Monte Carlo Markov Chain - Metropolis-Hastings - Estimation of parameters











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I have 6 parameters to estimate : $p=(theta=[a,b]$, $nu=[r_0,c_0,alpha,beta])$ with Bayesian and MCMC methods :



$$text{PSF}(r,c) = bigg(1 + dfrac{r^2 + c^2}{alpha^2}bigg)^{-beta}$$



and the modeling :



$$d(r,c) = a cdot text {PSF}_{alpha,beta} (r-r_0,c-c_0) + b + epsilon (r, c)$$



with $epsilon$ being a white gaussian noise.



This is the matrix form :



$$begin{bmatrix}d(1,1) \ d(1,2) \ d(1,3) \ vdots \ d(20,20) end{bmatrix}
= begin{bmatrix} text{PSF}_{alpha,beta}(1-r_0,1-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,2-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,3-c_0) & 1 \ vdots & vdots \ text{PSF}_{alpha,beta}(20-r_0,20-c_0) & 1 end{bmatrix} times begin{bmatrix}a \ b end{bmatrix}
+ begin{bmatrix}epsilon(1,1) \ epsilon(1,2) \ epsilon(1,3) \ vdots \ epsilon(20,20) end{bmatrix}$$



So we can write for the 1D data vector "d" :



$$d=H(nu),theta + epsilon$$ with $H$ the matrix defined above.



I know that we have the relation for posterior function ($d$=data and $p$ = vector of parameters :



$$f(p|d) = dfrac{f(p),f(d|p)}{int_{p}f(d|p),text{d}p}$$



We can also write :



$$f(p|d) propto text{Likelihood(d|p)},f(p)quad(1)$$ with $f(p)$ the prior function (that I can take as uniform distribution).



Now, how can I estimate the parameters $p$ from this relation $(1)$ with MCMC methods, especially in my case with Metropolis algorithm ?



With likelihood method, I used previously the following cost function to estimate these 6 parameters :



function cost = Crit_J(p,D)



% Compute the model corresponding to parameters p                                        

[R,C] = size(D);

[Cols,Rows] = meshgrid(1:C,1:R);                                                         

% Model

Model = (1+((Rows-p(3)).^2+(Cols-p(4)).^2)/p(5)^2).^(-p(6));                             

model = Model(:);                                                                        

d = D(:);

% Introduce H matrix 

H = [ model, ones(length(model),1)];                                                     

% Compute the cost function : taking absolute value                                      

cost = abs((d-H*[p(1),p(2)]')'*(d-H*[p(1),p(2)]'));                                      



end


And after, I perform these estimations with "Matlab fminsearch" function to find a local minimum.



Here, at first sight, I thought that I have to compute the distribution for each $p$ parameters generated randomly, but it seems to be more subtl.



Question 1) How can we proove that, starting from arbitrary values for parameters $p$, the Metropolis algorithm will converge to right estimations ?



Question 2) How to implement it for this concrete example ?



Any help is welcome, regards






maximum-likelihood monte-carlo log-likelihood







share|cite|improve this question


























    up vote
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    down vote

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    I have 6 parameters to estimate : $p=(theta=[a,b]$, $nu=[r_0,c_0,alpha,beta])$ with Bayesian and MCMC methods :



    $$text{PSF}(r,c) = bigg(1 + dfrac{r^2 + c^2}{alpha^2}bigg)^{-beta}$$



    and the modeling :



    $$d(r,c) = a cdot text {PSF}_{alpha,beta} (r-r_0,c-c_0) + b + epsilon (r, c)$$



    with $epsilon$ being a white gaussian noise.



    This is the matrix form :



    $$begin{bmatrix}d(1,1) \ d(1,2) \ d(1,3) \ vdots \ d(20,20) end{bmatrix}
    = begin{bmatrix} text{PSF}_{alpha,beta}(1-r_0,1-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,2-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,3-c_0) & 1 \ vdots & vdots \ text{PSF}_{alpha,beta}(20-r_0,20-c_0) & 1 end{bmatrix} times begin{bmatrix}a \ b end{bmatrix}
    + begin{bmatrix}epsilon(1,1) \ epsilon(1,2) \ epsilon(1,3) \ vdots \ epsilon(20,20) end{bmatrix}$$



    So we can write for the 1D data vector "d" :



    $$d=H(nu),theta + epsilon$$ with $H$ the matrix defined above.



