Decision-making function

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A decision function is a term from mathematical statistics , the sub-area of statistics that uses the methods of probability theory . A distinction is made between non- randomized decision functions , in which each observation is assigned an unambiguous decision, and randomized decision functions , in which the choice of decision is still dependent on chance. Decision functions are used in the context of statistical decision problems . These include both test problems and estimation problems and the determination of confidence intervals using range estimators .

Closely related to the decision function is the loss function which, after a decision has been made, indicates the loss in relation to the decision made if the real but unknown value deviates from this decision. The decision function and the loss function are then combined to form the risk function , which indicates the potential loss using a given decision function.

definition

A statistical model and a decision space are given . ${\displaystyle (X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle (\Omega ,\Sigma )}$

Non-randomized decision function

Then, within the framework of mathematical statistics is a function that - -measurable is a non-randomized decision function called. The set of all non-randomized decision functions is denoted by. ${\displaystyle d\colon X\to \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Sigma }$ ${\displaystyle D}$

Randomized decision function

A randomized decision function is then a Markov kernel from to , that is, for : ${\displaystyle \delta (x,S)}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \delta \colon X\times \Sigma \to [0,1]}$

• For each is a probability measure on . ${\displaystyle x\in X}$ ${\displaystyle \delta (x,\;\cdot \;)}$ ${\displaystyle (\Omega ,\Sigma )}$
• For each there is a measurable function. ${\displaystyle S\in \Sigma }$ ${\displaystyle \delta (\;\cdot \;,S)}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle \delta (x,S)}$is then the probability of making a decision from the crowd while observing . The set of all randomized decision functions is denoted by. ${\displaystyle x}$ ${\displaystyle S}$ ${\displaystyle {\mathcal {D}}}$

Representation of non-randomized decision functions

Any non- randomized decision function can naturally be represented as a randomized decision function. To this end, the Markov kernel is defined as ${\displaystyle d}$

${\displaystyle \delta _{d}(x,A)={\begin{cases}1&{\text{ falls }}d(x)\in A\\0&{\text{ falls }}d(x)\notin A\end{cases}}}$.

If you use the Dirac measure , the Markov kernel can be written even more compactly than ${\displaystyle \Delta _{x}}$

${\displaystyle \delta _{d}(x,A):=\Delta _{\{d(x)\}}(A)}$.

This allows for surjective embedding in. Each non-randomized decision function is therefore only a special case of a randomized decision function. ${\displaystyle D}$ ${\displaystyle {\mathcal {D}}}$

example

Corresponding decision functions can be specified for each of the three classes of statistical decision problems. For example, classic decision functions are the point estimators for determining an unknown parameter, the interval estimators for determining a confidence interval and the statistical tests .

Point estimator

For example, if one considers the product model , which models a 100 coin toss, and chooses the parameter space as the basic set for the decision space and the corresponding Borel σ-algebra as the σ-algebra , then the sample mean is ${\displaystyle (\{0,1\}^{100},{\mathcal {P}}(\{0,1\})^{100},(\operatorname {Ber_{\vartheta }} ^{\otimes n})_{\vartheta \in [0,1]})}$ ${\displaystyle \Theta =[0,1]}$ ${\displaystyle {\mathcal {B}}([0,1])}$

${\displaystyle M:\{0,1\}^{100}\to [0,1],\quad M(\omega )={\frac {1}{100}}\sum _{i=1}^{100}\omega _{i}}$

a decision function that assigns the decision for an estimated parameter of the Bernoulli distribution to each outcome of the experiment, which consists of a 100-digit sequence of zeros and ones . This is a non-randomized decision function. ${\displaystyle \vartheta }$

Reduction to strongly sufficient σ-algebras

Every decision function can be reduced in the following sense: if a strongly sufficient σ-algebra (which for Borel spaces corresponds to a sufficient σ-algebra in the conventional sense ), then the decision function from to can be replaced by a decision function from to , so that for the risk function ${\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \delta (x,S)}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle \delta ^{*}(x,{\tilde {S}})}$ ${\displaystyle (X,{\mathcal {S}})}$ ${\displaystyle (\Omega ,\Sigma )}$

${\displaystyle R(\vartheta ,\delta )=R(\vartheta ,\delta ^{*}){\text{ für alle }}\vartheta \in \Theta }$

applies. The strongly sufficient σ-algebra thus already contains all information necessary for the risk assessment.

Optimal decision-making functions

There are different optimality criteria for decision functions, which are partly based on order theory and partly also on game theory.

Allowed decision-making functions

The risk function can be used to define an order relation between the decision functions ${\displaystyle R_{\delta }(\vartheta )}$

${\displaystyle \delta _{1}\preceq \delta _{2}{\text{ genau dann, wenn }}R_{\delta _{1}}(\vartheta )\leq R_{\delta _{1}}(\vartheta ){\text{ für alle }}\vartheta \in \Theta }$.

If and , then one calls and equivalent and writes . ${\displaystyle \delta _{1}\preceq \delta _{2}}$ ${\displaystyle \delta _{1}\succeq \delta _{2}}$ ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$ ${\displaystyle \delta _{1}\sim \delta _{2}}$

Is now a subset of the decision-making functions, it means a decision function allowed with regard to if for any further decision function with valid that is. ${\displaystyle {\tilde {D}}\subset {\mathcal {D}}}$ ${\displaystyle \delta _{0}}$ ${\displaystyle {\tilde {D}}}$ ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{1}\preceq \delta _{0}}$ ${\displaystyle \delta _{1}\sim \delta _{0}}$

The admissible decision functions are thus the minimum elements of the set with regard to the order relation . ${\displaystyle {\tilde {D}}}$ ${\displaystyle \preceq }$

Minimax decision functions

A decision function is called a minimax decision function with respect to the quantity , if ${\displaystyle \delta _{0}}$ ${\displaystyle {\tilde {D}}\subset {\mathcal {D}}}$

${\displaystyle \sup _{\vartheta \in \Theta }R(\vartheta ,\delta _{0})=\inf _{\delta \in {\tilde {D}}}\sup _{\vartheta \in \Theta }R(\vartheta ,\delta )}$

applies. The minimax decision functions correspond to a minimax strategy for a player with a strategy set against a player with a strategy set in a two-person zero-sum game with the risk function as the payout function. ${\displaystyle {\tilde {D}}}$ ${\displaystyle \Theta }$

Bayesian decision functions

If the Bayesian risk of the decision function is related to the a priori distribution , then a decision function is called a Bayesian decision function regarding the a priori distribution if ${\displaystyle r(\mu ,\delta )}$ ${\displaystyle \delta }$ ${\displaystyle \mu }$ ${\displaystyle \delta _{0}}$ ${\displaystyle \mu }$

${\displaystyle r(\mu ,\delta _{0})\leq r(\mu ,\delta )}$

applies to all . ${\displaystyle \delta \in {\tilde {D}}}$

Relationships between the optimality criteria

Conclusions from admissible decision functions

• If the decision function is admissible and an equalizer , then it is a minimax decision function.

Conclusions from Minimax decision functions

• If the minimax decision function is a worst-case a priori distribution , then a Bayesian decision function is with respect to and is a saddle point of the Bayesian risk. ${\displaystyle \delta _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle \delta _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle (\delta _{0},\mu _{0})}$
• If the minimax decision function is unique, it is also permissible.

Conclusions from Bayesian decision functions

• If the Bayesian decision function is unique with respect to it, then it is admissible. ${\displaystyle \delta _{0}}$ ${\displaystyle \mu _{0}}$
• If the Bayesian decision function is an equalizer , it is also a minimax decision function.

literature

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