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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{conditional expectation} \begin{quote}% This article is under construction. \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory} [[!include measure theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{conditional_expectation_relative_to_a_random_variable}{Conditional expectation relative to a random variable}\dotfill \pageref*{conditional_expectation_relative_to_a_random_variable} \linebreak \noindent\hyperlink{conditional_expectation_relative_to_a_subalgebra}{Conditional expectation relative to a sub-$\sigma$-algebra}\dotfill \pageref*{conditional_expectation_relative_to_a_subalgebra} \linebreak \noindent\hyperlink{conditional_probability}{Conditional probability}\dotfill \pageref*{conditional_probability} \linebreak \noindent\hyperlink{conditional_distribution_conditional_density}{Conditional distribution, Conditional density}\dotfill \pageref*{conditional_distribution_conditional_density} \linebreak \noindent\hyperlink{integral_kernel_stochastic_kernel}{Integral kernel, Stochastic kernel}\dotfill \pageref*{integral_kernel_stochastic_kernel} \linebreak \noindent\hyperlink{integral_kernel}{Integral kernel}\dotfill \pageref*{integral_kernel} \linebreak \noindent\hyperlink{stochastic_kernel}{Stochastic kernel}\dotfill \pageref*{stochastic_kernel} \linebreak \noindent\hyperlink{coupling_koppelung}{Coupling (Koppelung)}\dotfill \pageref*{coupling_koppelung} \linebreak \noindent\hyperlink{discrete_case}{Discrete case}\dotfill \pageref*{discrete_case} \linebreak \noindent\hyperlink{stochastic_processes}{Stochastic processes}\dotfill \pageref*{stochastic_processes} \linebreak \noindent\hyperlink{martingale}{Martingale}\dotfill \pageref*{martingale} \linebreak \noindent\hyperlink{markow_process}{Markow Process}\dotfill \pageref*{markow_process} \linebreak \noindent\hyperlink{chapmankolmogorow_equation}{Chapman-Kolmogorow Equation}\dotfill \pageref*{chapmankolmogorow_equation} \linebreak \noindent\hyperlink{InQuantumProbability}{In quantum probability theory}\dotfill \pageref*{InQuantumProbability} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} In [[probability theory]] a \emph{conditional expectation value} or \emph{conditional expectation}, for short, is like an \emph{[[expectation value]]} of some [[random variable]]/[[observable]], but \emph{conditioned} on the assumption that a certain event is assumed to have occured. More technically: If $(\Omega,\mathfrak{A},P)$ is a [[probability space]], the \emph{conditional expectation} $E[X|\Sigma]$ of a (measurable) random variable $X$ with respect to some sub-$\sigma$-algebra $\Sigma\subseteq \mathfrak{A}$ is some measurable random variable which is a `'coarsened'` version of $X$. We can think of $E[X|\Sigma]$ as a random variable with the same [[domain]] but which is measured with a sigma algebra containing only restricted information on the original event since to some events in $\mathfrak{A}$ has been assigned probability $1$ or $0$ in a consistent way. \hypertarget{conditional_expectation_relative_to_a_random_variable}{}\subsection*{{Conditional expectation relative to a random variable}}\label{conditional_expectation_relative_to_a_random_variable} Let $(\Omega,\mathfrak{A},P)$ be a [[probability space]], let $Y$ be a measurable function into a [[measure space]] $(U,\Sigma,P^Y)$ equipped with the [[pushforward measure]] induced by $Y$, let $X:(\Omega,\mathfrak{A},P)\to(\mathbb{R},\mathcal{B}(\mathbb{R}), \lambda)$ be a real-valued [[random variable]]. Then for $X$ and $Y$ there exists a essentially unique (two sets are defined to be equivalent if their difference is a set of measure $0$) [[integrable function]] $g=:E[X|Y]$ such that the following [[commuting diagram|diagram commutes]]: \begin{displaymath} \itexarray{ (\Omega,\mathfrak{A},P)& \stackrel{Y}{\to}& (\U, \Sigma, P^Y) \\ \downarrow^{\mathrlap{X}} && \swarrow_{\mathrlap{g=:E[X|Y]}} \\ (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda) } \end{displaymath} where $g:y\mapsto E[X|Y=y]$. Here `'commutes'` shall mean that (1) $g$ is $\Sigma$-measurable. (2) the [[integrals]] over $X$ and $g\circ Y$ are equal. In this case $g=E[X|Y]$ is called a version of the \textbf{conditional expectation of $X$ provided $Y$}. In more detail (2) is equivalent to that for all $B\in \Sigma$ we have \begin{displaymath} \int_{Y^{-1}(B)}X(\omega)d P(\omega)=\int_B g(u)d P^Y (u) \end{displaymath} and to \begin{displaymath} \int_{Y^{-1}(B)}X(\omega)d P(\omega)=\int_{Y^{-1}(B)}(g\circ Y)(\omega)d P (\omega) \end{displaymath} (The equivalence of the last two formulas is given since we always have $\int_B g(u)d P^Y (u)=\int_{Y^{-1}(B)} (g\circ Y)(\omega)d P (\omega)$ by the substitution rule.) Note that it does \emph{not} follow from the preceding definition that the conditional expectation exists. This is a consequence of the [[Radon-Nikodym theorem]] as will be shown in the following section. (Note that the argument of the theorem applies to the definition of the conditional expectation by random variables if we consider the [[pushforward measure]] as given by a sub-$\sigma$-algebra of the original one. In this sense $E[X|Y]$ is a `'coarsened version'` of $X$ factored by the information (i.e. the $\sigma$-algebra) given by $Y$.) \hypertarget{conditional_expectation_relative_to_a_subalgebra}{}\subsection*{{Conditional expectation relative to a sub-$\sigma$-algebra}}\label{conditional_expectation_relative_to_a_subalgebra} Note that by construction of the pushforward-measure it suffices to define the conditional expectation only for the case where $\Sigma:=\mathfrak{S}\subseteq \mathfrak{A}$ is a sub-$\sigma$-algebra. (Note that we loose information with the notation $P^Y$; e.g $P^{id}_\mathfrak{A}$ is different from $P^{id}_\mathfrak{S}$) The diagram \begin{displaymath} \itexarray{ (\Omega,\mathfrak{A},P)& \stackrel{id}{\to}& (\Omega, \mathfrak{S}, P^{id}) \\ \downarrow^X&& \swarrow^{Z=:E[X|\mathfrak{S}]} \\ (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda) } \end{displaymath} is commutative (in our sense) iff (a) $Z$ is $\mathfrak{S}$-measurable (b) $\int_A Z d P=\int_A X d P$, $\forall A\in \mathfrak{S}$ We hence can write the conditional expectation as the equivalence class \begin{displaymath} E[X|\mathfrak{S}]=\{Z\in L^1 (\Omega, F,P)|\int_A ZdP=\int_A XdP\;\forall A\in \mathfrak{S}\} \end{displaymath} An element of this class is also called a \emph{version}. \begin{theorem} \label{}\hypertarget{}{} $E[X|\mathfrak{S}]$ exists and is unique almost surely. \end{theorem} \begin{proof} Existence: By \begin{displaymath} Q(A):=\int_A X(\omega)P(d\omega) \end{displaymath} $A\in \mathfrak{A}$ is defined a measure $Q$ on $(\Omega,\mathfrak{A},P)$ (if $X\ge 0$; if not consider the positive part $X^+$ and the negative part $X^-$ of $X=X^+ -X^-$ separate and use linearity of the integral). Let $P|_{\mathfrak{S}}$ be the restriction of $P$ to $\mathfrak{S}$. Then \begin{displaymath} Q\lt\lt P|_{\mathfrak{S}} \end{displaymath} meaning: $P|_{\mathfrak{S}}(M)=0\Rightarrow Q(M)=0$ for all $M\in\mathfrak{S}$. This is the condition of the [[Radon-Nikodym derivative|theorem of Radon-Nikodym]] (the other condition of the theorem that $P|_{\mathfrak{S}}$ is $\sigma$-finite is satisfied since $P$ is a probability measure). The theorem implies that $Q$ has a density w.r.t $P|_{\mathfrak{S}}$ which is $E[X|\mathfrak{S}]$. Uniqueness: If $g$ and $g^\prime$ are candidates, by linearity the integral over their difference is zero. \end{proof} \hypertarget{conditional_probability}{}\subsection*{{Conditional probability}}\label{conditional_probability} From elementary probability theory we know that $P(A)=E[1_A]$. For $A\in \mathfrak{S}$ we call $P(A|\mathfrak{S}):=E[1_A|\mathfrak{S}]$ the \emph{conditional probability of $A$ provided $B$}. \hypertarget{conditional_distribution_conditional_density}{}\subsection*{{Conditional distribution, Conditional density}}\label{conditional_distribution_conditional_density} \hypertarget{integral_kernel_stochastic_kernel}{}\subsection*{{Integral kernel, Stochastic kernel}}\label{integral_kernel_stochastic_kernel} In probability theory and statistics, a stochastic kernel is the transition function of a stochastic process. