Fabrizio Leisen
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Fabrizio Leisen has a total of 59 co-authors. Here are the top co-authors by number of shared publications:
- Cristiano Villa
University of Kent
8 shared publications - Roberto Casarin
Ca’ Foscari University of Venice, Venice, Italy
8 shared publications - Luca Rossini
University of Milan Italy
7 shared publications - Pietro Rigo
Dipartimento di Scienze Statistiche “P. Fortunati”, Università di Bologna, via delle Belle Arti 41, 40126 Bologna, Italy
6 shared publications - Emanuela Dreassi
Università di Firenze, Florence, Italy
5 shared publications - Federico Bassetti
Dipartimento di Matematica Università di Pavia Pavia Italy
5 shared publications
Latest Publications
- Open Access
- 10 August 2026
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- p. 1-27
Asymptotics of predictive distributions driven by sample means and variances
Let alpha Subscript n Baseline left parenthesis dot right parenthesis equals double struck upper P left parenthesis upper X Subscript n plus 1 Baseline element of dot vertical bar upper X 1 comma ellipsis comma upper X Subscript n Baseline right parenthesis $\alpha_n({\cdot})=\mathbb{P}\bigl(X_{n+1}\in{\cdot}\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence left parenthesis upper X 1 comma upper X 2 comma ellipsis right parenthesis $(X_1,X_2,\ldots)$ of p -dimensional random vectors. Suppose alpha Subscript n Baseline equals script upper N left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis $\alpha_n=\mathcal{N}(M_n,Q_n)$ , where upper M Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts upper X Subscript i $M_n=({1}/{n})\sum_{i=1}^nX_i$ and upper Q Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis Superscript down tack $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$ . Then there is a random probability measure alpha $\alpha$ on the Borel subsets of double struck upper R Superscript p $\mathbb{R}^p$ such that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ , where StartMetric dot EndMetric $\lVert{\cdot}\rVert$ is the total variation distance. An explicit expression for alpha $\alpha$ is provided and the convergence rate of StartMetric alpha Subscript n Baseline minus alpha EndMetric $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to n Superscript negative 1 divided by 2 $n^{-1/2}$ . Moreover, it is still true that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ even if alpha Subscript n Baseline equals script upper L left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis $\alpha_n=\mathcal{L}(M_n,Q_n)$ , where script upper L $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives alpha Subscript n $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed.- 10 August 2026
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- p. 1-27
Open AccessAsymptotics of predictive distributions driven by sample means and variances
Let alpha Subscript n Baseline left parenthesis dot right parenthesis equals double struck upper P left parenthesis upper X Subscript n plus 1 Baseline element of dot vertical bar upper X 1 comma ellipsis comma upper X Subscript n Baseline right parenthesis $\alpha_n({\cdot})=\mathbb{P}\bigl(X_{n+1}\in{\cdot}\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence left parenthesis upper X 1 comma upper X 2 comma ellipsis right parenthesis $(X_1,X_2,\ldots)$ of p -dimensional random vectors. Suppose alpha Subscript n Baseline equals script upper N left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis $\alpha_n=\mathcal{N}(M_n,Q_n)$ , where upper M Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts upper X Subscript i $M_n=({1}/{n})\sum_{i=1}^nX_i$ and upper Q Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis Superscript down tack $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$ . Then there is a random probability measure alpha $\alpha$ on the Borel subsets of double struck upper R Superscript p $\mathbb{R}^p$ such that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ , where StartMetric dot EndMetric $\lVert{\cdot}\rVert$ is the total variation distance. An explicit expression for alpha $\alpha$ is provided and the convergence rate of StartMetric alpha Subscript n Baseline minus alpha EndMetric $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to n Superscript negative 1 divided by 2 $n^{-1/2}$ . Moreover, it is still true that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ even if alpha Subscript n Baseline equals script upper L left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis $\alpha_n=\mathcal{L}(M_n,Q_n)$ , where script upper L $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives alpha Subscript n $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed. - Open Access
