arXiv subjects

David Steinsaltz

Publications and source records attributed to David Steinsaltz.

At least 19 recordsLinked to original sources

arXiv2025

A Bayesian Approach for Earthquake Impact Modelling

Immediately following a disaster event, such as an earthquake, estimates of the damage extent play a key role in informing the coordination of response and recovery efforts. We develop a novel impact estimation tool that leverages a generalised Bayesian approach to generate earthquake impact estimates across three impact types: mortality, population displacement, and building damage. Inference is performed within a likelihood-free framework, and a scoring-rule-based posterior avoids information loss from non-sufficient summary statistics. We propose an adaptation of existing scoring-rule-based loss functions that accommodates the use of an approximate Bayesian computation sequential Monte Carlo (ABC-SMC) framework. The fitted model achieves results comparable to those of two leading impact estimation tools in the prediction of total mortality when tested on a set of held-out past events. The proposed method provides four advantages over existing empirical approaches: modelling produces a gridded spatial map of the estimated impact, predictions benefit from the Bayesian quantification and interpretation of uncertainty, there is direct handling of multi-shock earthquake events, and the use of a joint model between impact types allows predictions to be updated as impact observations become available.

stat.AP↗

arXiv2025

Stable Distillation and High-Dimensional Hypothesis Testing

While powerful methods have been developed for high-dimensional hypothesis testing assuming orthogonal parameters, current approaches struggle to generalize to the more common non-orthogonal case. We propose Stable Distillation (SD), a simple paradigm for iteratively extracting independent pieces of information from observed data, assuming a parametric model. When applied to hypothesis testing for large regression models, SD orthogonalizes the effect estimates of non-orthogonal predictors by judiciously introducing noise into the observed outcomes vector, yielding mutually independent p-values across predictors. Generic regression and gene-testing simulations show that SD yields a scalable approach for non-orthogonal designs that exceeds or matches the power of existing methods against sparse alternatives. While we only present explicit SD algorithms for hypothesis testing in ordinary least squares and logistic regression, we provide general guidance for deriving and improving the power of SD procedures.

stat.ME↗

arXiv2024

The Rényi Outlier Test

Cox and Kartsonaki proposed a simple outlier test for a vector of p-values based on the Rényi transformation that is fast for large $p$ and numerically stable for very small p-values -- key properties for large data analysis. We propose and implement a generalization of this procedure we call the Rényi Outlier Test (ROT). This procedure maintains the key properties of the original but is much more robust to uncertainty in the number of outliers expected a priori among the p-values. The ROT can also account for two types of prior information that are common in modern data analysis. The first is the prior probability that a given p-value may be outlying. The second is an estimate of how far of an outlier a p-value might be, conditional on it being an outlier; in other words, an estimate of effect size. Using a series of pre-calculated spline functions, we provide a fast and numerically stable implementation of the ROT in our R package renyi.

stat.ME↗

arXiv2023

Bai-Pollicott Algorithm for Markovian Products of Positive Matrices

We consider the problem of estimating the top Lyapunov exponents for Markovian products of positive matrices. We define a new transfer operator as a matrix of classical transfer operators and prove its spectral properties. With the spectral properties, we generalize (and provide a more theoretically rigorous foundation for) an algorithm that was introduced informally by Bai and formally by Pollicott based on dynamical zeta functions.

math.PR↗

arXiv2022

Survival Regression with Proper Scoring Rules and Monotonic Neural Networks

We consider frequently used scoring rules for right-censored survival regression models such as time-dependent concordance, survival-CRPS, integrated Brier score and integrated binomial log-likelihood, and prove that neither of them is a proper scoring rule. This means that the true survival distribution may be scored worse than incorrect distributions, leading to inaccurate estimation. We prove that, in contrast to these scores, the right-censored log-likelihood is a proper scoring rule, i.e., the highest expected score is achieved by the true distribution. Despite this, modern feed-forward neural-network-based survival regression models are unable to train and validate directly on the right-censored log-likelihood, due to its intractability, and resort to the aforementioned alternatives, i.e., non-proper scoring rules. We therefore propose a simple novel survival regression method capable of directly optimizing log-likelihood using a monotonic restriction on the time-dependent weights, coined SurvivalMonotonic-net (SuMo-net). SuMo-net achieves state-of-the-art log-likelihood scores across several datasets with 20--100$\times$ computational speedup on inference over existing state-of-the-art neural methods, and is readily applicable to datasets with several million observations.

stat.ML↗

arXiv2020

A kernel- and optimal transport- based test of independence between covariates and right-censored lifetimes

We propose a nonparametric test of independence, termed optHSIC, between a covariate and a right-censored lifetime. Because the presence of censoring creates a challenge in applying the standard permutation-based testing approaches, we use optimal transport to transform the censored dataset into an uncensored one, while preserving the relevant dependencies. We then apply a permutation test using the kernel-based dependence measure as a statistic to the transformed dataset. The type 1 error is proven to be correct in the case where censoring is independent of the covariate. Experiments indicate that optHSIC has power against a much wider class of alternatives than Cox proportional hazards regression and that it has the correct type 1 control even in the challenging cases where censoring strongly depends on the covariate.

