Michele Caprio

University of Pennsylvania | UP · Department of Computer and Information Science

Given a set of probability measures $\mathcal{P}$ representing an agent's knowledge on the elements of a sigma-algebra $\mathcal{F}$, we can compute upper and lower bounds for the probability of any event $A\in\mathcal{F}$ of interest. A procedure generating a new assessment of beliefs is said to constrict $A$ if the bounds on the probability of $A...

We formulate an ergodic theory for the (almost sure) limit PE˜co of a sequence (PEnco) of successive dynamic imprecise probability kinematics (DIPK, introduced in [10]) updates of a set PE0co representing the initial beliefs of an agent. As a consequence, we formulate a strong law of large numbers.

In the context of a finite admixture model whose components and weights are unknown, if the number of identifiable components is a function of the amount of data collected, we use techniques from stochastic convex geometry to find the growth rate of its expected value. We also show that by placing a Dirichlet process prior on the densities supporte...

We state concentration and martingale inequalities for the output of the hidden layers of a stochastic deep neural network (SDNN), as well as for the output of the whole SDNN. These results allow us to introduce an expected classifier (EC), and to give probabilistic upper bound for the classification error of the EC. We also state the optimal numbe...

We formulate an ergodic theory for the (almost sure) limit of a sequence of successive dynamic imprecise probability kinematics (DIPK, introduced in Caprio and Gong, 2021) updates of a set representing the initial beliefs of an agent. As a consequence, we formulate a strong law of large numbers.

We introduce dynamic probability kinematics (DPK), a method for an agent to mechanically update subjective beliefs in the presence of partial information. We then generalize DPK to dynamic imprecise probability kinematics (DIPK), which allows the agent to express their initial beliefs via a set of probabilities to take ambiguity into account. We pr...

We propose a new, more general definition of extended probability measures. We study their properties and provide a behavioral interpretation. We put them to use in an inference procedure, whose environment is canonically represented by the probability space $(\Omega,\mathcal{F},P)$, when both $P$ and the composition of $\Omega$ are unknown. We dev...

We present two classes of abstract prearithmetics, {AM}M≥1 and {BM}M>0. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic R+=(R+,+,×,≤R+), and the second one is projective with respect to the extended real Diophantine arithmetic R¯=(R¯,+,×,≤R¯). In addition, we have that every AM and every BM is a comple...

We provide optimal variational upper and lower bounds for the augmented Kullback-Leibler divergence in terms of the augmented total variation distance between two probability measures defined on two Euclidean spaces having different dimensions.

We provide optimal lower and upper bounds for the augmented Kullback-Leibler divergence in terms of the augmented total variation distance between two probability measures defined on two Euclidean spaces having different dimensions. We call them refined Pinsker’s and reverse Pinsker’s inequalities, respectively.

Let $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$ . In addition, assume that the restriction of u on the fixed points of $\Phi $ has a unique maximizer $\eta ^\star $ . Then, we consider feasible paths $(x_0,x_1,\ldots )$ with values...

We present three classes of abstract prearithmetics, $\{\mathbf{A}_M\}_{M \geq 1}$, $\{\mathbf{A}_{-M,M}\}_{M \geq 1}$, and $\{\mathbf{B}_M\}_{M > 0}$. The first one is weakly projective with respect to the nonnegative real Diophantine arithmetic $\mathbf{R_+}=(\mathbb{R}_+,+,\times,\leq_{\mathbb{R}_+})$, the second one is weakly projective with re...

Let $\Phi$ be a correspondence from a normed vector space $X$ into itself, let $u: X\to \mathbf{R}$ be a function, and $\mathcal{I}$ be an ideal on $\mathbf{N}$. Also, assume that the restriction of $u$ on the fixed points of $\Phi$ has a unique maximizer $\eta^\star$. Then, we consider feasible paths $(x_0,x_1,\ldots)$ with values in $X$ such that...