    I know that we have the relation for posterior function ($d$=data and $p$ = vector of parameters :



    $$f(p|d) = dfrac{f(p),f(d|p)}{int_{p}f(d|p),text{d}p}$$



    We can also write :



    $$f(p|d) propto text{Likelihood(d|p)},f(p)quad(1)$$ with $f(p)$ the prior function (that I can take as uniform distribution).



    Now, how can I estimate the parameters $p$ from this relation $(1)$ with MCMC methods, especially in my case with Metropolis algorithm ?



    With likelihood method, I used previously the following cost function to estimate these 6 parameters :



    function cost = Crit_J(p,D)
    

    
% Compute the model corresponding to parameters p                                        
    
[R,C] = size(D);
    
[Cols,Rows] = meshgrid(1:C,1:R);                                                         
    
% Model
    
Model = (1+((Rows-p(3)).^2+(Cols-p(4)).^2)/p(5)^2).^(-p(6));                             
    
model = Model(:);                                                                        
    
d = D(:);
    
% Introduce H matrix 
    
H = [ model, ones(length(model),1)];                                                     
    
% Compute the cost function : taking absolute value                                      
    
cost = abs((d-H*[p(1),p(2)]')'*(d-H*[p(1),p(2)]'));                                      
    

    
end


    And after, I perform these estimations with "Matlab fminsearch" function to find a local minimum.



    Here, at first sight, I thought that I have to compute the distribution for each $p$ parameters generated randomly, but it seems to be more subtl.



    Question 1) How can we proove that, starting from arbitrary values for parameters $p$, the Metropolis algorithm will converge to right estimations ?



    Question 2) How to implement it for this concrete example ?



    Any help is welcome, regards






    maximum-likelihood monte-carlo log-likelihood







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have 6 parameters to estimate : $p=(theta=[a,b]$, $nu=[r_0,c_0,alpha,beta])$ with Bayesian and MCMC methods :



      $$text{PSF}(r,c) = bigg(1 + dfrac{r^2 + c^2}{alpha^2}bigg)^{-beta}$$



      and the modeling :



      $$d(r,c) = a cdot text {PSF}_{alpha,beta} (r-r_0,c-c_0) + b + epsilon (r, c)$$



      with $epsilon$ being a white gaussian noise.



      This is the matrix form :



      $$begin{bmatrix}d(1,1) \ d(1,2) \ d(1,3) \ vdots \ d(20,20) end{bmatrix}
      = begin{bmatrix} text{PSF}_{alpha,beta}(1-r_0,1-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,2-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,3-c_0) & 1 \ vdots & vdots \ text{PSF}_{alpha,beta}(20-r_0,20-c_0) & 1 end{bmatrix} times begin{bmatrix}a \ b end{bmatrix}
      + begin{bmatrix}epsilon(1,1) \ epsilon(1,2) \ epsilon(1,3) \ vdots \ epsilon(20,20) end{bmatrix}$$



      So we can write for the 1D data vector "d" :



      $$d=H(nu),theta + epsilon$$ with $H$ the matrix defined above.



      I know that we have the relation for posterior function ($d$=data and $p$ = vector of parameters :



      $$f(p|d) = dfrac{f(p),f(d|p)}{int_{p}f(d|p),text{d}p}$$



      We can also write :



      $$f(p|d) propto text{Likelihood(d|p)},f(p)quad(1)$$ with $f(p)$ the prior function (that I can take as uniform distribution).



      Now, how can I estimate the parameters $p$ from this relation $(1)$ with MCMC methods, especially in my case with Metropolis algorithm ?



