Transition core

This article deals with transition probabilities in probability theory. For information on transition probabilities in physics, see transition dipole moment and vibronic transition .

A transition kernel refers to special images between measurement spaces in probability theory that can be measured in the first argument and provide a measure in the second argument . Special cases of transition kernels are the so-called stochastic kernels , which are also called Markov kernels or probability kernels . With them the measure is always a measure of probability . If the measure is always a sub-probability measure , one also speaks of sub-Markov nuclei or substochastic nuclei .

In particular, the Markov kernels play an important role in probability theory, for example in the formulation of the regular conditional distribution or the theory of stochastic processes . In particular, they form the basis for the formulation of the transition probabilities of Markov chains or existence statements like the Ionescu-Tulcea theorem .

definition

Two measuring rooms and are given . A map is called a transition kernel from to if: ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0})}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle K\colon \Omega _{0}\times {\mathcal {A}}_{1}\to [0,\infty )}$ ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0})}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$

• For each is a measure on . ${\displaystyle x\in \Omega _{0}}$ ${\displaystyle K(x,\;\cdot \;)}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$
• For each is a - measurable function . ${\displaystyle A_{1}\in {\mathcal {A}}_{1}}$ ${\displaystyle K(\;\cdot \;,A_{1})}$ ${\displaystyle {\mathcal {A}}_{0}}$

If the measure is a σ-finite measure for all , one speaks of a σ-finite transition kernel, if it is always finite, one speaks of a finite transition kernel. If the measure for all is a probability measure , it is called a stochastic kernel or Markov kernel . If the measure for all is a sub-probability measure , it is called a substochastic kernel or sub-Markovian kernel . ${\displaystyle x\in \Omega _{0}}$ ${\displaystyle x\in \Omega _{0}}$ ${\displaystyle K}$ ${\displaystyle x\in \Omega _{0}}$ ${\displaystyle K}$

Note: With some definitions the arguments are written in reverse order, or also , based on conditional probabilities . ${\displaystyle K}$ ${\displaystyle K(A',x)}$ ${\displaystyle K(A'|x)}$

Elementary examples

• The Poisson distribution is a Markov kernel from to . Because the function with parameters is continuously in and therefore measurable. Furthermore, the Poisson distribution with parameter is a probability distribution for each . So it's a transition core. ${\displaystyle \operatorname {Poi} _{x}(A):=K(x,A)}$ ${\displaystyle (\mathbb {R} _{+},{\mathcal {B}}(\mathbb {R} _{+}))}$ ${\displaystyle (\mathbb {N} ,{\mathcal {P}}(\mathbb {N} ))}$ ${\displaystyle f_{A}(x)=\operatorname {Poi} _{x}(A)}$ ${\displaystyle A\in {\mathcal {P}}(\mathbb {N} )}$ ${\displaystyle x\in \mathbb {R} _{+}}$ ${\displaystyle x\in \mathbb {R} _{+}}$ ${\displaystyle x}$
• The stochastic matrix

${\displaystyle A={\begin{pmatrix}0&{\tfrac {1}{2}}&{\tfrac {1}{2}}\\{\tfrac {1}{2}}&{\tfrac {1}{2}}&0\\{\tfrac {1}{2}}&0&{\tfrac {1}{2}}\end{pmatrix}}}$

can be understood as a Markov nucleus from to . Because for each the -th line is a probability vector and thus a probability measure . In addition, it is a mapping between finite sets provided with the power set and thus measurable. ${\displaystyle (\{1,2,3\},{\mathcal {P}}(\{1,2,3\}))}$ ${\displaystyle (\{1,2,3\},{\mathcal {P}}(\{1,2,3\}))}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle \{1,2,3\}}$

properties

Dimensions through cores

Every measure on rules through ${\displaystyle \mu }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle K}$

${\displaystyle \nu (A')=\int _{\Omega }K(x,A')\mu (dx)}$

a measure up to. This dimension is usually referred to as. If a probability measure is true , then is also , i.e. is also a probability measure. ${\displaystyle \nu }$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle \mu K}$ ${\displaystyle \mu }$ ${\displaystyle \mu (\Omega )=1}$ ${\displaystyle \mu K(\Omega ')=1}$ ${\displaystyle \mu K}$

In the case , a measure for which applies is called a stationary measure . A stationary probability measure is also called a stationary distribution . ${\displaystyle (\Omega ,{\mathcal {A}})=(\Omega ',{\mathcal {A}}')}$ ${\displaystyle \mu }$ ${\displaystyle \mu =\mu K}$

Measurable functions through cores

Every non-negative measurable function orders through ${\displaystyle g\colon \Omega '\to \mathbb {R} }$ ${\displaystyle K}$

