Related papers: Fano's inequality for random variables
In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu…
We want to select the best systems out of a given set of systems (or rank them) with respect to their expected performance. The systems allow random observations only and we assume that the joint observation of the systems has a…
Testing the (in)equality of variances is an important problem in many statistical applications. We develop default Bayes factor tests to assess the (in)equality of two or more population variances, as well as a test for whether the…
In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the…
We extend the seminal van den Berg-Kesten Inequality on disjoint occurrence of two events to a setting with arbitrarily many events, where the quantity of interest is the maximum number that occur disjointly. This provides a handy tool for…
How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational…
We consider the variance of a function of $n$ independent random variables and provide new inequalities which, in particular, extend previous results obtained for symmetric functions in the i.i.d.~setting. For instance, we obtain various…
It is often claimed that Bayesian methods, in particular Bayes factor methods for hypothesis testing, can deal with optional stopping. We first give an overview, using elementary probability theory, of three different mathematical meanings…
We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.
The potential applications of boundary functionals of random processes, such as the extreme values of these processes, the moment of first reaching a fixed level, the value of the process at the moment of reaching the level, the moment of…
Starting from considerations about meaning and subsequent use of asymmetric uncertainty intervals of experimental results, we review the issue of uncertainty propagation. We show that, using a probabilistic approach (the so-called Bayesian…
Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with strong and parametric assumptions about the random effects distribution. There is marked disagreement in the literature…
We develop minimax optimal risk bounds for the general learning task consisting in predicting as well as the best function in a reference set G up to the smallest possible additive term, called the convergence rate. When the reference set…
We study the stochastic convergence of the Ces\`{a}ro mean of a sequence of random variables. These arise naturally in statistical problems that have a sequential component, where the sequence of random variables is typically derived from a…
Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law…
We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…
We present a new exponential inequality as a generalization of that of Sung \textit{et al.} \cite{sun2011} for $M$-acceptable random variables, and hence for extended negative ones. Our result is based on the simple real inequality $e^{x}…
For many standard models of random structure, first-order logic sentences exhibit a convergence phenomenon on random inputs. The most well-known example is for random graphs with constant edge probability, where the probabilities of…
Insurance losses due to flooding can be estimated by simulating and then summing losses over a large number of locations and a large set of hypothetical years of flood events. Replicated realisations lead to Monte Carlo return-level…
A simple method is shown to provide optimal variational bounds on $f$-divergences with possible constraints on relative information extremums. Known results are refined or proved to be optimal as particular cases.