likelihood

09/16/2024

• Likelihood
  • Likelihood is a function or measurement that evaluates how well a set of observed data is explained by a model or parameter.
  • example: there are two boxes \(\text{A, B}\), which contain the following balls, respectively:
    
  • For the observed data \(\vcenter{\hbox{$\Huge \bullet$}}\), likelihoods are: \[ L ( \text{A} \,|\, \vcenter{\hbox{$\Huge \bullet$}} ) = P ( \vcenter{\hbox{$\Huge \bullet$}} \,|\, \text{A} ) = 1/2 \quad \quad L ( \text{B} \,|\, \vcenter{\hbox{$\Huge \bullet$}} ) = P ( \vcenter{\hbox{$\Huge \bullet$}} \,|\, \text{B} ) = 2/3 \]
  
  
  • Here, the vertical axis represents the probability of each possible observation under a fixed distribution. The horizontal axis represents the likelihood, that is, how well an observed datum is explained when evaluated under different models or parameter values. \[ \begin{array}{l} \quad P ( \vcenter{\hbox{$\Huge \bullet$}} \,|\, \text{A} ) = 1/2 \quad \quad P ( \vcenter{\hbox{$\Huge \bullet$}} \,|\, \text{B} ) = 2/3 \,\bigg\} \text{likelihood} \\ \underbrace{\quad P(\bigcirc \mid \text{A}) = 1/2 \quad}_{\text{probability}} \quad P(\bigcirc \mid \text{B}) = 1/3 \end{array} \]

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