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Maximum Entropy Method – 3 – Lagrange Parameters

Written By: Jean-Paul Cipria - Juin• 07•16

Why do we use Gamma Probability Density ?
What are the right parameters to find the khi² density ?
Why do we used Gamma density for Entropy ?
How to correlate Bayesian Inferences to kind of Probability Density ?
… etc

Gamma and Khi2 Densities

Gamma and Khi2 Densities*

We are begining to answer …

First action is to be able to calculate integral « à la main » in french WITHOUT USING SIMULINK ! AND WITHOUT USING OBJECT PROGRAMMATION or C++ ! (Epistemologic regression … see article .. later …)

(*) : We use the word « density » f(a) and not « distribution » because the right functional term is the density (distribution) f(a) to be integrated to have a probability P(X<b). Then f(a) is also a mathematical distribution as cos(x) is a function. Then there is no information to say the word distribution !

%% Intégrale par méthode de Simpson
% ---------------------------------
A = Fonction*dx; % Nous multiplions tous les termes de la table par dx

I = 0; 
% On ajoute tous les termes.
for k=1:1:N
    D(k)=I; % Fonction de Répartition du Khi² : C'est l'intégrale point par point. Ou la probabilité de 0 à x.
    I = I  + A(k) + (A(k+1)-A(k))/2; % Simpson : Nous ajoutons la demi pente entre deux échantillons
end
D(N+1)=I; % Il manque un terme à D(N+1) car nous calculons la pente ou dérivée à l'orde N+1 au dernier coup de calcul

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References

(CIPRIA-2016) : Jean-Paul Cipria – FROM MAXIMAL ENTROPY METHOD TO BAYESIAN INFERENCE IN NEUROSCIENCES – MATLAB SIMULATION – (Extracts) – 2016

Abstract : This Matlab study shows how to link two physics concepts : Information Entropy and Bayesian Inference1. The entropy2 is used by physicists to view the « most probable » best informations brain pictures therefore Bayesian Inference is a statistic method to generalyze a data set to the « most probable » concept by the brain. The first part shows how to use maximal entropy method to find missing informations on the choosen transformation display. A second part displays some MEM pictures. The last part discuses about a statistical methods issued by stationary principle law on the neurosciences.
Keys : Maximum Entropy Method, MEM, Bayesian Inference.

© Jean-Paul Cipria
Senior Engineer-Physist

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