arXiv subjects

Sandra Fortini

Publications and source records attributed to Sandra Fortini.

At least 19 recordsLinked to original sources

arXiv2026

Merging of Bayes and quasi-Bayes empirical Bayes procedures for Poisson compound decisions

The Poisson compound decision problem is a long-standing problem in statistics, in which empirical Bayes methods are used to estimate Poisson means under a mixture model. We study this problem from the viewpoint of $g$-modeling, comparing two nonparametric strategies for estimating the unknown mixing distribution: a Bayesian empirical Bayes strategy, based on the Dirichlet process posterior, and a quasi-Bayesian empirical Bayes strategy, based on Newton's algorithm. The latter is computationally attractive, but its relationship with the Bayesian strategy requires theoretical justification. Under a Poisson mixture model with a ``true'', or oracle, mixing distribution, we establish concentration rates for the marginal probability mass functions induced by the Bayesian and quasi-Bayesian estimates. These rates are then translated into rates of decay for the corresponding regrets, interpreted as excess Bayes risks, and used to prove a frequentist merging result between the Bayesian and quasi-Bayesian empirical Bayes strategies. We also extend the analysis to the multidimensional Poisson compound decision problem. Numerical experiments on synthetic data illustrate that the quasi-Bayesian strategy achieves accuracy comparable to the Bayesian strategy, while requiring substantially fewer computational resources, especially in the multidimensional setting.

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arXiv2026

Quasi-Bayes empirical Bayes estimation of sums of random variables

The estimation of sums of functions of observable and unobservable variables is a long-standing problem in statistics with applications across many domains. Empirical Bayes methods provide a natural framework for this task under mixture models, but existing approaches often rely on restrictive parametric assumptions or apply only to limited classes of functionals in nonparametric settings. We propose a nonparametric methodology, referred to as quasi-Bayes empirical Bayes, that addresses these limitations through a recursive estimation of the mixing distribution based on Newton's algorithm. The resulting plug-in estimate of the target sum is computationally efficient, scalable, and applicable to a broad class of utility functions, while enabling uncertainty quantification via asymptotic credible intervals derived from a Gaussian central limit theorem. We establish large sample asymptotic theoretical guarantees by proving a merging between the quasi-Bayes and Bayes estimates and by showing consistency under a correctly specified frequentist model. Synthetic-data and real-data analyses demonstrate the practical accuracy and stability of the method, with performance comparable to, and in some cases better than, existing empirical Bayes procedures.

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arXiv2026

Uncertainty Decomposition for Bayes-Filtered Transformers via Bayesian Predictive Inference

Bayes-filtered transformers are transformers meta-learned on sequences from a prior predictive distribution to approximate the corresponding posterior predictive distribution. They output total predictive uncertainty in a single forward pass but never explicitly represent a posterior distribution, making the standard route to separating aleatoric from epistemic uncertainty unavailable. We address this challenge through the lens of Bayesian predictive inference (BPI). Our main result is a predictive Central Limit Theorem (CLT) for supervised settings under conditions that are among the weakest known in the BPI literature. The CLT characterises the posterior of the limiting predictive distribution given an observed context as asymptotically Gaussian; the variance of this Gaussian quantifies epistemic uncertainty. We apply the framework to TabPFN, a Bayes-filtered transformer that is a state-of-the-art foundation model for tabular prediction. The resulting credible bands achieve near-nominal frequentist coverage as context length grows, and the decomposition largely matches standard desiderata: epistemic uncertainty shrinks with context length and is highest in sparsely observed regions within the span of the context data, while aleatoric uncertainty dominates near decision boundaries where classes overlap.

