10 research outputs found
Frequentistic approximations to Bayesian prevision of exchangeable random elements
Given a sequence \xi_1, \xi_2,... of X-valued, exchangeable random elements,
let q(\xi^(n)) and p_m(\xi^(n)) stand for posterior and predictive
distribution, respectively, given \xi^(n) = (\xi_1,..., \xi_n). We provide an
upper bound for limsup b_n d_[[X]](q(\xi^(n)), \delta_\empiricn) and limsup b_n
d_[X^m](p_m(\xi^(n)), \empiricn^m), where \empiricn is the empirical measure,
b_n is a suitable sequence of positive numbers increasing to +\infty, d_[[X]]
and d_[X^m] denote distinguished weak probability distances on [[X]] and [X^m],
respectively, with the proviso that [S] denotes the space of all probability
measures on S. A characteristic feature of our work is that the aforesaid
bounds are established under the law of the \xi_n's, unlike the more common
literature on Bayesian consistency, where they are studied with respect to
product measures (p_0)^\infty, as p_0 varies among the admissible
determinations of a random probability measure
Uniform rates of the Glivenko-Cantelli convergence and their use in approximating Bayesian inferences
This paper deals with the problem of quantifying the approximation a
probability measure by means of an empirical (in a wide sense) random
probability measure, depending on the first n terms of a sequence of random
elements. In Section 2, one studies the range of oscillation near zero of the
Wasserstein distance
^{(p)}_{\pms} between \pfrak_0 and
\hat{\pfrak}_n, assuming that the \xitil_i's are i.i.d. with \pfrak_0 as
common law. Theorem 2.3 deals with the case in which \pfrak_0 is fixed as a
generic element of the space of all probability measures on (\rd,
\mathscr{B}(\rd)) and \hat{\pfrak}_n coincides with the empirical measure.
In Theorem 2.4 (Theorem 2.5, respectively) \pfrak_0 is a d-dimensional Gaussian
distribution (an element of a distinguished type of statistical exponential
family, respectively) and \hat{\pfrak}_n is another -dimensional Gaussian
distribution with estimated mean and covariance matrix (another element of the
same family with an estimated parameter, respectively). These new results
improve on allied recent works (see, e.g., [31]) since they also provide
uniform bounds with respect to , meaning that the finiteness of the p-moment
of the random variable \sup_{n \geq 1} b_n
^{(p)}_{\pms}(\pfrak_0,
\hat{\pfrak}_n) is proved for some suitable diverging sequence b_n of positive
numbers. In Section 3, under the hypothesis that the \xitil_i's are
exchangeable, one studies the range of the random oscillation near zero of the
Wasserstein distance between the conditional distribution--also called
posterior--of the directing measure of the sequence, given \xitil_1, \dots,
\xitil_n, and the point mass at \hat{\pfrak}_n. In a similar vein, a bound
for the approximation of predictive distributions is given. Finally, Theorems
from 3.3 to 3.5 reconsider Theorems from 2.3 to 2.5, respectively, according to
a Bayesian perspective
Predictive Constructions Based on Measure-Valued PĂłlya Urn Processes
Measure-valued PĂłlya urn processes (MVPP) are Markov chains with an additive structure
that serve as an extension of the generalized k-color PĂłlya urn model towards a continuum of pos-
sible colors. We prove that, for any MVPP on a Polish space , the normalized sequence
agrees with the marginal predictive distributions of some random process .
Moreover, , where is a random transition kernel on ; thus, if
represents the contents of an urn, then X n denotes the color of the ball drawn with distribution
and - the subsequent reinforcement. In the case , for some
non-negative random weights ... , the process is better understood as a randomly reinforced extension of Blackwell and MacQueenâs PĂłlya sequence. We study the asymptotic properties
of the predictive distributions and the empirical frequencies of under different assumptions
on the weights. We also investigate a generalization of the above models via a randomization of the
law of the reinforcement
On Complex Dependence Structures in Bayesian Nonparametrics: a Distanceâbased Approach
No abstract availableRandom vectors of measures are the main building block to a major portion of Bayesian nonparametric models. The introduction of infiniteâdimensional parameter spaces guarantees notable flexibility and generality to the models but makes their treatment and interpretation more demanding. To overcome these issues we seek a deep understanding of infiniteâdimensional random objects and their role in modeling complex dependence structures in the data. Comparisons with baseline models play a major role in the learning process and are expressed through the introduction of suitable distances. In particular, we define a distance between the laws of random vectors of measures that builds on the Wasserstein distance and combines intuitive geometric properties with analytical tractability. This is first used to evaluate approximation errors in posterior sampling schemes and then culminates in the definition of a new principled and non modelâspecific measure of dependence for partial exchangeability, going beyond current measures of linear dependence. The study of dependence is complemented by the investigation of asymptotic properties for partially exchangeable mixture models from a frequentist perspective. We extend Schwartz theory to a multisample framework by relying on natural distances between vectors of densities and leverage it to find optimal contraction rates for a wide class of hierarchical models
On Complex Dependence Structures in Bayesian Nonparametrics: a Distanceâbased Approach
No abstract availableRandom vectors of measures are the main building block to a major portion of Bayesian nonparametric models. The introduction of infiniteâdimensional parameter spaces guarantees notable flexibility and generality to the models but makes their treatment and interpretation more demanding. To overcome these issues we seek a deep understanding of infiniteâdimensional random objects and their role in modeling complex dependence structures in the data. Comparisons with baseline models play a major role in the learning process and are expressed through the introduction of suitable distances. In particular, we define a distance between the laws of random vectors of measures that builds on the Wasserstein distance and combines intuitive geometric properties with analytical tractability. This is first used to evaluate approximation errors in posterior sampling schemes and then culminates in the definition of a new principled and non modelâspecific measure of dependence for partial exchangeability, going beyond current measures of linear dependence. The study of dependence is complemented by the investigation of asymptotic properties for partially exchangeable mixture models from a frequentist perspective. We extend Schwartz theory to a multisample framework by relying on natural distances between vectors of densities and leverage it to find optimal contraction rates for a wide class of hierarchical models
Note on âFrequentistic approximations to Bayesian prevision of exchangeable random elementsâ [Int. J. Approx. Reason. 78 (2016) 138â152]
This note points some ambiguities in the notation adopted in âFrequentistic approximations to Bayesian prevision of exchangeable random elementsâ [Int. J. Approx. Reason. 78 (2016) 138â152] and provides the correct way to read those statements and proofs which are affected by the aforesaid ambiguities