Editorial

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Open AccessEditorial

Preface to the Special Issue on “Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini’s 75th Birthday”

by Emanuele Dolera

Mathematics 2022, 10(15), 2567; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152567 - 23 Jul 2022

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Abstract

It is my pleasure to write this Preface to the Special Issue of Mathematics entitled “Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini’s 75th Birthday” [...] Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

Research

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16 pages, 359 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Fisher, Bayes, and Predictive Inference

by Sandy Zabell

Mathematics 2022, 10(10), 1634; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101634 - 11 May 2022

Cited by 2 | Viewed by 1715

Abstract

We review historically the position of Sir R.A. Fisher towards Bayesian inference and, particularly, the classical Bayes–Laplace paradigm. We focus on his Fiducial Argument. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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27 pages, 472 KiB

Open AccessArticle

Asymptotic Efficiency of Point Estimators in Bayesian Predictive Inference

by Emanuele Dolera

Mathematics 2022, 10(7), 1136; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071136 - 01 Apr 2022

Cited by 1 | Viewed by 1265

Abstract

The point estimation problems that emerge in Bayesian predictive inference are concerned with random quantities which depend on both observable and non-observable variables. Intuition suggests splitting such problems into two phases, the former relying on estimation of the random parameter of the model, [...] Read more.

The point estimation problems that emerge in Bayesian predictive inference are concerned with random quantities which depend on both observable and non-observable variables. Intuition suggests splitting such problems into two phases, the former relying on estimation of the random parameter of the model, the latter concerning estimation of the original quantity from the distinguished element of the statistical model obtained by plug-in of the estimated parameter in the place of the random parameter. This paper discusses both phases within a decision theoretic framework. As a main result, a non-standard loss function on the space of parameters, given in terms of a Wasserstein distance, is proposed to carry out the first phase. Finally, the asymptotic efficiency of the entire procedure is discussed. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

10 pages, 307 KiB

Open AccessArticle

Single-Block Recursive Poisson–Dirichlet Fragmentations of Normalized Generalized Gamma Processes

by Lancelot F. James

Mathematics 2022, 10(4), 561; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040561 - 11 Feb 2022

Cited by 1 | Viewed by 1055

Abstract

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for

0 < α < 1,

and

θ > - α

, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters [...] Read more.

Dong, Goldschmidt and Martin (2006) (DGM) showed that, for

0 < α < 1,

and

θ > - α

, the repeated application of independent single-block fragmentation operators based on mass partitions following a two-parameter Poisson–Dirichlet distribution with parameters

(α, 1 - α)

to a mass partition having a Poisson–Dirichlet distribution with parameters

(α, θ)

leads to a remarkable nested family of Poisson—Dirichlet distributed mass partitions with parameters

(α, θ + r)

for

r = 0, 1, 2, \dots .

Furthermore, these generate a Markovian sequence of

α

-diversities following Mittag-Leffler distributions, whose ratios lead to independent Beta-distributed variables. These Markov chains are referred to as Mittag-Leffler Markov chains and arise in the broader literature involving Pólya urn and random tree/graph growth models. Here we obtain explicit descriptions of properties of these processes when conditioned on a mixed Poisson process when it equates to an integer

n,

which has interpretations in a species sampling context. This is equivalent to obtaining properties of the fragmentation operations of (DGM) when applied to mass partitions formed by the normalized jumps of a generalized gamma subordinator and its generalizations. We focus primarily on the case where

n = 0, 1 .

Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

12 pages, 294 KiB

Open AccessFeature PaperArticle

Partial Exchangeability for Contingency Tables

by Persi Diaconis

Mathematics 2022, 10(3), 442; https://doi.org/10.3390/math10030442 - 29 Jan 2022

Cited by 3 | Viewed by 1889

Abstract

A parameter free version of classical models for contingency tables is developed along the lines of de Finetti’s notions of partial exchangeability. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

19 pages, 829 KiB

Open AccessArticle

Trapping the Ultimate Success

by Alexander Gnedin and Zakaria Derbazi

Mathematics 2022, 10(1), 158; https://0-doi-org.brum.beds.ac.uk/10.3390/math10010158 - 05 Jan 2022

Cited by 3 | Viewed by 1749

Abstract

We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, [...] Read more.

We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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11 pages, 270 KiB

Open AccessArticle

A Central Limit Theorem for Predictive Distributions

by Patrizia Berti, Luca Pratelli and Pietro Rigo

Mathematics 2021, 9(24), 3211; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243211 - 12 Dec 2021

Cited by 1 | Viewed by 1668

Abstract

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions

f : S \to R

. Let [...] Read more.

Let S be a Borel subset of a Polish space and F the set of bounded Borel functions

f : S \to R

. Let

a_{n} (\cdot) = P (X_{n + 1} \in \cdot ∣ X_{1}, \dots, X_{n})

be the n-th predictive distribution corresponding to a sequence

(X_{n})

of S-valued random variables. If

(X_{n})

is conditionally identically distributed, there is a random probability measure

μ

on S such that

\int f d a_{n} \overset{a . s .}{⟶} \int f d μ

for all

f \in F

. Define

D_{n} (f) = d_{n} \{\int f d a_{n} - \int f d μ\}

for all

f \in F

, where

d_{n} > 0

is a constant. In this note, it is shown that, under some conditions on

(X_{n})

and with a suitable choice of

d_{n}

, the finite dimensional distributions of the process

D_{n} = \{D_{n} (f) : f \in F\}

stably converge to a Gaussian kernel with a known covariance structure. In addition,

E \{φ (D_{n} (f)) ∣ X_{1}, \dots, X_{n}\}

converges in probability for all

f \in F

and

φ \in C_{b} (R)

. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

27 pages, 1580 KiB

Open AccessArticle

Mixture of Species Sampling Models

by Federico Bassetti and Lucia Ladelli

Mathematics 2021, 9(23), 3127; https://0-doi-org.brum.beds.ac.uk/10.3390/math9233127 - 04 Dec 2021

Cited by 2 | Viewed by 1350

Abstract

We introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric [...] Read more.

