Bayesian risk

The Bayesian risk is a term used in mathematical statistics and a generalization of a risk function . The Bayesian risk clearly shows the potential damage when using a decision-making process, if one already has certain prior information about the initial situation. Bayesian risk is used, for example, to define Bayesian decision functions.

definition

A statistical model with a set of parameters and a risk function for a decision function are given . It is the set of probability measures on . Then the function is called ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta }$ ${\displaystyle R(\vartheta ,\delta )}$ ${\displaystyle \delta }$ ${\displaystyle \Theta _{W}}$ ${\displaystyle \Theta }$

${\displaystyle r(\mu ,\delta )=\int _{\Theta }R(\vartheta ,\delta )\mathrm {d} \mu (\vartheta )}$

the Bayesian risk of the decision function with respect to the a priori distribution . ${\displaystyle \delta }$ ${\displaystyle \mu \in \Theta _{W}}$

Remarks

• The transition from the risk function, which has the parameter as a variable , to the Bayesian risk function, which uses probability measures on the parameter set, can be interpreted as additional information that flows into the model. Instead of evaluating the risk for each individual value, one has information in the form of the a priori distribution about which values ​​this parameter assumes with greater or lesser probability. This makes it easier to estimate the overall risk. ${\displaystyle \vartheta }$
• The following applies to a fixed decision-making function ${\displaystyle \delta }$

${\displaystyle \sup _{\vartheta \in \Theta }R(\vartheta ,\delta )=\sup _{\mu \in \Theta _{W}}r(\mu ,\delta )}$.

In this way, for example, minimax decision functions can also be characterized using a priori probabilities.

Game theory interpretation

A statistical decision problem can be interpreted as a two-person zero-sum game of nature. First nature chooses its pure strategy from the parameter set by choosing the parameter , the statistician reacts to this with his mixed strategy , which corresponds to the choice of a decision function. The ordinary risk function then corresponds to the statistician's loss, which due to the zero-sum property is nature's gain. ${\displaystyle \vartheta }$ ${\displaystyle \Theta }$

The transition from the decision set to the set of probability measures on this set can then be viewed as a mixed extension of the game. Nature now uses mixed instead of pure strategies. Bayesian Risk is now the new payout feature to the expanded game with mixed strategies for both players. ${\displaystyle \Theta }$ ${\displaystyle \Theta _{W}}$

This new game has the advantage that far more general statements about the existence of equilibrium points apply.

The minimax strategies of this game are important for finding optimal decision-making functions. The minimax strategies for the statistician (player 2) are precisely the minimax decision functions , the minimax strategies of nature (player 1) are then also called the most unfavorable a priori distributions .

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .