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Patrizia Berti

Publications and source records attributed to Patrizia Berti.

18 recordsLinked to original sources
arXiv2023

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 $σ=(σ_0,σ_1,\ldots)$, called ``strategy" in what follows, where $σ_0(\cdot)=P(X_1\in\cdot)$ is the marginal distribution of $X_1$ and $σ_n(\cdot)=P(X_{n+1}\in\cdot\mid X_1,\ldots,X_n)$ the $n$-th predictive distribution. Because of the Ionescu-Tulcea theorem, $σ$ 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 Pólya urns, random change points, covariates and stationary sequences.

stat.ME↗
arXiv2022

New perspectives on knockoffs construction

Let $Λ$ 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 $Λ$ is proved; (ii) A certain subclass of $Λ$, 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 $X=x$.

math.ST↗
arXiv2022

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\simν\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{θν(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}{n+θ}\quad\quad\text{a.s.} \end{gather*} where $θ>0$ is a constant, $ν$ 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 $ν$ given any sub-$σ$-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 $μ$ is the weak limit of the empirical measures, conditions for $μ$ to be a.s. discrete, or a.s. non-atomic, or $μ\llν$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.

math.PR↗
arXiv2021

Bayesian predictive inference without a prior

Let $(X_n:n\ge 1)$ be a sequence of random observations. Let $σ_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ be the $n$-th predictive distribution and $σ_0(\cdot)=P(X_1\in\cdot)$ the marginal distribution of $X_1$. In a Bayesian framework, to make predictions on $(X_n)$, one only needs the collection $σ=(σ_n:n\ge 0)$. Because of the Ionescu-Tulcea theorem, $σ$ 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, $σ$ 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 $σ_{n+1}$ is a simple recursive update of $σ_n$. Various new $σ$ satisfying (i)-(ii) are introduced and investigated. For such $σ$, the asymptotics of $σ_n$, as $n\rightarrow\infty$, is determined. In some cases, the probability distribution of $(X_n)$ is also evaluated.

math.ST↗
arXiv2015

Asymptotics for randomly reinforced urns with random barriers

An urn contains black and red balls. Let $Z_n$ be the proportion of black balls at time $n$ and $0\leq L L$, then $b_n$ is replaced together with a random number $R_n$ of red balls. Otherwise, no additional balls are added, and $b_n$ alone is replaced. In this paper, we assume $R_n=B_n$. Then, under mild conditions, it is shown that $Z_n\overset{a.s.}\longrightarrow Z$ for some random variable $Z$, and \begin{gather*} D_n:=\sqrt{n}\,(Z_n-Z)\longrightarrow\mathcal{N}(0,σ^2)\quad\text{conditionally a.s.} \end{gather*} where $σ^2$ is a certain random variance. Almost sure conditional convergence means that \begin{gather*} P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)\overset{weakly}\longrightarrow\mathcal{N}(0,\,σ^2)\quad\text{a.s.} \end{gather*} where $P\bigl(D_n\in\cdot\mid\mathcal{G}_n\bigr)$ is a regular version of the conditional distribution of $D_n$ given the past $\mathcal{G}_n$. Thus, in particular, one obtains $D_n\longrightarrow\mathcal{N}(0,σ^2)$ stably. It is also shown that $L<Z<U$ a.s. and $Z$ has non-atomic distribution.

math.PR↗
arXiv2015

Central limit theorems for an Indian buffet model with random weights

The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let $L_n$ be the number of dishes experimented by the first $n$ customers, and let $\overline{K}_n=(1/n)\sum_{i=1}^nK_i$ where $K_i$ is the number of dishes tried by customer $i$. The asymptotic distributions of $L_n$ and $\overline{K}_n$, suitably centered and scaled, are obtained. The convergence turns out to be stable (and not only in distribution). As a particular case, the results apply to the standard (i.e., nongeneralized) Indian buffet process.

math.PR↗
arXiv2014

Two versions of the fundamental theorem of asset pricing

Let $L$ be a convex cone of real random variables on the probability space $(Ω,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $$ P \sim P_0,\quad E_P \abs{X}< \infty\, \text{ and } \, E_P(X) \leq 0\, \text{ for all }X \in L $$ is investigated. Two results are provided. In the first, $P$ is a finitely additive probability, while $P$ is $σ$-additive in the second. If $L$ is a linear space then $-X\in L$ whenever $X\in L$, so that $E_P(X)\leq 0$ turns into $E_P(X)=0$. Hence, the results apply to various significant frameworks, including equivalent martingale measures and equivalent probability measures with given marginals.