math.ST↗

arXiv2020

Regeneration-enriched Markov processes with application to Monte Carlo

We study a class of Markov processes that combine local dynamics, arising from a fixed Markov process, with regenerations arising at a state-dependent rate. We give conditions under which such processes possess a given target distribution as their invariant measures, thus making them amenable for use within Monte Carlo methodologies. Since the regeneration mechanism can compensate the choice of local dynamics, while retaining the same invariant distribution, great flexibility can be achieved in selecting local dynamics, and the mathematical analysis is simplified. We give straightforward conditions for the process to possess a central limit theorem, and additional conditions for uniform ergodicity and for a coupling from the past construction to hold, enabling exact sampling from the invariant distribution. We further consider and analyse a natural approximation of the process which may arise in the practical simulation of some classes of continuous-time dynamics.

math.PR↗

arXiv2020

Consistency of permutation tests for HSIC and dHSIC

The Hilbert--Schmidt Independence Criterion (HSIC) is a popular measure of the dependency between two random variables. The statistic dHSIC is an extension of HSIC that can be used to test joint independence of $d$ random variables. Such hypothesis testing for (joint) independence is often done using a permutation test, which compares the observed data with randomly permuted datasets. The main contribution of this work is proving that the power of such independence tests converges to 1 as the sample size converges to infinity. This answers a question that was asked in (Pfister, 2018) Additionally this work proves correct type 1 error rate of HSIC and dHSIC permutation tests and provides guidance on how to select the number of permutations one uses in practice. While correct type 1 error rate was already proved in (Pfister, 2018), we provide a modified proof following (Berrett, 2019), which extends to the case of non-continuous data. The number of permutations to use was studied e.g. by (Marozzi, 2004) but not in the context of HSIC and with a slight difference in the estimate of the $p$-value and for permutations rather than vectors of permutations. While the last two points have limited novelty we include these to give a complete overview of permutation testing in the context of HSIC and dHSIC.

math.ST↗

arXiv2020

On Transfer Operators for Markovian Products of Invertible Random Matrices

In this article we consider the Markovian products of invertible (not necessarily positive) matrices chosen from a strongly irreducible, contracting, finite set of matrices. We construct Markovian transfer operators and prove the spectral property which draws a connection between the top Lyapunov exponent associated to the random matrix product problem and the spectrum of the corresponding Markovian transfer operator.

math.PR↗

arXiv2019

Improved Concentration Bounds for Gaussian Quadratic Forms

For a wide class of monotonic functions $f$, we develop a Chernoff-style concentration inequality for quadratic forms $Q_f \sim \sum\limits_{i=1}^n f(η_i) (Z_i + δ_i)^2$, where $Z_i \sim N(0,1)$. The inequality is expressed in terms of traces that are rapid to compute, making it useful for bounding p-values in high-dimensional screening applications. The bounds we obtain are significantly tighter than those that have been previously developed, which we illustrate with numerical examples.

math.ST↗

arXiv2019

A note on the jump locations of Markov processes

For a continuous-time Markov process, we characterize the law of the first jump location when started from an arbitrary initial distribution, in terms of the invariant distribution of an auxiliary Markov process. This could be of interest in the burgeoning fields of piecewise-deterministic Markov chain Monte Carlo methods and quasi-stationary Monte Carlo methods.

math.PR↗

arXiv2019

An approximation scheme for quasi-stationary distributions of killed diffusions

In this paper we study the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is killed at a smooth rate and then regenerated at a random location, distributed according to the weighted empirical occupation measure. We show that the weighted occupation measures almost surely comprise an asymptotic pseudo-trajectory for a certain deterministic measure-valued semiflow, after suitably rescaling the time, and that with probability one they converge to the quasi-stationary distribution of the killed diffusion. These results provide theoretical justification for a scalable quasi-stationary Monte Carlo method for sampling from Bayesian posterior distributions.

math.PR↗

arXiv2018

Theoretical properties of quasi-stationary Monte Carlo methods

This paper gives foundational results for the application of quasi-stationarity to Monte Carlo inference problems. We prove natural sufficient conditions for the quasi-limiting distribution of a killed diffusion to coincide with a target density of interest. We also quantify the rate of convergence to quasi-stationarity by relating the killed diffusion to an appropriate Langevin diffusion. As an example, we consider in detail a killed Ornstein--Uhlenbeck process with Gaussian quasi-stationary distribution.

stat.ME↗

arXiv2018

Stochastic growth rates for populations in random environments with rare migration

