arXiv subjects
Publications and source records attributed to Luca Pratelli.
Two misunderstandings, frequently arising in Bayesian predictive inference, are discussed. The first deals with the data generating mechanism, while the second consists in overestimating the role played by asymptotic exchangeability. Some consequences of such misunderstandings are highlighted through examples.
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.
Let $(\mathcal{X},\mathcal{F},\mu)$ and $(\mathcal{Y},\mathcal{G},\nu)$ be probability spaces and $(Z_n)$ a sequence of random variables with values in $(\mathcal{X}\times\mathcal{Y},\,\mathcal{F}\otimes\mathcal{G})$. Let $\Gamma(\mu,\nu)$ be the collection of all probability measures $p$ on $\mathcal{F}\otimes\mathcal{G}$ such that $$p\bigl(A\times\mathcal{Y}\bigr)=\mu(A)\quad\text{and}\quad p\bigl(\mathcal{X}\times B\bigr)=\nu(B)\quad\text{for all }A\in\mathcal{F}\text{ and }B\in\mathcal{G}.$$ In this paper, we build some probability measures $\Pi$ on $\Gamma(\mu,\nu)$. In addition, for each such $\Pi$, we assume that $(Z_n)$ is exchangeable with de Finetti's measure $\Pi$ and we evaluate the conditional distribution $\Pi(\cdot\mid Z_1,\ldots,Z_n)$. In Bayesian nonparametrics, if $(Z_1,\ldots, Z_n)$ are the available data, $\Pi$ and $\Pi(\cdot\mid Z_1,\ldots, Z_n)$ can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals $\mu$ and $\nu$. Finally, analogous results are obtained for the set $\Gamma(\mu)$ of those probability measures on $\mathcal{F}\otimes\mathcal{G}$ with marginal $\mu$ on $\mathcal{F}$ (but arbitrary marginal on $\mathcal{G}$). That is, we introduce some priors on $\Gamma(\mu)$ and we evaluate the corresponding posteriors.
Let $X=(X_1,\ldots,X_p)$ be a $p$-variate random vector and $F$ a fixed finite set. In a number of applications, mainly in genetics, it turns out that $X_i\in F$ for each $i=1,\ldots,p$. Despite the latter fact, to obtain a knockoff $\widetilde{X}$ (in the sense of \cite{CFJL18}), $X$ is usually modeled as an absolutely continuous random vector. While comprehensible from the point of view of applications, this approximate procedure does not make sense theoretically, since $X$ is supported by the finite set $F^p$. In this paper, explicit formulae for the joint distribution of $(X,\widetilde{X})$ are provided when $P(X\in F^p)=1$ and $X$ is exchangeable or partially exchangeable. In fact, when $X_i\in F$ for all $i$, there seem to be various reasons for assuming $X$ exchangeable or partially exchangeable. The robustness of $\widetilde{X}$, with respect to the de Finetti's measure $\pi$ of $X$, is investigated as well. Let $\mathcal{L}_\pi(\widetilde{X}\mid X=x)$ denote the conditional distribution of $\widetilde{X}$, given $X=x$, when the de Finetti's measure is $\pi$. It is shown that $$\norm{\mathcal{L}_{\pi_1}(\widetilde{X}\mid X=x)-\mathcal{L}_{\pi_2}(\widetilde{X}\mid X=x)}\le c(x)\,\norm{\pi_1-\pi_2}$$ where $\norm{\cdot}$ is total variation distance and $c(x)$ a suitable constant. Finally, a numerical experiment is performed. Overall, the knockoffs of this paper outperform the alternatives (i.e., the knockoffs obtained by giving $X$ an absolutely continuous distribution) as regards the false discovery rate but are slightly weaker in terms of power.
Many flexible families of positive random variables exhibit non-closed forms of the density and distribution functions and this feature is considered unappealing for modelling purposes. However, such families are often characterized by a simple expression of the corresponding Laplace transform. Relying on the Laplace transform, we propose to carry out parameter estimation and goodness-of-fit testing for a general class of non-standard laws. We suggest a novel data-driven inferential technique, providing parameter estimators and goodness-of-fit tests, whose large-sample properties are derived. The implementation of the method is specifically considered for the positive stable and Tweedie distributions. A Monte Carlo study shows good finite-sample performance of the proposed technique for such laws.