${\displaystyle f(x)=\int _{\Omega '}g(y)K(x,dy)}$

a non-negative measurable function . This function is usually referred to as. With the shorthand applies to all dimensions on and all non-negative measurable functions the equation . ${\displaystyle f\colon \Omega \to \mathbb {R} }$ ${\displaystyle Kg}$ ${\displaystyle \mu f=\int _{\Omega }f(x)\,\mu (dx)}$ ${\displaystyle \mu }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle g\colon \Omega '\to \mathbb {R} }$ ${\displaystyle (\mu K)g=\mu (Kg)}$

Discreet case

In the discrete case, where and are finite or countable sets, it suffices to state the probabilities with which one can get from the state into the state . With the names of the general case then applies . These probabilities form a transition matrix which has the property that all elements lie between and and that the row sums have the value . Such a matrix is ​​called a stochastic matrix. It assigns each probability distribution of a discrete density , the probability density ${\displaystyle \Omega }$ ${\displaystyle \Omega '}$ ${\displaystyle p_{i,j}}$ ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle p_{i,j}=K(i,\{j\})}$ ${\displaystyle M=(p_{i,j})_{i\in \Omega ,j\in \Omega '}}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle \sum _{j\in \Omega '}p_{i,j}}$ ${\displaystyle 1}$ ${\displaystyle \Omega }$ ${\displaystyle \rho =(\rho _{i})_{i\in \Omega }}$

${\displaystyle \rho M={\Bigl (}\sum _{i\in \Omega }\rho _{i}p_{i,j}{\Bigr )}_{j\in \Omega '}}$

a probability distribution up to, that is , it is calculated with the usual matrix multiplication , with counting densities being understood as row vectors. ${\displaystyle \Omega '}$ ${\displaystyle \rho M}$

If is a nonnegative function, interpreted as a column vector with nonnegative entries, then applies ${\displaystyle g\colon \Omega '\to \mathbb {R} }$ ${\displaystyle (g_{j})_{j\in \Omega '}}$

${\displaystyle Kg={\Bigl (}\sum _{j\in \Omega '}p_{i,j}g_{j}{\Bigr )}_{i\in \Omega }}$.

That means, in the discrete case, it is also calculated with the usual matrix multiplication , interpreted as a column vector with indices in . ${\displaystyle Kg}$ ${\displaystyle \Omega }$

Note: In some definitions, rows and columns of the matrix are used in reverse.

Operations of transition cores

Concatenation

If there are three measuring spaces and two substochastic kernels from to and from to , then the chain of kernels and a mapping is ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0}),\;(\Omega _{1},{\mathcal {A}}_{1}),\,(\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle K_{1}}$ ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0})}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle K_{2}}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$

${\displaystyle K_{1}\cdot K_{2}\colon \Omega _{0}\times {\mathcal {A}}_{2}\to [0,\infty )}$

defined by

${\displaystyle (K_{1}\cdot K_{2})(x_{0},A_{2})=\int _{\Omega _{1}}K_{1}(x_{0},\mathrm {d} x_{1})K_{2}(x_{1},A_{2})}$.

The concatenation is then a substochastic kernel from to . If and are stochastic, then is also stochastic. ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle K_{1}\cdot K_{2}}$

Products

Given are the dimensional spaces and and two finite transition kernels from to and from to . Then you define the product of the kernels and ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1}),(\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle (\Omega _{3},{\mathcal {A}}_{3})}$ ${\displaystyle K_{1}}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle K_{2}}$ ${\displaystyle (\Omega _{1}\times \Omega _{2},{\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2})}$ ${\displaystyle (\Omega _{3},{\mathcal {A}}_{3})}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$

${\displaystyle K_{1}\otimes K_{2}\colon \Omega _{1}\times ({\mathcal {A}}_{2}\otimes {\mathcal {A}}_{3})\to [0,\infty )}$

as

${\displaystyle (\omega _{1},A)\mapsto \int _{\Omega _{2}}K_{1}(\omega _{1},\mathrm {d} \omega _{2})\int _{\Omega _{3}}K_{2}((\omega _{1},\omega _{2}),\mathrm {d} \omega _{3})\chi _{A}((\omega _{2},\omega _{3}))}$.

