Mauch MIREX 2010

function CPD = maximize_params(CPD, temp)

% MAXIMIZE_PARAMS Set the params of a CPD to their ML values (Gaussian)

% CPD = maximize_params(CPD, temperature)

%

% Temperature is currently only used for entropic prior on Sigma

% For details, see "Fitting a Conditional Gaussian Distribution", Kevin Murphy, tech. report,

% 1998, available at www.cs.berkeley.edu/~murphyk/papers.html

% Refering to table 2, we use equations 1/2 to estimate the covariance matrix in the untied/tied case,

% and equation 9 to estimate the weight matrix and mean.

% We do not implement spherical Gaussians - the code is already pretty complicated!

if ~adjustable_CPD(CPD), return; end

%assert(approxeq(CPD.nsamples, sum(CPD.Wsum)));

assert(~any(isnan(CPD.WXXsum)))

assert(~any(isnan(CPD.WXYsum)))

assert(~any(isnan(CPD.WYYsum)))

[self_size cpsize dpsize] = size(CPD.weights);

% Append 1s to the parents, and derive the corresponding cross products.

% This is used when estimate the means and weights simultaneosuly,

% and when estimatting Sigma.

% Let x2 = [x 1]'

XY = zeros(cpsize+1, self_size, dpsize); % XY(:,:,i) = sum_l w(l,i) x2(l) y(l)' 

XX = zeros(cpsize+1, cpsize+1, dpsize); % XX(:,:,i) = sum_l w(l,i) x2(l) x2(l)' 

YY = zeros(self_size, self_size, dpsize); % YY(:,:,i) = sum_l w(l,i) y(l) y(l)' 

for i=1:dpsize

  XY(:,:,i) = [CPD.WXYsum(:,:,i) % X*Y

	       CPD.WYsum(:,i)']; % 1*Y

  % [x  * [x' 1]  = [xx' x

  %  1]              x'  1]

  XX(:,:,i) = [CPD.WXXsum(:,:,i) CPD.WXsum(:,i);

	       CPD.WXsum(:,i)'   CPD.Wsum(i)];

  YY(:,:,i) = CPD.WYYsum(:,:,i);

end

w = CPD.Wsum(:);

% Set any zeros to one before dividing

% This is valid because w(i)=0 => WYsum(:,i)=0, etc

w = w + (w==0);

if CPD.clamped_mean

  % Estimating B2 and then setting the last column (the mean) to the clamped mean is *not* equivalent

  % to estimating B and then adding the clamped_mean to the last column.

  if ~CPD.clamped_weights

    B = zeros(self_size, cpsize, dpsize);

    for i=1:dpsize

      if det(CPD.WXXsum(:,:,i))==0

	B(:,:,i) = 0;

      else

	% Eqn 9 in table 2 of TR

	%B(:,:,i) = CPD.WXYsum(:,:,i)' * inv(CPD.WXXsum(:,:,i));

	B(:,:,i) = (CPD.WXXsum(:,:,i) \ CPD.WXYsum(:,:,i))';

      end

    end

    %CPD.weights = reshape(B, [self_size cpsize dpsize]);

    CPD.weights = B;

  end

elseif CPD.clamped_weights % KPM 1/25/02

  if ~CPD.clamped_mean % ML estimate is just sample mean of the residuals

    for i=1:dpsize

      CPD.mean(:,i) = (CPD.WYsum(:,i) - CPD.weights(:,:,i) * CPD.WXsum(:,i)) / w(i);

    end

  end

else % nothing is clamped, so estimate mean and weights simultaneously

  B2 = zeros(self_size, cpsize+1, dpsize);

  for i=1:dpsize

    if det(XX(:,:,i))==0  % fix by U. Sondhauss 6/27/99

      B2(:,:,i)=0;          

    else                    

      % Eqn 9 in table 2 of TR

      %B2(:,:,i) = XY(:,:,i)' * inv(XX(:,:,i));

      B2(:,:,i) = (XX(:,:,i) \ XY(:,:,i))';

    end                   

    CPD.mean(:,i) = B2(:,cpsize+1,i);

    CPD.weights(:,:,i) = B2(:,1:cpsize,i);

  end

end

% Let B2 = [W mu]

if cpsize>0

  B2(:,1:cpsize,:) = reshape(CPD.weights, [self_size cpsize dpsize]);

end

B2(:,cpsize+1,:) = reshape(CPD.mean, [self_size dpsize]);

% To avoid singular covariance matrices,

% we use the regularization method suggested in "A Quasi-Bayesian approach to estimating

% parameters for mixtures of normal distributions", Hamilton 91.

% If the ML estimate is Sigma = M/N, the MAP estimate is (M+gamma*I) / (N+gamma),

% where gamma >=0 is a smoothing parameter (equivalent sample size of I prior)

gamma = CPD.cov_prior_weight;

if ~CPD.clamped_cov

  if CPD.cov_prior_entropic % eqn 12 of Brand AI/Stat 99

    Z = 1-temp;

    % When temp > 1, Z is negative, so we are dividing by a smaller

    % number, ie. increasing the variance.

  else

    Z = 0;

  end

  if CPD.tied_cov

    S = zeros(self_size, self_size);

    % Eqn 2 from table 2 in TR

    for i=1:dpsize

      S = S + (YY(:,:,i) - B2(:,:,i)*XY(:,:,i));

    end

    %denom = max(1, CPD.nsamples + gamma + Z);

    denom = CPD.nsamples + gamma + Z;

    S = (S + gamma*eye(self_size)) / denom;

    if strcmp(CPD.cov_type, 'diag')

      S = diag(diag(S));

    end

    CPD.cov = repmat(S, [1 1 dpsize]);

  else 

    for i=1:dpsize      

      % Eqn 1 from table 2 in TR

      S = YY(:,:,i) - B2(:,:,i)*XY(:,:,i);

      %denom = max(1, w(i) + gamma + Z); % gives wrong answers on mhmm1

      denom = w(i) + gamma + Z;

      S = (S + gamma*eye(self_size)) / denom;

      CPD.cov(:,:,i) = S;

    end

    if strcmp(CPD.cov_type, 'diag')

      for i=1:dpsize      

	CPD.cov(:,:,i) = diag(diag(CPD.cov(:,:,i)));

      end

    end

  end

end

check_covars = 0;

min_covar = 1e-5;

if check_covars % prevent collapsing to a point

  for i=1:dpsize

    if min(svd(CPD.cov(:,:,i))) < min_covar

      disp(['resetting singular covariance for node ' num2str(CPD.self)]);

      CPD.cov(:,:,i) = CPD.init_cov(:,:,i);

    end

  end

end