Weak convergence of predictive distributions
Let $(X_n)$ be a sequence of random variables with values in a standard Borel space $S$. We investigate the condition \begin{gather}\label{x56w1q} E\bigl\{f(X_{n+1})\mid X_1,\ldots,X_n\bigr\}\,\quad\text{converges in probability,}\tag{*} \\\text{as }n\rightarrow\infty,\text{ for each bounded Borel function }f:S\rightarrow\mathbb{R}.\notag \end{gather} Some consequences of \eqref{x56w1q} are highlighted and various sufficient conditions for it are obtained. In particular, \eqref{x56w1q} is characterized in terms of stable convergence. Since \eqref{x56w1q} holds whenever $(X_n)$ is conditionally identically distributed, three weak versions of the latter condition are investigated as well. For each of such versions, our main goal is proving (or disproving) that \eqref{x56w1q} holds. Several counterexamples are given.Open AccessWeak convergence of predictive distributions
Let $(X_n)$ be a sequence of random variables with values in a standard Borel space $S$. We investigate the condition \begin{gather}\label{x56w1q} E\bigl\{f(X_{n+1})\mid X_1,\ldots,X_n\bigr\}\,\quad\text{converges in probability,}\tag{*} \\\text{as }n\rightarrow\infty,\text{ for each bounded Borel function }f:S\rightarrow\mathbb{R}.\notag \end{gather} Some consequences of \eqref{x56w1q} are highlighted and various sufficient conditions for it are obtained. In particular, \eqref{x56w1q} is characterized in terms of stable convergence. Since \eqref{x56w1q} holds whenever $(X_n)$ is conditionally identically distributed, three weak versions of the latter condition are investigated as well. For each of such versions, our main goal is proving (or disproving) that \eqref{x56w1q} holds. Several counterexamples are given. - Open Access
- 1 January 2024
- Published by Institute of Mathematical Statistics in Electronic Journal of Statistics
- Vol. 18 (1) , 119-144
Generating knockoffs via conditional independence
Let X be a p-variate random vector and X˜ a knockoff copy of X (in the sense of [9]). A new approach for constructing X˜ (henceforth, NA) has been introduced in [8]. NA has essentially three advantages: (i) To build X˜ is straightforward; (ii) The joint distribution of (X,X˜) can be written in closed form; (iii) X˜ is often optimal under various criteria. However, for NA to apply, X1,…,Xp should be conditionally independent given some random element Z. Our first result is that any probability measure μ on Rp can be approximated by a probability measure μ0 of the form μ0(A1×⋯×Ap)=E{∏i=1pP(X i∈Ai∣Z)}. The approximation is in total variation distance when μ is absolutely continuous, and an explicit formula for μ0 is provided. If X∼μ0, then X1,…,Xp are conditionally independent. Hence, with a negligible error, one can assume X∼μ0 and build X˜ through NA. Our second result is a characterization of the knockoffs X˜ obtained via NA. It is shown that X˜ is of this type if and only if the pair (X,X˜) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti’s theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X˜ given X is obtained in a few cases. In one of such cases, it is assumed Xi∈{0,1} for all i.
- 1 January 2024
- Published by Institute of Mathematical Statistics in Electronic Journal of Statistics
- Vol. 18 (1) , 119-144
Open AccessGenerating knockoffs via conditional independence
Let X be a p-variate random vector and X˜ a knockoff copy of X (in the sense of [9]). A new approach for constructing X˜ (henceforth, NA) has been introduced in [8]. NA has essentially three advantages: (i) To build X˜ is straightforward; (ii) The joint distribution of (X,X˜) can be written in closed form; (iii) X˜ is often optimal under various criteria. However, for NA to apply, X1,…,Xp should be conditionally independent given some random element Z. Our first result is that any probability measure μ on Rp can be approximated by a probability measure μ0 of the form μ0(A1×⋯×Ap)=E{∏i=1pP(X i∈Ai∣Z)}. The approximation is in total variation distance when μ is absolutely continuous, and an explicit formula for μ0 is provided. If X∼μ0, then X1,…,Xp are conditionally independent. Hence, with a negligible error, one can assume X∼μ0 and build X˜ through NA. Our second result is a characterization of the knockoffs X˜ obtained via NA. It is shown that X˜ is of this type if and only if the pair (X,X˜) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti’s theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X˜ given X is obtained in a few cases. In one of such cases, it is assumed Xi∈{0,1} for all i.