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arXiv2025

Leveraging External Data for Testing Experimental Therapies with Biomarker Interactions in Randomized Clinical Trials

In oncology the efficacy of novel therapeutics often differs across patient subgroups, and these variations are difficult to predict during the initial phases of the drug development process. The relation between the power of randomized clinical trials and heterogeneous treatment effects has been discussed by several authors. In particular, false negative results are likely to occur when the treatment effects concentrate in a subpopulation but the study design did not account for potential heterogeneous treatment effects. The use of external data from completed clinical studies and electronic health records has the potential to improve decision-making throughout the development of new therapeutics, from early-stage trials to registration. Here we discuss the use of external data to evaluate experimental treatments with potential heterogeneous treatment effects. We introduce a permutation procedure to test, at the completion of a randomized clinical trial, the null hypothesis that the experimental therapy does not improve the primary outcomes in any subpopulation. The permutation test leverages the available external data to increase power. Also, the procedure controls the false positive rate at the desired $\alpha$-level without restrictive assumptions on the external data, for example, in scenarios with unmeasured confounders, different pre-treatment patient profiles in the trial population compared to the external data, and other discrepancies between the trial and the external data. We illustrate that the permutation test is optimal according to an interpretable criteria and discuss examples based on asymptotic results and simulations, followed by a retrospective analysis of individual patient-level data from a collection of glioblastoma clinical trials.

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arXiv2024

Quasi-Bayes empirical Bayes: a sequential approach to the Poisson compound decision problem

The Poisson compound decision problem is a long-standing problem is statistics, for which empirical Bayes methods are commonly used to estimate Poisson means in static or batch settings. We consider this problem in a streaming, or online, framework. Building on a quasi-Bayesian approach based on Newton's algorithm, we develop a sequential estimate that is easy to evaluate, computationally efficient, and has constant per-observation cost as the data accrue. We establish frequentist guarantees for the proposed estimate, including consistency and asymptotic optimality, with optimality understood as vanishing excess Bayes risk, or regret. Empirical performance is assessed through simulation studies and comparisons with benchmark procedures.

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arXiv2024

Quasi-Bayesian sequential deconvolution

Density deconvolution is the inverse problem of estimating a probability density from observations contaminated by additive noise. Traditionally studied in static or batch settings, it increasingly arises with streaming data, where existing frequentist and Bayesian procedures face substantial computational bottlenecks. We develop a quasi-Bayesian nonparametric method for sequential density deconvolution based on Newton's recursive algorithm. The resulting estimate is straightforward to evaluate and scalable to massive datasets, as its per-observation computational cost remains constant as new data arrive. The quasi-Bayesian interpretation enables uncertainty quantification: local and uniform central limit theorems yield asymptotic credible intervals and bands, respectively. Under a frequentist data-generating model, we establish $L^1$-consistency for the proposed estimate and show that it asymptotically agrees with the estimate that would be obtained if the uncontaminated variables were directly observed. Further, under additional regularity conditions, we derive an $L^1$-Wasserstein convergence rate and establish merging, at an explicit rate, with the Bayesian nonparametric posterior mean estimate under a Dirichlet process mixture model. Synthetic-data experiments, together with an acquisition-ordered flow-cytometry application, demonstrate accuracy comparable to Bayesian nonparametric and Fourier deconvolution methods, while offering a substantial computational advantage.

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arXiv2024

Exchangeability, prediction and predictive modeling in Bayesian statistics

There is currently a renewed interest in the Bayesian predictive approach to statistics. This paper offers a review on foundational concepts and focuses on predictive modeling, which by directly reasoning on prediction, bypasses inferential models or may characterize them. We detail predictive characterizations in exchangeable and partially exchangeable settings, for a large variety of data structures, and hint at new directions. The underlying concept is that Bayesian predictive rules are probabilistic learning rules, formalizing through conditional probability how we learn on future events given the available information. This concept has implications in any statistical problem and in inference, from classic contexts to less explored challenges, such as providing Bayesian uncertainty quantification to predictive algorithms in data science, as we show in the last part of the paper. The paper gives a historical overview, but also includes a few new results, presents some recent developments and poses some open questions.

