Maximum a Posteriori (MAP) and Maximum Likelihood (ML)

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Amine STOFF 2014-2-2

1
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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/114546-maximum-a-posteriori-map-and-maximum-likelihood-ml

评论： Sonny Revell 2021-3-4

Hello everyone. I have a serious problem here with developping this matlab code. I'm new with Matlab.

We want to solve a problem of binary classification . Forthis we have two distributions that have a degree of overlap determined. We want to apply the Bayesian hypothesis test to discriminate between the two classes.

P(Wi|x)=(P(Wi)*P(x|Wi))/P(x)

In this case, we use a parametric estimation of the fdp (probability density functions), and we will consider fdp's from normal distributions. The work involved is as followings:

1- generate 2 gausiens distributions with a degree of overlap determined.

2- Estimate the parameters of distributions: the mean vector and matrix covariance.

3- Conceive the Bayesian test : P(Wi|x)>P(Wj|x).decide the class i

4- Draw the decision boundary

For the two first questions i already solved it by this matlab code :

      clear all, close all , clc
      sigma=1; % sigma is changed depending on the dispersion desired
      N=500;
      P1=sigma*randn(2,500)+[2;2]*ones(1,500);
      P2=sigma*randn(2,500)+[-2;-2]*ones(1,500);
      T1=ones(1,500);
      T2=zeros(1,500);
      %Data generation
      plot(P1(1,:),P1(2,:),'ro',P2(1,:),P2(2,:),'go')
      hold on
      %we generates the random values
      P=[P1,P2]; 
      T=[T1,T2];
      ind=randperm(2*N);
      P=P(:,ind);
      T=T(ind);
      Mx1 = mean(P1,2);
      Mx2 = mean(P2,2);
      for i=1:N
          p=P(:,i)
          s1= exp(-(p-Mx1)'*(p-Mx1)/(2*sigma^2)) % First Maximum Likelihood classifier 
          s2= exp(-(p-Mx2)'*(p-Mx2)/(2*sigma^2))
      For the last question here is the code of the function that defines the boundary:
      function []=frontiere(w,P,T,x)
      %--------------------------------------------------------------------------
      % function []=frontiere(W,P,T,x)
      % This function displays the border of classification determined by the Bayesian 
      %discriminant designed to separate between two distribution
      % W: weight matrices model
      % P: dataset
      % T: desired output (Target)
      % x: coordinates which determines the central position of distributions.
      %--------------------------------------------------------------------------
      %disp(' ')
      %disp(' Determining the decision boundary')
      %disp(' ')
      X = [-x:0.02:x]; Y = [-x:0.02:x];
      for i=1:size(X,2)
      for j=1:size(Y,2)
      p = [X(i);Y(j)];
      % IN THIS PART WE MUST IMPLEMENT THE BAYESIAN DISCRIMINANT
      % a2 = output of Bayes classifier
      end
      end
      plotpv(P,T==1);
      hold on
      contour(X,Y,a2,1);
      drawnow