In Basili and Pratelli (2024), a novel and coherent concept of interval probability measures has been introduced, providing a method for representing imprecise probabilities and uncertainty. Within the framework of set algebra, we introduced the concepts of weak complementation and interval probability measures associated with a family of random variables, which effectively capture the inherent uncertainty in any event. This paper conducts a comprehensive analysis of these concepts within a specific probability space. Additionally, we elaborate on an updating rule for events, integrating essential concepts of statistical independence, dependence, and stochastic dominance.
Let $\alpha_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence $(X_1,X_2,\ldots)$ of $p$-dimensional random vectors. Suppose $$\alpha_n= \mathcal{N} _p (M_n,Q_n)$$ where $M_n=\frac{1}{n}\sum_{i=1}^nX_i$ and $Q_n=\frac{1}{n}\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^t$. Then, there is a random probability measure $\alpha$ on the Borel subsets of $\mathbb{R}^p$ such that $\lVert\alpha_n-\alpha\rVert\overset{a.s.}\longrightarrow 0$ where $\lVert\cdot\rVert$ is total variation distance. An explicit expression for $\alpha$ is provided and the convergence rate of $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to $n^{-1/2}$. Moreover, it is still true that $\lVert\alpha_n-\alpha\rVert\overset{a.s.}\longrightarrow 0$ even if $\alpha_n=\mathcal{L}(M_n,Q_n)$ where $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions (introduced in [13]) is investigated and a numerical experiment is performed.
This paper introduces a novel concept of interval probability measures that enables the representation of imprecise probabilities, or uncertainty, in a natural and coherent manner. Within an algebra of sets, we introduce a notion of weak complementation denoted as $\psi$. The interval probability measure of an event $H$ is defined with respect to the set of indecisive eventualities $(\psi(H))^c$, which is included in the standard complement $H^c$. We characterize a broad class of interval probability measures and define their properties. Additionally, we establish an updating rule with respect to $H$, incorporating concepts of statistical independence and dependence. The interval distribution of a random variable is formulated, and a corresponding definition of stochastic dominance between two random variables is introduced. As a byproduct, a formal solution to the century-old Keynes-Ramsey controversy is presented.
Let $(\Omega,\mathcal{F})$ be a standard Borel space and $\mathcal{P}(\mathcal{F})$ the collection of all probability measures on $\mathcal{F}$. Let $E\subset\Omega\times\Omega$ be a measurable equivalence relation, that is, $E\in\mathcal{F}\otimes\mathcal{F}$ and the relation on $\Omega$ defined as $x\sim y$ $\Leftrightarrow$ $(x,y)\in E$ is reflexive, symmetric and transitive. It is shown that there are two $\sigma$-fields $\mathcal{G}_0$ and $\mathcal{G}_1$ on $\Omega$ such that, for all $\mu,\,\nu\in\mathcal{P}(\mathcal{F})$, $$\inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_1}\quad\text{and}\quad\min_{P\in\Gamma(\mu,\nu_0)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}_0}.$$ Here, $\nu_0\in\mathcal{P}(\mathcal{F})$ is a suitable probability measure satisfying $\nu_0=\nu$ on $\mathcal{G}_0$. Moreover, $\mathcal{G}_0\subset\mathcal{F}$ while $\mathcal{G}_1\subset\widehat{\mathcal{F}}$, where $\widehat{\mathcal{F}}$ is the universally measurable $\sigma$-field with respect to $\mathcal{F}$. However, for all $\mu,\,\nu\in\mathcal{P}(\mathcal{F})$, there is a $\sigma$-field $\mathcal{G}(\mu,\nu)\subset\mathcal{F}$ such that $$\inf_{P\in\Gamma(\mu,\nu)}(1-P(E))=\norm{\mu-\nu}_{\mathcal{G}(\mu,\nu)}.$$
A new approach based on censoring and moment criterion is introduced for parameter estimation of count distributions when the probability generating function is available even though a closed form of the probability mass function and/or finite moments do not exist.