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arXiv2023

Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions

There is a growing literature on the study of large-width properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed parameters or weights, and Gaussian stochastic processes. Motivated by some empirical and theoretical studies showing the potential of replacing Gaussian distributions with Stable distributions, namely distributions with heavy tails, in this paper we investigate large-width properties of deep Stable NNs, i.e. deep NNs with Stable-distributed parameters. For sub-linear activation functions, a recent work has characterized the infinitely wide limit of a suitable rescaled deep Stable NN in terms of a Stable stochastic process, both under the assumption of a ``joint growth" and under the assumption of a ``sequential growth" of the width over the NN's layers. Here, assuming a ``sequential growth" of the width, we extend such a characterization to a general class of activation functions, which includes sub-linear, asymptotically linear and super-linear functions. As a novelty with respect to previous works, our results rely on the use of a generalized central limit theorem for heavy tails distributions, which allows for an interesting unified treatment of infinitely wide limits for deep Stable NNs. Our study shows that the scaling of Stable NNs and the stability of their infinitely wide limits may depend on the choice of the activation function, bringing out a critical difference with respect to the Gaussian setting.

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arXiv2023

Non-asymptotic approximations of Gaussian neural networks via second-order Poincar\'e inequalities

There is a recent and growing literature on large-width asymptotic and non-asymptotic properties of deep Gaussian neural networks (NNs), namely NNs with weights initialized as Gaussian distributions. For a Gaussian NN of depth $L\geq1$ and width $n\geq1$, it is well-known that, as $n\rightarrow+\infty$, the NN's output converges (in distribution) to a Gaussian process. Recently, some quantitative versions of this result, also known as quantitative central limit theorems (QCLTs), have been obtained, showing that the rate of convergence is $n^{-1}$, in the $2$-Wasserstein distance, and that such a rate is optimal. In this paper, we investigate the use of second-order Poincar\'e inequalities as an alternative approach to establish QCLTs for the NN's output. Previous approaches consist of a careful analysis of the NN, by combining non-trivial probabilistic tools with ad-hoc techniques that rely on the recursive definition of the network, typically by means of an induction argument over the layers, and it is unclear if and how they still apply to other NN's architectures. Instead, the use of second-order Poincar\'e inequalities rely only on the fact that the NN is a functional of a Gaussian process, reducing the problem of establishing QCLTs to the algebraic problem of computing the gradient and Hessian of the NN's output, which still applies to other NN's architectures. We show how our approach is effective in establishing QCLTs for the NN's output, though it leads to suboptimal rates of convergence. We argue that such a worsening in the rates is peculiar to second-order Poincar\'e inequalities, and it should be interpreted as the "cost" for having a straightforward, and general, procedure for obtaining QCLTs.

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arXiv2022

Large-width asymptotics for ReLU neural networks with $\alpha$-Stable initializations

There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized as Gaussian distributions. Two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. In this paper, we consider these problems for $\alpha$-Stable NNs, namely NNs whose weights are initialized as $\alpha$-Stable distributions with $\alpha\in(0,2]$. First, for $\alpha$-Stable NNs with a ReLU activation function, we show that if the NN's width goes to infinity then a rescaled NN converges weakly to an $\alpha$-Stable stochastic process. As a difference with respect to the Gaussian setting, our result shows that the choice of the activation function affects the scaling of the NN, that is: to achieve the infinitely wide $\alpha$-Stable process, the ReLU activation requires an additional logarithmic term in the scaling with respect to sub-linear activations. Then, we study the large-width training dynamics of $\alpha$-Stable ReLU-NNs, characterizing the infinitely wide dynamics in terms of a random kernel, referred to as the $\alpha$-Stable NTK, and showing that, for a sufficiently large width, the gradient descent achieves zero training error at a linear rate. The randomness of the $\alpha$-Stable NTK is a further difference with respect to the Gaussian setting, that is: within the $\alpha$-Stable setting, the randomness of the NN at initialization does not vanish in the large-width regime of the training.

