Optimal estimation

<p>In applied statistics, optimal estimation is a <a href="/facts/Regularization_(mathematics)/K601A3y2">regularized</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> <a href="/facts/Inverse_problem/Fe01pQ9E">inverse method</a> based on <a href="/facts/Bayes%27_theorem/wQqFB2Dt">Bayes' theorem</a>.
It is used very commonly in the <a href="/facts/Geoscience/rQsXCSbM">geosciences</a>, particularly for <a href="/facts/Atmospheric_sounding/05PaHaRW">atmospheric sounding</a>.
A matrix inverse problem looks like this:
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A
        
        
          
            
              x
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        =
        
          
            
              y
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    {\displaystyle \mathbf {A} {\vec {x}}={\vec {y}}}

<p>The essential concept is to transform the matrix, A, into a <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a> and the variables, 
  
    
      
        
          
            
              x
              →
            
          
        
      
    
    {\displaystyle {\vec {x}}}
  
 and 
  
    
      
        
          
            
              y
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    {\displaystyle {\vec {y}}}
  
 into probability distributions by assuming Gaussian statistics and using empirically-determined covariance matrices.
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Optimal estimation open-in-new

Optimal estimation