368 research outputs found
A Maximum Entropy Procedure to Solve Likelihood Equations
In this article, we provide initial findings regarding the problem of solving likelihood equations by means of a maximum entropy (ME) approach. Unlike standard procedures that require equating the score function of the maximum likelihood problem at zero, we propose an alternative strategy where the score is instead used as an external informative constraint to the maximization of the convex Shannon\u2019s entropy function. The problem involves the reparameterization of the score parameters as expected values of discrete probability distributions where probabilities need to be estimated. This leads to a simpler situation where parameters are searched in smaller (hyper) simplex space. We assessed our proposal by means of empirical case studies and a simulation study, the latter involving the most critical case of logistic regression under data separation. The results suggested that the maximum entropy reformulation of the score problem solves the likelihood equation problem. Similarly, when maximum likelihood estimation is difficult, as is the case of logistic regression under separation, the maximum entropy proposal achieved results (numerically) comparable to those obtained by the Firth\u2019s bias-corrected approach. Overall, these first findings reveal that a maximum entropy solution can be considered as an alternative technique to solve the likelihood equation
Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach
Quantum tori are limits of finite dimensional C*-algebras for the quantum
Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening
of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic
structure. In this paper, we propose a proof of the continuity of the family of
quantum and fuzzy tori which relies on explicit representations of the
C*-algebras rather than on more abstract arguments, in a manner which takes
full advantage of the notion of bridge defining the quantum propinquity.Comment: 41 Pages. This paper is the second half of ArXiv:1302.4058v2. The
latter paper has been divided in two halves for publications purposes, with
the first half now the current version of 1302.4058, which has been accepted
in Trans. Amer. Math. Soc. This second half is now a stand-alone paper, with
a brief summary of 1302.4058 and a new introductio
Approximation of fuzzy numbers by convolution method
In this paper we consider how to use the convolution method to construct
approximations, which consist of fuzzy numbers sequences with good properties,
for a general fuzzy number. It shows that this convolution method can generate
differentiable approximations in finite steps for fuzzy numbers which have
finite non-differentiable points. In the previous work, this convolution method
only can be used to construct differentiable approximations for continuous
fuzzy numbers whose possible non-differentiable points are the two endpoints of
1-cut. The constructing of smoothers is a key step in the construction process
of approximations. It further points out that, if appropriately choose the
smoothers, then one can use the convolution method to provide approximations
which are differentiable, Lipschitz and preserve the core at the same time.Comment: Submitted to Fuzzy Sets and System at Sep 18 201
Relations Between Low-lying Quantum Wave Functions and Solutions of the Hamilton-Jacobi Equation
We discuss a new relation between the low lying Schroedinger wave function of
a particle in a one-dimentional potential V and the solution of the
corresponding Hamilton-Jacobi equation with -V as its potential. The function V
is , and can have several minina (V=0). We assume the problem to be
characterized by a small anhamornicity parameter and a much smaller
quantum tunneling parameter between these different minima.
Expanding either the wave function or its energy as a formal double power
series in and , we show how the coefficients of
in such an expansion can be expressed in terms of definite
integrals, with leading order term determined by the classical solution of the
Hamilton-Jacobi equation. A detailed analysis is given for the particular
example of quartic potential .Comment: LaTex, 48 pages, no figur
A set-valued framework for birth-and-growth process
We propose a set-valued framework for the well-posedness of birth-and-growth
process. Our birth-and-growth model is rigorously defined as a suitable
combination, involving Minkowski sum and Aumann integral, of two very general
set-valued processes representing nucleation and growth respectively. The
simplicity of the used geometrical approach leads us to avoid problems arising
by an analytical definition of the front growth such as boundary regularities.
In this framework, growth is generally anisotropic and, according to a
mesoscale point of view, it is not local, i.e. for a fixed time instant, growth
is the same at each space point
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