- 1 January 2024
- Published by Statistica Sinica (Institute of Statistical Science) in Statistica Sinica
- Vol. 33 (4) , 2405-2429
Bayesian Predictive Inference Without a Prior
- 1 January 2024
- Published by Statistica Sinica (Institute of Statistical Science) in Statistica Sinica
- Vol. 33 (4) , 2405-2429
Bayesian Predictive Inference Without a Prior
- Open Access
Generating knockoffs via conditional independence
Let $X$ be a $p$-variate random vector and $\widetilde{X}$ a knockoff copy of $X$ (in the sense of \cite{CFJL18}). A new approach for constructing $\widetilde{X}$ (henceforth, NA) has been introduced in \cite{JSPI}. NA has essentially three advantages: (i) To build $\widetilde{X}$ is straightforward; (ii) The joint distribution of $(X,\widetilde{X})$ can be written in closed form; (iii) $\widetilde{X}$ is often optimal under various criteria. However, for NA to apply, $X_1,\ldots, X_p$ should be conditionally independent given some random element $Z$. Our first result is that any probability measure $\mu$ on $\mathbb{R}^p$ can be approximated by a probability measure $\mu_0$ of the form $$\mu_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p P(X_i\in A_i\mid Z)\Bigr\}.$$ The approximation is in total variation distance when $\mu$ is absolutely continuous, and an explicit formula for $\mu_0$ is provided. If $X\sim\mu_0$, then $X_1,\ldots,X_p$ are conditionally independent. Hence, with a negligible error, one can assume $X\sim\mu_0$ and build $\widetilde{X}$ through NA. Our second result is a characterization of the knockoffs $\widetilde{X}$ obtained via NA. It is shown that $\widetilde{X}$ is of this type if and only if the pair $(X,\widetilde{X})$ can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of $\widetilde{X}$ given $X$ is obtained in a few cases. In one of such cases, it is assumed $X_i\in\{0,1\}$ for all $i$.Open AccessGenerating knockoffs via conditional independence
Let $X$ be a $p$-variate random vector and $\widetilde{X}$ a knockoff copy of $X$ (in the sense of \cite{CFJL18}). A new approach for constructing $\widetilde{X}$ (henceforth, NA) has been introduced in \cite{JSPI}. NA has essentially three advantages: (i) To build $\widetilde{X}$ is straightforward; (ii) The joint distribution of $(X,\widetilde{X})$ can be written in closed form; (iii) $\widetilde{X}$ is often optimal under various criteria. However, for NA to apply, $X_1,\ldots, X_p$ should be conditionally independent given some random element $Z$. Our first result is that any probability measure $\mu$ on $\mathbb{R}^p$ can be approximated by a probability measure $\mu_0$ of the form $$\mu_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p P(X_i\in A_i\mid Z)\Bigr\}.$$ The approximation is in total variation distance when $\mu$ is absolutely continuous, and an explicit formula for $\mu_0$ is provided. If $X\sim\mu_0$, then $X_1,\ldots,X_p$ are conditionally independent. Hence, with a negligible error, one can assume $X\sim\mu_0$ and build $\widetilde{X}$ through NA. Our second result is a characterization of the knockoffs $\widetilde{X}$ obtained via NA. It is shown that $\widetilde{X}$ is of this type if and only if the pair $(X,\widetilde{X})$ can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of $\widetilde{X}$ given $X$ is obtained in a few cases. In one of such cases, it is assumed $X_i\in\{0,1\}$ for all $i$. - Open Access
A probabilistic view on predictive constructions for Bayesian learning
Given a sequence $X=(X_1,X_2,\ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,\ldots,X_n)$ for each $n\ge 0$. To this end, in principle, she only needs to select a collection $\sigma=(\sigma_0,\sigma_1,\ldots)$, called ``strategy" in what follows, where $\sigma_0(\cdot)=P(X_1\in\cdot)$ is the marginal distribution of $X_1$ and $\sigma_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $\sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence $X$ is determined. The strategies concern generalized P\'olya urns, random change points, covariates and stationary sequences.Open AccessA probabilistic view on predictive constructions for Bayesian learning
Given a sequence $X=(X_1,X_2,\ldots)$ of random observations, a Bayesian forecaster aims to predict $X_{n+1}$ based on $(X_1,\ldots,X_n)$ for each $n\ge 0$. To this end, in principle, she only needs to select a collection $\sigma=(\sigma_0,\sigma_1,\ldots)$, called ``strategy" in what follows, where $\sigma_0(\cdot)=P(X_1\in\cdot)$ is the marginal distribution of $X_1$ and $\sigma_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $\sigma$ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence $X$ is determined. The strategies concern generalized P\'olya urns, random change points, covariates and stationary sequences. - Open Access
Kernel based Dirichlet sequences
Let $X=(X_1,X_2,\ldots)$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose \begin{gather*} X_1\sim\nu\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{\theta\nu(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}{n+\theta}\quad\quad\text{a.s.} \end{gather*} where $\theta>0$ is a constant, $\nu$ a probability measure on $\mathcal{B}$, and $K$ a random probability measure on $\mathcal{B}$. Then, $X$ is exchangeable whenever $K$ is a regular conditional distribution for $\nu$ given any sub-$\sigma$-field of $\mathcal{B}$. Under this assumption, $X$ enjoys all the main properties of classical Dirichlet sequences, including Sethuraman's representation, conjugacy property, and convergence in total variation of predictive distributions. If $\mu$ is the weak limit of the empirical measures, conditions for $\mu$ to be a.s. discrete, or a.s. non-atomic, or $\mu\ll\nu$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.Open AccessKernel based Dirichlet sequences