We introduce mixtures of species sampling sequences (mSSS) and discuss how these sequences are related to various types of Bayesian models. As a particular case, we recover species sampling sequences with general (not necessarily diffuse) base measures. These models include some “spike-and-slab” non-parametric priors recently introduced to provide sparsity. Furthermore, we show how mSSS arise while considering hierarchical species sampling random probabilities (e.g., the hierarchical Dirichlet process). Extending previous results, we prove that mSSS are obtained by assigning the values of an exchangeable sequence to the classes of a latent exchangeable random partition. Using this representation, we give an explicit expression of the Exchangeable Partition Probability Function of the partition generated by an mSSS. Some special cases are discussed in detail—in particular, species sampling sequences with general base measures and a mixture of species sampling sequences with Gibbs-type latent partition. Finally, we give explicit expressions of the predictive distributions of an mSSS. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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11 pages, 692 KiB

Open AccessArticle

The Rescaled Pólya Urn and the Wright—Fisher Process with Mutation

by Giacomo Aletti and Irene Crimaldi

Mathematics 2021, 9(22), 2909; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222909 - 15 Nov 2021

Cited by 1 | Viewed by 1296

Abstract

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation [...] Read more.

In recent papers the authors introduce, study and apply a variant of the Eggenberger—Pólya urn, called the “rescaled” Pólya urn, which, for a suitable choice of the model parameters, exhibits a reinforcement mechanism mainly based on the last observations, a random persistent fluctuation of the predictive mean and the almost sure convergence of the empirical mean to a deterministic limit. In this work, motivated by some empirical evidence, we show that the multidimensional Wright—Fisher diffusion with mutation can be obtained as a suitable limit of the predictive means associated to a family of rescaled Pólya urns. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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15 pages, 357 KiB

Open AccessArticle

On Johnson’s “Sufficientness” Postulates for Feature-Sampling Models

by Federico Camerlenghi and Stefano Favaro

Mathematics 2021, 9(22), 2891; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222891 - 13 Nov 2021

Cited by 1 | Viewed by 1294

Abstract

In the 1920s, the English philosopher W.E. Johnson introduced a characterization of the symmetric Dirichlet prior distribution in terms of its predictive distribution. This is typically referred to as Johnson’s “sufficientness” postulate, and it has been the subject of many contributions in Bayesian [...] Read more.

In the 1920s, the English philosopher W.E. Johnson introduced a characterization of the symmetric Dirichlet prior distribution in terms of its predictive distribution. This is typically referred to as Johnson’s “sufficientness” postulate, and it has been the subject of many contributions in Bayesian statistics, leading to predictive characterization for infinite-dimensional generalizations of the Dirichlet distribution, i.e., species-sampling models. In this paper, we review “sufficientness” postulates for species-sampling models, and then investigate analogous predictive characterizations for the more general feature-sampling models. In particular, we present a “sufficientness” postulate for a class of feature-sampling models referred to as Scaled Processes (SPs), and then discuss analogous characterizations in the general setup of feature-sampling models. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

19 pages, 385 KiB

Open AccessArticle

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

by Sandra Fortini, Sonia Petrone and Hristo Sariev

Mathematics 2021, 9(22), 2845; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222845 - 10 Nov 2021

Cited by 2 | Viewed by 1681

Abstract

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP [...] Read more.

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP

{(μ_{n})}_{n \geq 0}

on a Polish space

X

, the normalized sequence

{(μ_{n} / μ_{n} (X))}_{n \geq 0}

agrees with the marginal predictive distributions of some random process

{(X_{n})}_{n \geq 1}

. Moreover,

μ_{n} = μ_{n - 1} + R_{X_{n}}

,

n \geq 1

, where

x \mapsto R_{x}

is a random transition kernel on

X

; thus, if

μ_{n - 1}

represents the contents of an urn, then

X_{n}

denotes the color of the ball drawn with distribution

μ_{n - 1} / μ_{n - 1} (X)

and

R_{X_{n}}

—the subsequent reinforcement. In the case

R_{X_{n}} = W_{n} δ_{X_{n}}

, for some non-negative random weights

W_{1}, W_{2}, \dots

, the process

{(X_{n})}_{n \geq 1}

is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of

{(X_{n})}_{n \geq 1}

under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

12 pages, 308 KiB

Open AccessArticle

A Compound Poisson Perspective of Ewens–Pitman Sampling Model

by Emanuele Dolera and Stefano Favaro

Mathematics 2021, 9(21), 2820; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212820 - 06 Nov 2021

Cited by 2 | Viewed by 1252

Abstract

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set

{1, \dots, n}

, with

n \in N

, which is indexed by real parameters

α

and

θ

such that either [...] Read more.

The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set

{1, \dots, n}

, with

n \in N

, which is indexed by real parameters

α

and

θ

such that either

α \in [0, 1)

and

θ > - α

, or

α < 0

and

θ = - m α

for some

m \in N

. For

α = 0

, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either

α \in (0, 1)

, or

α < 0

. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s

α

diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of

α

-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case. Full article

(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)

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