit-normal prior for the branching ratio $\eta$ . These choices are practical, aiming to place sufficient mass near zero and within the unit interval to improve NN performance for small parameter values without significantly impacting generalizability.

Key Results
Simulation experiments demonstrate that the proposed NN estimator performs comparably to the state-of-the-art PMMH estimator in terms of accuracy (Mean Squared Error) and coverage probabilities.

Accuracy: For exponential kernels, the NN estimates exhibit minimal empirical bias, closely matching PMMH results across various aggregation levels ( $\Delta$ ).
Efficiency: The NN approach offers a dramatic reduction in computational time compared to PMMH. While PMMH requires iterative sampling and is computationally expensive, the NN provides near-instantaneous inference after the initial training phase.
Comparison to Alternatives: The NN estimator outperforms spectral methods (Whittle estimator) and biased reconstruction methods (MCEM) in both accuracy and stability.
Real-World Application: The method is applied to weekly measles data in Tokyo and Salmonella data in New South Wales. The results replicate findings from previous studies (Cheysson and Lang, 2022; Chen et al., 2025) and successfully model time-varying background rates for seasonal fluctuations.

Significance and Claims
The paper claims that the proposed method successfully bridges the gap between the high accuracy of likelihood-based methods (like PMMH) and the computational efficiency required for practical applications.

Amortised Inference: A primary contribution is the demonstration of "amortised inference," where the computational cost is shifted entirely to the training phase. This allows for the analysis of long sample paths or real-time data without retraining or additional model fitting.
Robustness to Data Constraints: The method handles unequally sized intervals and time-varying background rates, overcoming limitations of existing spectral and approximate likelihood methods.
Uncertainty Quantification: Unlike many approximate methods that lack standard error estimates, the NN provides marginal credible intervals directly through quantile estimation.
Simplicity: The summary statistic relies on simple imputation and autoregression, requiring minimal expert knowledge to design compared to complex latent path reconstruction schemes.

The authors conclude that while the PMMH algorithm remains a gold standard for accuracy, the NN estimator offers a highly scalable, fast, and accurate alternative for discretely observed Hawkes processes, particularly in settings where speed and scalability are critical.