math.PR↗
arXiv2013

Exchangeable sequences driven by an absolutely continuous random measure

Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $α_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $α$ a random probability measure on $S$ such that $α_n\stackrel{\mathrm{weak}}{\longrightarrow}α$ a.s. Two (related) problems are addressed. One is to give conditions for $α\llλ$ a.s., where $λ$ is a (nonrandom) $σ$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal {L}(X_1,\...,X_n)$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alp ha_n-α\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_n)$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029-2052].

math.PR↗
arXiv2012

An Anscombe-type theorem

Let (X_n) be a sequence of random variables (with values in a separable metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to converge stably (in particular, in distribution) are provided. Some examples, where such conditions work but those already existing fail, are given as well. Key words and phrases: Anscombe theorem, Exchangeability, Random indices, Random sums, Stable convergence

math.PR↗
arXiv2010

Finitely additive equivalent martingale measures

Let $L$ be a linear space of real bounded random variables on the probability space $(Ω,\mathcal{A},P_0)$. There is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\sim P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $c\,E_Q(X)\leq\text{ess sup}(-X)$, $X\in L$, for some constant $c>0$ and (countably additive) probability $Q$ on $\mathcal{A}$ such that $Q\sim P_0$. A necessary condition for such a $P$ to exist is $\bar{L-L_\infty^+}\,\cap L_\infty^+=\{0\}$, where the closure is in the norm-topology. If $P_0$ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\ll P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $\text{ess sup}(X)\geq 0$ for all $X\in L$.

math.PR↗
arXiv2010

Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{μ_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)\sum_{i=1}^nδ_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

math.ST↗
arXiv2009

Central Limit Theorems for Multicolor Urns with Dominated Colors

An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.

math.PR↗
arXiv2009

Trivial intersection of $σ$-fields and Gibbs sampling

Let $(Ω,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying $P(F)\in\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=σ(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-$σ$-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose $X$ and $Y$ are the coordinate projections on $(Ω,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be the Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if $\overline{σ(X)}\cap\overline{σ(Y)}=\mathcal{N}$, or equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$. The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset $S_0\subsetΩ$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure.

math.PR↗
arXiv2007

0--1 laws for regular conditional distributions

Let $(Ω,\mathcal{B},P)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$σ$-field, and $μ$ a regular conditional distribution for $P$ given $\mathcal{A}$. Necessary and sufficient conditions for $μ(ω)(A)$ to be 0--1, for all $A\in\mathcal{A}$ and $ω\in A_0$, where $A_0\in\mathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathcal{A}$ is a tail sub-$σ$-field. Let $H(ω)$ denote the $\mathcal{A}$-atom including the point $ω\inΩ$. Necessary and sufficient conditions for $μ(ω)(H(ω))$ to be 0--1, for all $ω\in A_0$, are also given. If $(Ω,\mathcal{B})$ is a standard space, the latter 0--1 law is true for various classically interesting sub-$σ$-fields $\mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$σ$-fields connected with continuous time processes.

math.PR↗
arXiv2007

A conditional 0-1 law for the symmetric sigma-field

Let (Ω,\mathcal{B},P) be a probability space, \mathcal{A} a sub-sigma-field of \mathcal{B}, and μa regular conditional distribution for P given \mathcal{A}. For various, classically interesting, choices of \mathcal{A} (including tail and symmetric) the following 0-1 law is proved: There is a set A_0 in \mathcal{A} such that P(A_0)=1 and μ(ω)(A) is 0 or 1 for all A in \mathcal{A} and ωin A_0. Provided \mathcal{B} is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.

math.PR↗
arXiv2004

Limit theorems for a class of identically distributed random variables

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\varnothing,Ω} and G_n=σ(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq 1} is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_{n\geq 1} is exchangeable if and only if (X_{τ(n)})_{n\geq 1} is c.i.d. for any finite permutation τof {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n)\sum_{k=1}^nX_k converges a.s. and in L^1 whenever (X_n)_{n\geq 1} is (real-valued) c.i.d. and E[| X_1| ]<\infty. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[X_{n+1}| G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

math.PR↗