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arXiv2021

Deep Stable neural networks: large-width asymptotics and convergence rates

In modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotic properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and Gaussian stochastic processes (SPs). Such an interplay has proved to be critical in Bayesian inference under Gaussian SP priors, kernel regression for infinitely wide deep NNs trained via gradient descent, and information propagation within infinitely wide NNs. Motivated by empirical analyses that show the potential of replacing Gaussian distributions with Stable distributions for the NN's weights, in this paper we present a rigorous analysis of the large-width asymptotic behaviour of (fully connected) feed-forward deep Stable NNs, i.e. deep NNs with Stable-distributed weights. We show that as the width goes to infinity jointly over the NN's layers, i.e. the ``joint growth" setting, a rescaled deep Stable NN converges weakly to a Stable SP whose distribution is characterized recursively through the NN's layers. Because of the non-triangular structure of the NN, this is a non-standard asymptotic problem, to which we propose an inductive approach of independent interest. Then, we establish sup-norm convergence rates of the rescaled deep Stable NN to the Stable SP, under the ``joint growth" and a ``sequential growth" of the width over the NN's layers. Such a result provides the difference between the ``joint growth" and the ``sequential growth" settings, showing that the former leads to a slower rate than the latter, depending on the depth of the layer and the number of inputs of the NN. Our work extends some recent results on infinitely wide limits for deep Gaussian NNs to the more general deep Stable NNs, providing the first result on convergence rates in the ``joint growth" setting.

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arXiv2021

Infinite-color randomly reinforced urns with dominant colors

We define and prove limit results for a class of dominant P\'olya sequences, which are randomly reinforced urn processes with color-specific random weights and unbounded number of possible colors. Under fairly mild assumptions on the expected reinforcement, we show that the predictive and the empirical distributions converge almost surely (a.s.) in total variation to the same random probability measure $\tilde{P}$; moreover, $\tilde{P}(\mathcal{D})=1$ a.s., where $\mathcal{D}$ denotes the set of dominant colors for which the expected reinforcement is maximum. In the general case, the predictive probabilities and the empirical frequencies of any $\delta$-neighborhood of $\mathcal{D}$ converge a.s. to one. That is, although non-dominant colors continue to be regularly observed, their distance to $\mathcal{D}$ converges in probability to zero. We refine the above results with rates of convergence. We further hint potential applications of dominant P\'olya sequences in randomized clinical trials and species sampling, and use our central limit results for Bayesian inference.

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arXiv2021

Approximating the Operating Characteristics of Bayesian Uncertainty Directed Trial Designs

Bayesian response adaptive clinical trials are currently evaluating experimental therapies for several diseases. Adaptive decisions, such as pre-planned variations of the randomization probabilities, attempt to accelerate the development of new treatments. The design of response adaptive trials, in most cases, requires time consuming simulation studies to describe operating characteristics, such as type I/II error rates, across plausible scenarios. We investigate large sample approximations of pivotal operating characteristics in Bayesian Uncertainty directed trial Designs (BUDs). A BUD trial utilizes an explicit metric u to quantify the information accrued during the study on parameters of interest, for example the treatment effects. The randomization probabilities vary during time to minimize the uncertainty summary u at completion of the study. We provide an asymptotic analysis (i) of the allocation of patients to treatment arms and (ii) of the randomization probabilities. For BUDs with outcome distributions belonging to the natural exponential family with quadratic variance function, we illustrate the asymptotic normality of the number of patients assigned to each arm and of the randomization probabilities. We use these results to approximate relevant operating characteristics such as the power of the BUD. We evaluate the accuracy of the approximations through simulations under several scenarios for binary, time-to-event and continuous outcome models.

