Sufficient σ-algebra

A sufficient σ-algebra is a special set system in mathematical statistics that is used to formalize the compression of data without loss of information by means of sufficient statistics .

definition

A statistical model and a partial σ-algebra are given . Let the conditional expectation be given using the probability measure . The σ-algebra then called suffizient for if for every one - measurable function exists so that ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle \operatorname {E} _{P}(\cdot |{\mathcal {S}})}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle f_{A}}$

${\displaystyle f_{A}=\operatorname {E} _{P}(\mathbf {1} _{A}|{\mathcal {S}}){\text{ für alle }}P\in {\mathcal {P}}}$.

Remarks

A deficit of the term sufficiency is that if σ-algebras are with and is sufficient (with regard to a given distribution class), then in general it does not follow that it is also sufficient. One would expect this intuitively, because if the smaller σ-algebra is sufficient to enable lossless data compression, then this should also apply to the larger one, since it contains the smaller one, in which all the relevant information is already available . It should be noted here that the data compression here corresponds to the omission of the sets from the larger σ-algebra. ${\displaystyle {\mathcal {S}}_{1},{\mathcal {S}}_{2}}$ ${\displaystyle {\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}}$ ${\displaystyle {\mathcal {S}}_{1}}$ ${\displaystyle {\mathcal {S}}_{2}}$

Formally, this deficit can be seen as follows: if sufficient, the following applies according to the definition of the conditional expected value ${\displaystyle {\mathcal {S}}_{1}}$

${\displaystyle \int _{S}f_{A}\,\mathrm {d} P=P(A\cap S)}$

for everyone , but not necessarily for everyone . ${\displaystyle S\in {\mathcal {S}}_{1}}$ ${\displaystyle S\in {\mathcal {S}}_{2}}$

Explanation

The meaning of the term becomes clear when one restricts the probability measures from to . Then applies ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$

${\displaystyle P(A)=\int P(A|{\mathcal {S}})\mathrm {d} P|_{\mathcal {S}}=\int f_{A}\mathrm {d} P|_{\mathcal {S}}}$.

But since it does not depend on, the probability measures can only differ if their restrictions on are different. This means that all possible information that the probability measures can deliver is already contained in. ${\displaystyle f_{A}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$

Operational stability

• If and is sufficient for , then is sufficient for if and only if is sufficient for . ${\displaystyle {\mathcal {S}}^{*}\subset {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}^{*}}$ ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {S}}^{*}}$ ${\displaystyle (\Omega ,{\mathcal {S}},{\mathcal {P}}|_{\mathcal {S}})}$
• Let be the sets of all -zero sets . Are and sufficient and is , then is also sufficient. ${\displaystyle {\mathcal {N}}_{\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}_{1}}$ ${\displaystyle {\mathcal {S}}_{2}}$ ${\displaystyle {\mathcal {N}}_{\mathcal {P}}\subset {\mathcal {S}}_{1}}$ ${\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}}$
• Is sufficient and is a countably generated σ-algebra , then is also sufficient. It follows directly from this that countably generated over-σ-algebras of sufficient σ-algebras are again sufficient. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle \sigma ({\mathcal {S}},{\mathcal {E}})}$

Sufficiency and dominated distribution classes

Using the Halmos-Savage theorem , some stronger statements can be made for dominated distribution classes: ${\displaystyle {\mathcal {P}}}$

• Be sufficient and . Then every σ-algebra is with ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}^{*}}$

${\displaystyle {\mathcal {S}}\subset {\mathcal {S}}^{*}\subset {\mathcal {A}}}$

also sufficient.

• ${\displaystyle {\mathcal {S}}}$is sufficient regarding if and only if sufficient regarding is for all . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {P}}^{*}:=\{P_{i},P_{j}\}}$ ${\displaystyle P_{i},P_{j}\in {\mathcal {P}}}$
• If for the distribution classes on is dominated and is sufficient, then is also sufficient with regard to . ${\displaystyle i=1,2}$ ${\displaystyle {\mathcal {P}}_{i}}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle {\mathcal {S}}_{i}\subset {\mathcal {A}}_{i}}$ ${\displaystyle {\mathcal {S}}_{1}\otimes {\mathcal {S}}_{2}}$ ${\displaystyle {\mathcal {P}}_{1}\otimes {\mathcal {P}}_{2}}$

Another possibility for checking the sufficiency of a σ-algebra in the presence of a dominated distribution class is the Neyman criterion .

Related terms

The best-known term that can be derived from the sufficiency of a σ-algebra is sufficient statistics . A statistic is called sufficient if the σ-algebra it generates is sufficient. ${\displaystyle T}$ ${\displaystyle \sigma (T)}$

A modification of the sufficiency term treated here is strong sufficiency , which is defined using Markov kernels . The two terms agree on Borel spaces . An increase in sufficiency is minimal sufficiency : a σ-algebra is minimally sufficient if it is contained in every other sufficient σ-algebra except for -zero sets. Accordingly, a minimally sufficient σ-algebra is the maximum possible data reduction. ${\displaystyle {\mathcal {P}}}$

Another related but contrary term is that of a complete distribution class . This is a distribution class so that a distinction can be made between all functions. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {L}}(\Omega ,{\mathcal {S}},{\mathcal {P}})}$

The opposite of the term sufficiency is freedom of distribution . It formalizes that a σ-algebra carries no information or that a statistic does not convey any information.

The three theorems of Basu suggest a connection between sufficiency, completeness and freedom of distribution .

literature

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