      With likelihood method, I used previously the following cost function to estimate these 6 parameters :



      function cost = Crit_J(p,D)
      

      
% Compute the model corresponding to parameters p                                        
      
[R,C] = size(D);
      
[Cols,Rows] = meshgrid(1:C,1:R);                                                         
      
% Model
      
Model = (1+((Rows-p(3)).^2+(Cols-p(4)).^2)/p(5)^2).^(-p(6));                             
      
model = Model(:);                                                                        
      
d = D(:);
      
% Introduce H matrix 
      
H = [ model, ones(length(model),1)];                                                     
      
% Compute the cost function : taking absolute value                                      
      
cost = abs((d-H*[p(1),p(2)]')'*(d-H*[p(1),p(2)]'));                                      
      

      
end


      And after, I perform these estimations with "Matlab fminsearch" function to find a local minimum.



      Here, at first sight, I thought that I have to compute the distribution for each $p$ parameters generated randomly, but it seems to be more subtl.



      Question 1) How can we proove that, starting from arbitrary values for parameters $p$, the Metropolis algorithm will converge to right estimations ?



      Question 2) How to implement it for this concrete example ?



      Any help is welcome, regards






      maximum-likelihood monte-carlo log-likelihood







      share|cite|improve this question













      I have 6 parameters to estimate : $p=(theta=[a,b]$, $nu=[r_0,c_0,alpha,beta])$ with Bayesian and MCMC methods :



      $$text{PSF}(r,c) = bigg(1 + dfrac{r^2 + c^2}{alpha^2}bigg)^{-beta}$$



      and the modeling :



      $$d(r,c) = a cdot text {PSF}_{alpha,beta} (r-r_0,c-c_0) + b + epsilon (r, c)$$



      with $epsilon$ being a white gaussian noise.



      This is the matrix form :



      $$begin{bmatrix}d(1,1) \ d(1,2) \ d(1,3) \ vdots \ d(20,20) end{bmatrix}
      = begin{bmatrix} text{PSF}_{alpha,beta}(1-r_0,1-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,2-c_0) & 1 \ text{PSF}_{alpha,beta}(1-r_0,3-c_0) & 1 \ vdots & vdots \ text{PSF}_{alpha,beta}(20-r_0,20-c_0) & 1 end{bmatrix} times begin{bmatrix}a \ b end{bmatrix}
      + begin{bmatrix}epsilon(1,1) \ epsilon(1,2) \ epsilon(1,3) \ vdots \ epsilon(20,20) end{bmatrix}$$



      So we can write for the 1D data vector "d" :



      $$d=H(nu),theta + epsilon$$ with $H$ the matrix defined above.



      I know that we have the relation for posterior function ($d$=data and $p$ = vector of parameters :



      $$f(p|d) = dfrac{f(p),f(d|p)}{int_{p}f(d|p),text{d}p}$$



      We can also write :



      $$f(p|d) propto text{Likelihood(d|p)},f(p)quad(1)$$ with $f(p)$ the prior function (that I can take as uniform distribution).



      Now, how can I estimate the parameters $p$ from this relation $(1)$ with MCMC methods, especially in my case with Metropolis algorithm ?



      With likelihood method, I used previously the following cost function to estimate these 6 parameters :



      function cost = Crit_J(p,D)
      

      
% Compute the model corresponding to parameters p                                        
      
[R,C] = size(D);
      
[Cols,Rows] = meshgrid(1:C,1:R);                                                         
      
% Model
      
Model = (1+((Rows-p(3)).^2+(Cols-p(4)).^2)/p(5)^2).^(-p(6));                             
      
model = Model(:);                                                                        
      
d = D(:);
      
% Introduce H matrix 
      
H = [ model, ones(length(model),1)];                                                     
      
% Compute the cost function : taking absolute value                                      
      
cost = abs((d-H*[p(1),p(2)]')'*(d-H*[p(1),p(2)]'));                                      
      

      
end


      And after, I perform these estimations with "Matlab fminsearch" function to find a local minimum.



      Here, at first sight, I thought that I have to compute the distribution for each $p$ parameters generated randomly, but it seems to be more subtl.



      Question 1) How can we proove that, starting from arbitrary values for parameters $p$, the Metropolis algorithm will converge to right estimations ?



      Question 2) How to implement it for this concrete example ?



      Any help is welcome, regards






      maximum-likelihood monte-carlo log-likelihood




      maximum-likelihood monte-carlo log-likelihood






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 20 hours ago









      youpilat13

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