De Finetti's theorem

The set of de Finetti (also representation theorem of de Finetti and de Finetti's representation theorem ) is a set of the stochastics on interchangeable families of random variables named after its discoverer Bruno de Finetti .

The theorem says that the distribution of an interchangeable sequence of Bernoulli-distributed random variables can be viewed as an integral over conditionally independent Bernoulli-distributed random variables.

Formulation of the sentence

Let be an infinite sequence of interchangeable Bernoulli-distributed random variables with parameters and density . Then there is a probability distribution with a distribution function , so that for each and every realization : ${\displaystyle (X_{i})}$ ${\displaystyle p}$ ${\displaystyle \pi _{p}}$ ${\displaystyle F}$ ${\displaystyle N}$ ${\displaystyle (x_{1},\dots ,x_{N})\in \{0,1\}^{N}}$

${\displaystyle P(X_{1}=x_{1},\dots ,X_{N}=x_{N})=\int \prod \limits _{i=1}^{N}\pi _{p}(x_{i})\mathrm {d} F(p)=\int _{0}^{1}p^{m}(1-p)^{N-m}\mathrm {d} F(p)}$,

where the number of "successful" Bernoulli attempts is. ${\displaystyle m}$

Consideration as weighting

In other words, we can say that there exists a random variable on with distribution function , so that the given independent conditionally are, that is ${\displaystyle Y}$ ${\displaystyle [0,1]}$ ${\displaystyle F_{y}}$ ${\displaystyle (X_{i})}$ ${\displaystyle Y}$

${\displaystyle P(X_{1}=x_{1},\dots ,X_{N}=x_{N}\mid Y=p)=P(X_{1}=x_{1}\mid Y=p)\cdots P(X_{N}=x_{N}\mid Y=p)}$

and it also applies to everyone

${\displaystyle P(X_{i}=1\mid Y=p)=p}$.

Furthermore, de Finetti's law of large numbers applies

${\displaystyle \sum \limits _{i=1}^{n}{\frac {X_{i}}{n}}\to Y\quad {\text{f.s. wenn }}n\to \infty }$.

Web links

• Exchangeability and de Finetti's theorem