Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.10971

Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models

Kevin Rupp·Rudolf Schill·Jonas Süskind·Peter Georg·Maren Klever·Andreas Lösch·Lars Grasedyck·Tilo Wettig·Rainer Spang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivation: We consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix $Q$ which depends on a parameter $\theta$. Computing the probability distribution over states at time $t$ requires the matrix exponential $\exp(tQ)$, and inferring $\theta$ from data requires its derivative $\partial\exp\!(tQ)/\partial\theta$. Both are challenging to compute when the state space and hence the size of $Q$ is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store $Q$. However, when $Q$ can be written as a sum of tensor products, computing $\exp(tQ)$ becomes feasible by the uniformization method, which does not require explicit storage of $Q$. Results: Here we provide an analogous algorithm for computing $\partial\exp\!(tQ)/\partial\theta$, the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that $Q$ can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Availability: Implementation and data are available at https://github.com/spang-lab/TenSIR.

Explore related subjects

stat.MLcs.LGq-bio.PE

Keep this discovery

BibTeXRIS

Kevin Rupp, Rudolf Schill, Jonas Süskind, Peter Georg, Maren Klever, Andreas Lösch, Lars Grasedyck, Tilo Wettig, Rainer Spang. 2021-12-21. Differentiated uniformization: A new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models. https://arxiv.org/abs/2112.10971

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Hilbert-Valued Functional Decomposition Framework for Explaining Time-Dependent Outputs

Feature-based explanations quantify features' influence on model predictions, but are primarily designed for scalar outputs. In many applications, however, outputs are functional or multivariate, such as time-dependent trajectories in demand forecasting. Consequently, existing approaches typically explain each output location independently, ignoring dependencies across the output components. We address this limitation by developing a unified framework for feature-based explanations of time-dependent outputs. Specifically, we generalize functional decomposition to Hilbert-valued prediction functions and extend an existing feature-based explanation framework to this setting. Our framework introduces kernel-based output representations that enable time-dependency-aware explanations at multiple levels of temporal granularity, including time-specific, time-resolved, and time-aggregated, while providing a unified view in which existing methods arise as special cases. We validate our framework on synthetic and real-world data, including intraday financial market volatility prediction and energy demand forecasting.

stat.ML↗
arXiv2026

Risk-Averse Decision Making with Multi-Level Reliability Guarantees

Many applications in engineering, including wireless broadcasting, require designs that provide performance certificates at different target outage levels. This paper studies the problem of maximizing the weighted average of such certificates in the presence of uncertainty about the true system state. The problem is shown to be equivalent to an optimization over nested prediction sets, connecting to the literature on conformal prediction and extending prior art on single-level risk-averse decision making. Furthermore, we derive a dual formulation that decouples optimization across input values. Numerical experiments on a diversity-based wireless transmission system illustrate the cost of enforcing multi-level certificates with a single shared policy and trace the Pareto trade-off between multiple reliability levels.

stat.ML↗
arXiv2026

A distribution-free certification framework for trustworthy crash-severity prediction

Crash-severity models inform screening, dispatch and site prioritization, yet are deployed without a finite-sample statement of what one prediction means. Off-the-shelf guarantees fail here, because the features that make crash severity distinctive defeat them: the KABCO outcome is ordinal, the recorded label is a field assessment agreeing with medical severity about half the time, erring in a structured way, and deployment crosses jurisdictions and years calibration never saw. We develop a certification layer that wraps any severity model unmodified, with distribution-free guarantees using this structure: contiguous ordinal sets that read as "B or worse"; per-class validity for any pre-declared partition, with an oracle efficiency characterization; transfer of coverage to unobserved true severity through a declared reporting band, with a worst-case sharpness result; a one-sided certificate under deployment shift; and severity-weighted risk control. The guarantees compose with an attributable slack budget. The same analysis bounds what certification can achieve. A certified set's informativeness is governed by a functional of the true law that no base model can evade and that cannot be lower-bounded distribution-free; given a declared misreporting channel identified from record-linkage data, a nonvacuous lower bound on that floor becomes computable. On 5.2 million Texas records across seven base models spanning four decades, the layer attaches identical validity and certifies, on the vulnerable road users, a model-independent floor on set width that no base model beats, separating it from a remainder that stays bounded but distribution-free unidentifiable. The framework is released as an open-source package with theorem-level tests.

stat.ML↗