The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and off-diagonal elements the migration rate. The randomness of the matrices then represents stochasticity of environmental conditions. We consider the case where the off-diagonal elements are small, representing a situation where migration has been introduced into an otherwise sessile meta-population. We examine the asymptotic behaviour of the long-term growth rate. When there is a single site with the highest growth rate, under the assumption of Gaussian log growth rates at the individual sites (or having Gaussian-like tails) we show that the behavior near zero is like a power of $ε$, and derive upper and lower bounds for the power in terms of the difference in the growth rates and the distance between the sites. In particular, when the difference in mean log growth rate between two sites is sufficiently small, or the variance of the difference between the sites sufficiently large, migration will always be favored by natural selection, in the sense that introducing a small amount of migration will increase the growth rate of the population relative to the zero-migration case.

q-bio.PE↗

arXiv2018

Stability of fixed life histories to perturbation by rare diapause

We follow up on a companion work that considered growth rates of populations growing at different sites, with different randomly varying growth rates at each site, in the limit as migration between sites goes to 0. We extend this work here to the special case where the maximum average log growth rate is achieved at two different sites. The primary motivation is to cover the case where `sites' are understood as age classes for the same individuals. The theory then calculates the effect on growth rate of introducing a rare delay in development, a diapause, into an otherwise fixed-length semelparous life history. Whereas the increase in stochastic growth rate due to rare migrations was found to grow as a power of the migration rate, we show that under quite general conditions that in the diapause model --- or in the migration model with two or more sites having equal individual stochastic growth rates --- the increase in stochastic growth rate due to diapause at rate $\epsilon$ behaves like $(\log \epsilon^{-1})^{-1}$ as $\epsilon\downarrow 0$. In particular, this implies that a small random disruption to the deterministic life history will always be favored by natural selection, in the sense that it will increase the stochastic growth rate relative to the zero-delay deterministic life history.

q-bio.PE↗

arXiv2015

Stochastic growth rates for life histories with rare migration or diapause

The growth of a population divided among spatial sites, with migration between the sites, is sometimes modelled by a product of random matrices, with each diagonal elements representing the growth rate in a given time period, and off-diagonal elements the migration rate. If the sites are reinterpreted as age classes, the same model may apply to a single population with age-dependent mortality and reproduction. We consider the case where the off-diagonal elements are small, representing a situation where there is little migration or, alternatively, where a deterministic life-history has been slightly disrupted, for example by introducing a rare delay in development. We examine the asymptotic behaviour of the long-term growth rate. We show that when the highest growth rate is attained at two different sites in the absence of migration (which is always the case when modelling a single age-structured population) the increase in stochastic growth rate due to a migration rate $ε$ is like $(\log ε^{-1})^{-1}$ as $ε\downarrow 0$, under fairly generic conditions. When there is a single site with the highest growth rate the behavior is more delicate, depending on the tails of the growth rates. For the case when the log growth rates have Gaussian-like tails we show that the behavior near zero is like a power of $ε$, and derive upper and lower bounds for the power in terms of the difference in the growth rates and the distance between the sites.

q-bio.PE↗

arXiv2012

Quasilimiting behavior for one-dimensional diffusions with killing

This paper extends and clarifies results of Steinsaltz and Evans [Trans. Amer. Math. Soc. 359 (2007) 1285-1234], which found conditions for convergence of a killed one-dimensional diffusion conditioned on survival, to a quasistationary distribution whose density is given by the principal eigenfunction of the generator. Under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable. When the killing at infinity is larger than the principal eigenvalue, then the eigenfunction is always integrable. When the killing at infinity is smaller, the eigenfunction is integrable only when the unkilled process is recurrent; otherwise, the process conditioned on survival converges to 0 density on any bounded interval.

math.PR↗

arXiv2011

A mutation-selection model for general genotypes with recombination

We investigate a continuous time, probability measure-valued dynamical system that describes the process of mutation-selection balance in a context where the population is infinite, there may be infinitely many loci, and there are weak assumptions on selective costs. Our model arises when we incorporate very general recombination mechanisms into a previous model of mutation and selection from Steinsaltz, Evans and Wachter (2005) and take the relative strength of mutation and selection to be sufficiently small. The resulting dynamical system is a flow of measures on the space of loci. Each such measure is the intensity measure of a Poisson random measure on the space of loci: the points of a realization of the random measure record the set of loci at which the genotype of a uniformly chosen individual differs from a reference wild type due to an accumulation of ancestral mutations. Our motivation for working in such a general setting is to provide a basis for understanding mutation-driven changes in age-specific demographic schedules that arise from the complex interaction of many genes, and hence to develop a framework for understanding the evolution of aging. We establish the existence and uniqueness of the dynamical system, provide conditions for the existence and stability of equilibrium states, and prove that our continuous-time dynamical system is the limit of a sequence of discrete-time infinite population mutation-selection-recombination models in the standard asymptotic regime where selection and mutation are weak relative to recombination and both scale at the same infinitesimal rate in the limit.

q-bio.PE↗