Point processes are powerful tools to model user activities and have a plethora of applications in social sciences. Predicting user activities based on point processes is a central problem. However, existing works are mostly problem specific, use heuristics, or simplify the stochastic nature of point processes. In this paper, we propose a framework that provides an unbiased estimator of the probability mass function of point processes. In particular, we design a key reformulation of the prediction problem, and further derive a differential-difference equation to compute a conditional probability mass function. Our framework is applicable to general point processes and prediction tasks, and achieves superb predictive and efficiency performance in diverse real-world applications compared to state-of-arts.

Whenusedtomodelprobability distributions, theyexhibit tractable likelihoods and admit efficient learning algorithms.

Discriminative Random Walks (DRWs) are a simple yet powerful tool for semi-supervised node classification, but their theoretical foundations remain fragmentary. We revisit DRWs through the lens of information geometry, treating the family of class-specific hitting-time laws on an absorbing Markov chain as a statistical manifold. Starting from a log-linear edge-weight model, we derive closed-form expressions for the hitting-time probability mass function, its full moment hierarchy, and the observed Fisher information. The Fisher matrix of each seed node turns out to be rank-one, taking the quotient by its null space yields a low-dimensional, globally flat manifold that captures all identifiable directions of the model. Leveraging the geometry, we introduce a sensitivity score for unlabeled nodes that bounds, and in one-dimensional cases attains, the maximal first-order change in DRW betweenness under unit Fisher perturbations. The score can lead to principled strategies for active label acquisition, edge re-weighting, and explanation.

Point processes are powerful tools to model user activities and have a plethora of applications in social sciences. Predicting user activities based on point processes is a central problem. However, existing works are mostly problem specific, use heuristics, or simplify the stochastic nature of point processes. In this paper, we propose a framework that provides an unbiased estimator of the probability mass function of point processes. In particular, we design a key reformulation of the prediction problem, and further derive a differential-difference equation to compute a conditional probability mass function. Our framework is applicable to general point processes and prediction tasks, and achieves superb predictive and efficiency performance in diverse real-world applications compared to state-of-arts.

However, existing works are mostly problem specific, use heuristics, or simplify the stochastic nature of point processes.

Before proving Theorem 1, we will argue regularity. We will establish the three conditions: symmetry, shift-invariance, and monotonicity. Pr[M( q) = r] = Pr[M(Π q) = π(r)], which implies M is symmetric as desired. We first prove two lemmas. The second lemma gives a useful fact about partial sums of a non-decreasing sequence.

I ( U; O) = H (O) (10) A.2 Algorithm The proposed training algorithm of an SNN is presented in Algo.1.Algorithm 1 The proposed training algorithm of an SNN.Input: Initialized SNN; training dataset; total training epochs, I; training iterations per epoch, J .

Driving behavior big data leverages multi-sensor telematics to understand how people drive and powers applications such as risk evaluation, insurance pricing, and targeted intervention. Usage-based insurance (UBI) built on these data has become mainstream. Telematics-captured near-miss events (NMEs) provide a timely alternative to claim-based risk, but weekly NMEs are sparse, highly zero-inflated, and behaviorally heterogeneous even after exposure normalization. Analyzing multi-sensor telematics and ADAS warnings, we show that the traditional statistical models underfit the dataset. We address these challenges by proposing a set of zero-inflated Poisson (ZIP) frameworks that learn latent behavior groups and fit offset-based count models via EM to yield calibrated, interpretable weekly risk predictions. Using a naturalistic dataset from a fleet of 354 commercial drivers over a year, during which the drivers completed 287,511 trips and logged 8,142,896 km in total, our results show consistent improvements over baselines and prior telematics models, with lower AIC/BIC values in-sample and better calibration out-of-sample. We also conducted sensitivity analyses on the EM-based grouping for the number of clusters, finding that the gains were robust and interpretable. Practically, this supports context-aware ratemaking on a weekly basis and fairer premiums by recognizing heterogeneous driving styles.