A Direct Derivation of the Exact Fisher Information Matrix for Bivariate Bessel Distribution of Type I

ABSTRACT We provide some details of the implementation of optimal design algorithm in the PkStaMp library which is intended for constructing optimal sampling schemes for pharmacokinetic (PK) and pharmacodynamic (PD) studies. We discuss different types of approximation of individual Fisher information matrix and describe a user-defined option of the library.

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Tensor Sylvester matrices and the Fisher information matrix of VARMAX processes

Linear Algebra and its Applications ◽

10.1016/j.laa.2009.06.027 ◽

2010 ◽

Vol 432 (8) ◽

pp. 1975-1989 ◽

Cited By ~ 1

Author(s):

André Klein ◽

Peter Spreij

Keyword(s):

Fisher Information ◽

Fisher Information Matrix ◽

Information Matrix

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Singularities Affect Dynamics of Learning in Neuromanifolds

Neural Computation ◽

10.1162/neco.2006.18.5.1007 ◽

2006 ◽

Vol 18 (5) ◽

pp. 1007-1065 ◽

Cited By ~ 55

Author(s):

Shun-ichi Amari ◽

Hyeyoung Park ◽

Tomoko Ozeki

Keyword(s):

Model Selection ◽

Parameter Space ◽

Fisher Information ◽

Fisher Information Matrix ◽

Information Matrix ◽

Gaussian Mixture ◽

Multilayer Perceptrons ◽

Hierarchical Systems ◽

Space Forms ◽

Training Error

The parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units. A parameter space forms a geometrical manifold, called the neuromanifold in the case of neural networks. Such a model is identified with a statistical model, and a Riemannian metric is given by the Fisher information matrix. However, the matrix degenerates at singularities. Such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities, ARMA time-series model, and many other cases. The standard statistical paradigm of the Cramér-Rao theorem does not hold, and the singularity gives rise to strange behaviors in parameter estimation, hypothesis testing, Bayesian inference, model selection, and in particular, the dynamics of learning from examples. Prevailing theories so far have not paid much attention to the problem caused by singularity, relying only on ordinary statistical theories developed for regular (nonsingular) models. Only recently have researchers remarked on the effects of singularity, and theories are now being developed. This article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures. We demonstrate our recent results on these problems. Simple toy models are also used to show explicit solutions. We explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically, because the Fisher information matrix degenerates, that the model selection criteria such as AIC, BIC, and MDL fail to hold in these models, that a smooth Bayesian prior becomes singular in such models, and that the trajectories of dynamics of learning are strongly affected by the singularity, causing plateaus or slow manifolds in the parameter space. The natural gradient method is shown to perform well because it takes the singular geometrical structure into account. The generalization error and the training error are studied in some examples.

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