Let $X=(X_1,X_2,\ldots)$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose \begin{gather*} X_1\sim\nu\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{\theta\nu(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}{n+\theta}\quad\quad\text{a.s.} \end{gather*} where $\theta>0$ is a constant, $\nu$ a probability measure on $\mathcal{B}$, and $K$ a random probability measure on $\mathcal{B}$. Then, $X$ is exchangeable whenever $K$ is a regular conditional distribution for $\nu$ given any sub-$\sigma$-field of $\mathcal{B}$. Under this assumption, $X$ enjoys all the main properties of classical Dirichlet sequences, including Sethuraman's representation, conjugacy property, and convergence in total variation of predictive distributions. If $\mu$ is the weak limit of the empirical measures, conditions for $\mu$ to be a.s. discrete, or a.s. non-atomic, or $\mu\ll\nu$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance. - 10 April 2021
- Published by Taylor & Francis in Journal of Statistical Computation and Simulation
- Vol. 91 (14) , 2899-2916
A Pólya–Gamma sampler for a generalized logistic regression
In this paper, we introduce a novel Bayesian data augmentation approach for estimating the parameters of the generalized logistic regression model. We propose a Pólya–Gamma sampler algorithm that allows us to sample from the exact posterior distribution, rather than relying on approximations. A simulation study illustrates the flexibility and accuracy of the proposed approach to capture heavy and light tails in binary response data of different dimensions. The algorithm performance is tested on simulated data. Furthermore, the methodology is applied to two different real datasets, where we demonstrate that the Pólya–Gamma sampler provides more precise estimates than the empirical likelihood method, outperforming approximate approaches.
- 10 April 2021
- Published by Taylor & Francis in Journal of Statistical Computation and Simulation
- Vol. 91 (14) , 2899-2916
A Pólya–Gamma sampler for a generalized logistic regression
In this paper, we introduce a novel Bayesian data augmentation approach for estimating the parameters of the generalized logistic regression model. We propose a Pólya–Gamma sampler algorithm that allows us to sample from the exact posterior distribution, rather than relying on approximations. A simulation study illustrates the flexibility and accuracy of the proposed approach to capture heavy and light tails in binary response data of different dimensions. The algorithm performance is tested on simulated data. Furthermore, the methodology is applied to two different real datasets, where we demonstrate that the Pólya–Gamma sampler provides more precise estimates than the empirical likelihood method, outperforming approximate approaches.
- 10 February 2021
- Published by Taylor & Francis in Journal of the American Statistical Association
- Vol. 117 (539) , 1530-1539
Survival Regression Models With Dependent Bayesian Nonparametric Priors
We present a novel Bayesian nonparametric model for regression in survival analysis. Our model builds on the classical neutral to the right model of Doksum (1974) and on the Cox proportional hazards model of Kim and Lee (2003). The use of a vector of dependent Bayesian nonparametric priors allows us to efficiently model the hazard as a function of covariates whilst allowing nonproportionality. The model can be seen as having competing latent risks. We characterize the posterior of the underlying dependent vector of completely random measures and study the asymptotic behavior of the model. We show how an MCMC scheme can provide Bayesian inference for posterior means and credible intervals. The method is illustrated using simulated and real data.
- 10 February 2021
- Published by Taylor & Francis in Journal of the American Statistical Association
- Vol. 117 (539) , 1530-1539
Survival Regression Models With Dependent Bayesian Nonparametric Priors
We present a novel Bayesian nonparametric model for regression in survival analysis. Our model builds on the classical neutral to the right model of Doksum (1974) and on the Cox proportional hazards model of Kim and Lee (2003). The use of a vector of dependent Bayesian nonparametric priors allows us to efficiently model the hazard as a function of covariates whilst allowing nonproportionality. The model can be seen as having competing latent risks. We characterize the posterior of the underlying dependent vector of completely random measures and study the asymptotic behavior of the model. We show how an MCMC scheme can provide Bayesian inference for posterior means and credible intervals. The method is illustrated using simulated and real data.
- 1 December 2020
- Published by Wiley in Australian & New Zealand Journal of Statistics
- Vol. 62 (4) , 444-466
A loss-based prior for Gaussian graphical models
- 1 December 2020
- Published by Wiley in Australian & New Zealand Journal of Statistics
- Vol. 62 (4) , 444-466
A loss-based prior for Gaussian graphical models