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arXiv2021

Large-width functional asymptotics for deep Gaussian neural networks

In this paper, we consider fully connected feed-forward deep neural networks where weights and biases are independent and identically distributed according to Gaussian distributions. Extending previous results (Matthews et al., 2018a;b; Yang, 2019) we adopt a function-space perspective, i.e. we look at neural networks as infinite-dimensional random elements on the input space $\mathbb{R}^I$. Under suitable assumptions on the activation function we show that: i) a network defines a continuous Gaussian process on the input space $\mathbb{R}^I$; ii) a network with re-scaled weights converges weakly to a continuous Gaussian process in the large-width limit; iii) the limiting Gaussian process has almost surely locally $\gamma$-H\"older continuous paths, for $0 < \gamma <1$. Our results contribute to recent theoretical studies on the interplay between infinitely wide deep neural networks and Gaussian processes by establishing weak convergence in function-space with respect to a stronger metric.

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arXiv2021

Infinite-channel deep stable convolutional neural networks

The interplay between infinite-width neural networks (NNs) and classes of Gaussian processes (GPs) is well known since the seminal work of Neal (1996). While numerous theoretical refinements have been proposed in the recent years, the interplay between NNs and GPs relies on two critical distributional assumptions on the NN's parameters: A1) finite variance; A2) independent and identical distribution (iid). In this paper, we consider the problem of removing A1 in the general context of deep feed-forward convolutional NNs. In particular, we assume iid parameters distributed according to a stable distribution and we show that the infinite-channel limit of a deep feed-forward convolutional NNs, under suitable scaling, is a stochastic process with multivariate stable finite-dimensional distributions. Such a limiting distribution is then characterized through an explicit backward recursion for its parameters over the layers. Our contribution extends results of Favaro et al. (2020) to convolutional architectures, and it paves the way to expand exciting recent lines of research that rely on classes of GP limits.

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arXiv2020

Stable behaviour of infinitely wide deep neural networks

We consider fully connected feed-forward deep neural networks (NNs) where weights and biases are independent and identically distributed as symmetric centered stable distributions. Then, we show that the infinite wide limit of the NN, under suitable scaling on the weights, is a stochastic process whose finite-dimensional distributions are multivariate stable distributions. The limiting process is referred to as the stable process, and it generalizes the class of Gaussian processes recently obtained as infinite wide limits of NNs (Matthews at al., 2018b). Parameters of the stable process can be computed via an explicit recursion over the layers of the network. Our result contributes to the theory of fully connected feed-forward deep NNs, and it paves the way to expand recent lines of research that rely on Gaussian infinite wide limits.

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arXiv2019

Quasi-Bayes properties of a recursive procedure for mixtures

Bayesian methods are often optimal, yet increasing pressure for fast computations, especially with streaming data, brings renewed interest in faster, possibly sub-optimal, solutions. The extent to which these algorithms approximate Bayesian solutions is a question of interest, but often unanswered. We propose a methodology to address this question in predictive settings, when the algorithm can be reinterpreted as a probabilistic predictive rule. We specifically develop the proposed methodology for a recursive procedure for online learning in nonparametric mixture models, often refereed to as Newton's algorithm. This algorithm is simple and fast; however, its approximation properties are unclear. By reinterpreting it as a predictive rule, we can show that it underlies a statistical model which is, asymptotically, a Bayesian, exchangeable mixture model. In this sense, the recursive rule provides a quasi-Bayes solution. While the algorithm only offers a point estimate, our clean statistical formulation allows us to provide the asymptotic posterior distribution and asymptotic credible intervals for the mixing distribution. Moreover, it gives insights for tuning the parameters, as we illustrate in simulation studies, and paves the way to extensions in various directions. Beyond mixture models, our approach can be applied to other predictive algorithms.

math.ST↗

arXiv2016

On a notion of partially conditionally identically distributed sequences

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence that is weaker than exchangeability, but is equivalent to exchangeability for stationary sequences. In this article we extend this notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, that is equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies in the foundations of Bayesian statistics. Natural examples are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.

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