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We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time…
Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as Bayesian inference, estimators or hypothesis…
In this paper we propose a new concept of differentiability for interval-valued functions. This concept is based on the properties of the Hausdorff-Pompeiu metric and avoids using the generalized Hukuhara difference.
Existing machine learning methods for causal inference usually estimate quantities expressed via the mean of potential outcomes (e.g., average treatment effect). However, such quantities do not capture the full information about the…
Recently a new type of central limit theorem for belief functions was given in Epstein et al. [9]. In this paper, we generalize the central limit theorem in Epstein et al. [9] to accommodate general bounded random variables. These results…
Random integral mappings $I^{h,r}_{(a,b]}$ give isomorphisms between the sub-semigroups of the classical $(ID, \ast)$ and the free-infinite divisible $(ID,\boxplus)$ probability measures. This allows us to introduce new examples of such…
This work proposes a view of probability as a relative measure rather than an absolute one. To demonstrate this concept, we focus on finite outcome spaces and develop three fundamental axioms that establish requirements for relative…
Following the theory of information measures based on the cumulative distribution function, we propose the fractional generalized cumulative entropy, and its dynamic version. These entropies are particularly suitable to deal with…
We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form…
In this article we present very intuitive, easy to follow, yet mathematically rigorous, approach to the so called data fitting process. Rather than minimizing the distance between measured and simulated data points, we prefer to find such…
Information functionals allow to quantify the degree of randomness of a given probability distribution, either absolutely (through min/max entropy principles) or relative to a prescribed reference one. Our primary aim is to analyze the…
Analogy with Bayesian inference is used to formulate constraints within a scheme for functional integration proposed by Cartier and DeWitt-Morette. According to the analogy, functional counterparts of conditional and conjugate probability…
When knowledge is obtained from a database, it is only possible to deduce confidence intervals for probability values. With confidence intervals replacing point values, the results in the set covering model include interval constraints for…
A hypothesis testing and an interval estimation are studied for the common mean of several lognormal populations. Two methods are given based on the concept of generalized p-value and generalized confidence interval. These new methods are…
This study intends to introduce kernel mean embedding of probability measures over infinite-dimensional separable Hilbert spaces induced by functional response statistical models. The embedded function represents the concentration of…
We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the…
Density ratio estimation in high dimensions can be reframed as integrating a certain quantity, the time score, over probability paths which interpolate between the two densities. In practice, the time score has to be estimated based on…
The Rao-Blackwell theorem is utilized to analyze and improve the scalability of inference in large probabilistic models that exhibit symmetries. A novel marginal density estimator is introduced and shown both analytically and empirically to…
A generalization of the classic Gaussian random variable to the family of Multi- Gaussian (MG) random variables characterized by shape parameter M > 0, in addition to the mean and the standard deviation, is introduced. The probability…
The approximation of integral functionals with respect to a stationary Markov process by a Riemann-sum estimator is studied. Stationarity and the functional calculus of the infinitesimal generator of the process are used to get a better…