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 $i$.
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.
For each $n\ge 1$, let $X_{n,1},\ldots,X_{n,N_n}$ be real random variables and $S_n=\sum_{i=1}^{N_n}X_{n,i}$. Let $m_n\ge 1$ be an integer. Suppose $(X_{n,1},\ldots,X_{n,N_n})$ is $m_n$-dependent, $E(X_{ni})=0$, $E(X_{ni}^2)<\infty$ and $\sigma_n^2:=E(S_n^2)>0$ for all $n$ and $i$. Then, \begin{gather*} d_W\Bigl(\frac{S_n}{\sigma_n},\,Z\Bigr)\le 30\,\bigl\{c^{1/3}+12\,U_n(c/2)^{1/2}\bigr\}\quad\quad\text{for all }n\ge 1\text{ and }c>0, \end{gather*} where $d_W$ is Wasserstein distance, $Z$ a standard normal random variable and $$U_n(c)=\frac{m_n}{\sigma_n^2}\,\sum_{i=1}^{N_n}E\Bigl[X_{n,i}^2\,1\bigl\{\abs{X_{n,i}}>c\,\sigma_n/m_n\bigr\}\Bigr].$$ Among other things, this estimate of $d_W\bigl(S_n/\sigma_n,\,Z\bigr)$ yields a similar estimate of $d_{TV}\bigl(S_n/\sigma_n,\,Z\bigr)$ where $d_{TV}$ is total variation distance.
A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted alternatives to the Poisson distribution. Actually, the family of test statistics is based on the difference between the plug-in estimator of the Poisson cumulative distribution function and the empirical cumulative distribution function. The test statistics have an intuitive and simple form and are asymptotically normally distributed, allowing a straightforward implementation of the test. The finite sample properties of the test are investigated by means of an extensive simulation study. The test shows satisfactory behaviour compared to other tests with known limit distribution.
A goodness-of-fit test for one-parameter count distributions with finite second moment is proposed. The test statistic is derived from the $L^1$ distance of a function of the probability generating function of the model under the null hypothesis and that of the random variable actually generating data, when the latter belongs to a suitable wide class of alternatives. The test statistic has a rather simple form and it is asymptotically normally distributed under the null hypothesis, allowing a straightforward implementation of the test. Moreover, the test is consistent for alternative distributions belonging to the class, but also for all the alternative distributions whose probability of zero is different from that under the null hypothesis. Thus, the use of the test is proposed and investigated also for alternatives not in the class. The finite-sample properties of the test are assessed by means of an extensive simulation study.
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.
Let $(X_n:n\ge 1)$ be a sequence of random observations. Let $\sigma_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the $n$-th predictive distribution and $\sigma_0(\cdot)=P(X_1\in\cdot)$ the marginal distribution of $X_1$. In a Bayesian framework, to make predictions on $(X_n)$, one only needs the collection $\sigma=(\sigma_n:n\ge 0)$. 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 has to be selected. In this paper, $\sigma$ is subjected to two requirements: (i) The resulting sequence $(X_n)$ is conditionally identically distributed, in the sense of Berti, Pratelli and Rigo (2004); (ii) Each $\sigma_{n+1}$ is a simple recursive update of $\sigma_n$. Various new $\sigma$ satisfying (i)-(ii) are introduced and investigated. For such $\sigma$, the asymptotics of $\sigma_n$, as $n\rightarrow\infty$, is determined. In some cases, the probability distribution of $(X_n)$ is also evaluated.
Let $\Lambda$ be the collection of all probability distributions for $(X,\widetilde{X})$, where $X$ is a fixed random vector and $\widetilde{X}$ ranges over all possible knockoff copies of $X$ (in the sense of \cite{CFJL18}). Three topics are developed in this paper: (i) A new characterization of $\Lambda$ is proved; (ii) A certain subclass of $\Lambda$, defined in terms of copulas, is introduced; (iii) The (meaningful) special case where the components of $X$ are conditionally independent is treated in depth. In real problems, after observing $X=x$, each of points (i)-(ii)-(iii) may be useful to generate a value $\widetilde{x}$ for $\widetilde{X}$ conditionally on $X=x$.