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# Direct A-Posteriori Parametric Statistical Modeling by Neural Network Functions

IP.com Disclosure Number: IPCOM000100863D
Original Publication Date: 1990-Jun-01
Included in the Prior Art Database: 2005-Mar-16
Document File: 2 page(s) / 64K

IBM

## Related People

Bakis, R: AUTHOR [+3]

## Abstract

We give a direct construction of a-posteriori distributions which uses a parametric family of probability distributions and a deterministic multilayer perceptron.

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Direct A-Posteriori Parametric Statistical Modeling by Neural Network Functions

We give a direct construction of a-posteriori
distributions which uses a parametric family of probability
distributions and a deterministic multilayer perceptron.

Let (X,Y) be a random pair where X is to be used to predict
Y. If Y is a real vector, we have a regression setup, and if Y
has finitely many values, then we have a classification setup.  The
usual generative model for classification specifies positive a-priori
class probabilities.
Prob(Y = y) = p(y) y = 1,...,n that sum to one and conditional
probability distributions

(Image Omitted)

Prob(X  A Y = 1),..., Prob(X  A y = n) for the observable
predictor X.  The classifier is then constructed from these by Bayes
theorem which defines the a-posteriori distribution of classes by
Prob(Y = y X = x) = p (y) Prob (X = x Y =
y)  _                              n
S p(i)Prob(X = x Y = i)
i=1
In these models the "generative" distributions Prob(X = x Y = y) are
assumed to belong to a convenient parametric family and the models
are trained by estimating the parameter based on some likelihood
function of data.  The data are assumed to have been generated by the
model.  For example, if the conditional distributions of X are
Gaussian and the data consists of (X,Y) pairs, then the parameters
are class probabilities, mean vectors and covariance matrices while
the (maximum likelihood) estimates are relative frequencies, sample
mean vectors and sample covariance matrices.

We now give a direct (doubly) parametric construction of the
a-posteriori probability distributi...