The product is then a σ-finite transition kernel from to . If both kernels are stochastic (or substochastic), then the product of the kernels is also stochastic (or substochastic). ${\displaystyle K_{1}\otimes K_{2}}$ ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\Omega _{2}\times \Omega _{3},{\mathcal {A}}_{2}\otimes {\mathcal {A}}_{3})}$

If there is only one core of after , the core is understood as the core of , which is independent of the first component. ${\displaystyle K_{2}}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle (\Omega _{3},{\mathcal {A}}_{3})}$ ${\displaystyle (\Omega _{1}\times \Omega _{2},{\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2})}$

Further examples

• If a probability measure is on , then there is a ( independent) transition probability. ${\displaystyle \nu \colon {\mathcal {A}}'\to [0,1]}$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle K(x,A')=\nu (A')}$ ${\displaystyle x\in \Omega }$
• For and the Dirac measure in the point is defined by a transition probability from to , which is also called the unit kernel. It applies to all dimensions on and to all non-negative measurable functions . ${\displaystyle (\Omega ,{\mathcal {A}})=(\Omega ',{\mathcal {A}}')}$ ${\displaystyle \delta _{x}}$ ${\displaystyle x\in \Omega }$ ${\displaystyle I(x,A)=\delta _{x}(A)}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \mu I=\mu }$ ${\displaystyle \mu }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle If=f}$ ${\displaystyle f\colon \Omega \to \mathbb {R} }$
• If a function is nonnegative and measurable with respect to the product σ-algebra and a measure of with for all , then becomes through ${\displaystyle k\colon \Omega \times \Omega '\to \mathbb {R} }$ ${\displaystyle {\mathcal {A}}\otimes {\mathcal {A}}'}$ ${\displaystyle \nu }$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle \int _{\Omega '}k(x,y)\,\nu (dy)=1}$ ${\displaystyle x\in \Omega }$

${\displaystyle K(x,A')=\int _{A'}k(x,y)\,\nu (dy)}$

defines a transition probability. So here is the probability measure on with the probability density . ${\displaystyle K(x,\;\cdot \;)}$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle \nu }$ ${\displaystyle k(x,\;\cdot \;)}$

• Let be fixed and the binomial distribution with parameters and , understood as a probability measure . Then through ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle B_{n,p}}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle \Omega '=\{0,1,\dotsc ,n\}}$

${\displaystyle K(p,A')=B_{n,p}(A')}$

a transition probability from to is defined. For example, if a beta distribution is on , then the associated beta binomial distribution is on . ${\displaystyle (\Omega ,{\mathcal {A}})=([0,1],{\mathcal {B}}([0,1]))}$ ${\displaystyle (\Omega ',{\mathcal {P}}(\Omega '))}$ ${\displaystyle \beta _{a,b}}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \beta _{a,b}K}$ ${\displaystyle \Omega '}$

Representation as Daniell-continuous illustrations and composition

Each Markov core of to is in the area of the numeric, non-negative functions via ${\displaystyle K}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle E^{*}}$ ${\displaystyle f\colon \Omega \to [0,\infty ]}$

${\displaystyle (Tf)(\omega ):=\int f(\omega ')K(\omega ,d\omega ')}$

an image with the following properties is assigned: ${\displaystyle T\colon E^{*}\to E^{*}}$

1. ${\displaystyle f\geq 0\Rightarrow Tf\geq 0}$for each (positivity), ${\displaystyle f\in E^{*}}$
2. ${\displaystyle f_{n}\uparrow f\Rightarrow Tf_{n}\uparrow Tf}$for every monotonically increasing sequence in (Daniell continuity, after Percy John Daniell ), ${\displaystyle (f_{n})}$ ${\displaystyle E^{*}}$
3. ${\displaystyle T(f+g)=Tf+Tg}$ (Additivity).

For each image with these properties there is in turn exactly one core for which the image formed in this way represents. ${\displaystyle T}$ ${\displaystyle T}$

A definition for the composition of the associated cores can be derived from the composition of these figures : By ${\displaystyle T_{1}\circ T_{2}}$

${\displaystyle K_{1}K_{2}(\omega ,A)=\int K_{1}(\omega ,\mathrm {d} \omega ')K_{2}(\omega ',A)}$

is a stochastic core defined from to , which is called the composition of and . In the discrete case, the multiplication of the two transition matrices corresponds . ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle K_{1}K_{2}}$

Special applications

Markov cores are widely used in modeling about with the help of Markov and hidden Markov models . In quantum physics , transition probabilities between quantum mechanical states are often examined. Markov kernels are also used in mathematical statistics to define a decision function within the framework of a general statistical decision problem , which assigns a decision to each outcome of an experiment. The decision can be both a parameter estimate and the choice of a confidence interval or the decision for or against a hypothesis.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Heinz Bauer : Probability Theory. De Gruyter, Berlin 2002, ISBN 3-11-017236-4 .
• Erhan Çınlar: Probability and Stochastics. Springer, New York et al. 2011, ISBN 978-0-387-87858-4 .