Aldo Tagliani

On the application of maximum entropy to the moments problem

The maximum‐entropy approach to the solution of classical inverse problem of moments, in which one seeks to reconstruct a function p(x) [where x∈(0,+∞)] from the values of a finite set N+1 of its moments, is studied. It is shown that for N≥4 such a function always exists, while for N=2 and N=3 the acceptable values of the moments are singled out analytically. The paper extends to the general case where the results were previously bounded to the case N=2.

Entropy-convergence in Stieltjes and Hamburger moment problem

In the classical Stieltjes and Hamburger moment problem the sequence of maximum entropy approximants, whose first M moments are equal to given ones, is considered. It is proved that whenever an infinite moment problem is determined, then maximum entropy approximants converge in entropy to the function characterized by given moments. Entropy-convergence is proved by using exclusively existence and uniqueness conditions.

Maximum entropy in the finite Stieltjes and Hamburger moment problem

A necessary and sufficient condition for the existence of the maximum entropy (ME) function, defined in an infinite or semiinfinite interval, is provided. The conclusions reached show that, except in a few particular cases, the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first M+1 moments are assigned. Even if the conclusions reached are very similar to the Hausdorff case, the specificity of the Hamburger and Stieltjes cases demands a different handling. A sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided.

Principle of maximum entropy and probability distributions: definition of applicability field

Abstract With reference to the Principle of Maximum Entropy, the probability distribution exp (−∞ 0 3 a j x x ) is considered for a random variable X defined within the range ( 0,∞ ), the first three moments being known. A guideline is given in order to define the values that the first three moments (or equivalently, the coefficient of variation and of skewness) may possible assume, so that this principle may be applicable. A practical rule is provided to quantitatively identify the form that distribution and hazard rate will take on, once coefficient of variation and skewness are assigned.

Maximum entropy in the Hamburger moments problem

The maximum‐entropy approach to the solution of the Hamburger inverse problem of moments, in which one seeks to recover a positive density function p(x) [where x∈(−∞,+∞)] from the values of a finite N+1 of its moments, is considered. The obtained results show that unexpected upper bounds for the moments do not exist in the general Hamburger finite moment problem, unlike in the symmetric case previously considered. Some physical examples, illustrating the use of partial information to determine the approximate function, are presented.

On the existence of maximum entropy distributions with four and more assigned moments

Abstract It is known that the probability distribution e − ∑ 0 4 a j X j satisfy the Maximum Entropy Principle (MEP) if the available data consist in four moments of probability density function. Two problems are typically associated with use of MEP: the definition of the range of acceptable values for the moments M i ; the evaluation of the coefficients a j . Both problems have already been accurately resolved by analytical procedures when the first two moments of the distribution are known. In this work, the analytical solution in the case of four known moments is provided and a criterion for confronting the general case (whatever the number of known moments) is expounded. The first four moments are expressed in nondimensional form through the expectation and the coefficients of variation, skewness and kurtosis. The range of their acceptable values is obtained from the analytical properties of the differential equations which govern the problem and from the Schwarz inequality.

Maximum entropy in the generalized moment problem

The generalized moment problem in the framework of the maximum entropy approach is considered. A proof for the existence conditions of the solution is provided.

Inverse two-sided Laplace transform for probability density functions

We present a method for the numerical inversion of two-sided Laplace transform of a probability density function. The method assumes the knowledge of the first M derivatives at the origin of the function to be antitransformed. The approximate analytical form is obtained by resorting to maximum entropy principle. Both entropy and L1-norm convergence are proved. Some numerical examples are illustrated.

Maximum entropy and lacunary Stieltjes moment problem

The lacunary Stieltjes moment problem in the framework of the maximum entropy approach is considered. A proof for the existence conditions of the solution is provided.

Inverse two-sided z transform and moment problem

Numerical inversion of two-sided z transform F(z) is considered. The numerical inversion requires F(z) analyticity within an annular region which includes the unit circle |z|=1 and consequently the availability of a finite number of moments, related to the successive derivatives of F(z) at z=1. The approximate analytical form is obtained by resorting to the maximum entropy principle. Entropy and then L1 norm convergence are proved. Some numerical examples are illustrated.