Decision rule

<h2 id="formal-definition">Formal definition</h2>
<p>Given an observable random variable <i>X</i> over the <a href="/facts/Probability_space/dEcv2VrU">probability space</a> 
  
    
      
        
          (
          
            
              X
            
          
          ,
          Σ
          ,
          
            P
            
              θ
            
          
          )
        
      
    
    {\displaystyle \scriptstyle ({\mathcal {X}},\Sigma ,P_{\theta })}
  
, determined by a parameter <i>θ</i> ∈ <i>Θ</i>, and a set <i>A</i> of possible actions, a (deterministic) decision rule is a function <i>δ</i> : 
  
    
      
        
          
            
              X
            
          
        
      
    
    {\displaystyle \scriptstyle {\mathcal {X}}}
  
→ <i>A</i>.
</p>
<h2 id="examples-of-decision-rules">Examples of decision rules</h2>
<ul><li>An <a href="/facts/Estimator/CbkjcKvN">estimator</a> is a decision rule used for estimating a parameter. In this case the set of actions is the parameter space, and a loss function details the cost of the discrepancy between the true value of the parameter and the estimated value. For example, in a linear model with a single scalar parameter 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
, the domain of 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 may extend over 
  
    
      
        
          
            R
          
        
      
    
    {\displaystyle {\mathcal {R}}}
  
 (all real numbers). An associated decision rule for estimating 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 from some observed data might be, "choose the value of the 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
, say 
  
    
      
        
          
            
              θ
              ^
            
          
        
      
    
    {\displaystyle {\hat {\theta }}}
  
, that minimizes the sum of squared error between some observed responses, and responses predicted from the corresponding covariates given that you chose 
  
    
      
        
          
            
              θ
              ^
            
          
        
      
    
    {\displaystyle {\hat {\theta }}}
  
." Thus, the cost function is the sum of squared error, and one would aim to minimize this cost. Once the cost function is defined, 
  
    
      
        
          
            
              θ
              ^
            
          
        
      
    
    {\displaystyle {\hat {\theta }}}
  
 could be chosen, for instance, using some optimization algorithm.</li>
<li>Out of sample <a href="/facts/Prediction/0w4XXB33">prediction</a> in <a href="/facts/Regression_analysis/n6z5Tf7K">regression</a> and <a href="/facts/Statistical_classification/jXXHRkXR">classification</a> models.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Admissible_decision_rule/Kmw4R8Gz">Admissible decision rule</a></li>
<li><a href="/facts/Bayes_estimator/BpFRxYBg">Bayes estimator</a></li>
<li><a href="/facts/Classification_rule/n9IbC4Xo">Classification rule</a></li>
<li><a href="/facts/Scoring_rule/h5nm87vd">Scoring rule</a></li></ul>

Decision rule open-in-new

Decision rule