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function. \hypertarget{integral_kernel}{}\subsubsection*{{Integral kernel}}\label{integral_kernel} An \emph{integral transform} $T$ is an assignation of the form \begin{displaymath} (Tf)(u)=\int K(t,u)f(t)dt \end{displaymath} where the function of two variables $K( \dots ,\cdots)$ is called \emph{integral kernel of the transform $T$}. \hypertarget{stochastic_kernel}{}\subsubsection*{{Stochastic kernel}}\label{stochastic_kernel} Let $(\Omega_1,\mathfrak{A}_1)$ be a measure space, let $(\Omega_2,\mathfrak{A}_2)$ be a measurable space. A map $Q: \Omega_1\times \mathfrak{A}_2$ satisfying (1) $Q(-, A):\Omega_1\to [0,1]$ is $\mathfrak{A}_1$ measurable $\forall A_2\in \mathfrak{A}_2$ (2) $Q(\omega,-):\mathfrak{A}_2\to [0,1]$ is a probability measure on $(\Omega_2,\mathfrak{A}_2)$, $\forall \omega_1\in \Omega_1$ is called a \emph{stochastic kernel} or \emph{transition kernel} (or \emph{Markov kernel} - which we avoid since it is confusing) \emph{from $(\Omega_1,\mathfrak{A}_1)$ to $(\Omega_2,\mathfrak{A}_2)$.} Then $Q$ induces a function between the classes of measures on $(\Omega_1, \mathfrak{A}_1)$ and on $(\Omega_2, \mathfrak{A}_2)$ \begin{displaymath} \overline{Q}: \begin{cases} M(\Omega_1, \mathfrak{A}_1)& \to& M(\Omega_2, \mathfrak{A}_2) \\ \mu& \mapsto& (A\mapsto \int_{\Omega_1} Q(-, A) d\mu) \end{cases} \end{displaymath} If $\mu$ is a probability measure, then so is $\overline{Q}(\mu)$. The symbol $Q(\omega, A)$ is sometimes written as $Q(A|\omega)$ in optical proximity to a conditional probability. The stochastic kernel is hence in particular an integral kernel. In a discrete stochastic process (see below) the transition function is a stochastic kernel (more precisely it is the function $\overline{Q}$ induced by a kernel $Q$). \hypertarget{coupling_koppelung}{}\paragraph*{{Coupling (Koppelung)}}\label{coupling_koppelung} Let $(\Omega_1,\mathfrak{A}_1, P_1)$ be a probability space, let $(\Omega_2,\mathfrak{A}_2)$ be a measure space, let $Q:\Omega_1\times \mathfrak{A}_2\to [0,1]$ be a stochastic kernel from $(\Omega_1,\mathfrak{A}_1, P_1)$ to $(\Omega_2,\mathfrak{A}_2)$. Then by \begin{displaymath} P(A):=\int_{\Omega_1}(\int_{\Omega_2} 1_A (\omega_1,\omega_2 Q(\omega_1,\omega_2))P_1(d \omega_1) \end{displaymath} is defined a probability measure on $\mathfrak{A}_1\otimes\mathfrak{A}_2$ which is called \emph{coupling}. $P=:P\otimes Q$ is unique with the property \begin{displaymath} P(A_1\times A_2)=\int_{A_1} Q(\omega_1, A_2) P_1(d\omega_1) \end{displaymath} \begin{theorem} \label{}\hypertarget{}{} Let (with the above settings) $Y:\Omega_1\to \Omega_2$ be $(\mathfrak{A}_1,\mathfrak{A}_2)$-measurable, let $X$ be a $d$-dimensional random vector. Then there exists a stochastic kernel from $(\Omega_1, \mathfrak{A}_1)$ to $(\mathbb{R}^d,\mathcal{B}(\mathbb{R})^d)$ such that \begin{displaymath} P^{X,Y}=P^Y\otimes Q \end{displaymath} and $Q$ is (a version of) the conditional distribution of $X$ provided $Y$, i.e. \begin{displaymath} Q(y,-)=P^X(-|Y=y) \end{displaymath} \end{theorem} This theorem says that that $Q$ (more precisely $y\mapsto Q(y,-)$) fits in the diagram \begin{displaymath} \itexarray{ (\Omega_1,\mathfrak{A}_1,P)& \stackrel{Y}{\to}& (\Omega_2,\mathfrak{A}_2, P^Y) \\ \downarrow^X&& \swarrow^{Q} \\ (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda) } \end{displaymath} and $E[X|Y]=Q$. \hypertarget{discrete_case}{}\paragraph*{{Discrete case}}\label{discrete_case} In the discrete case, i.e. if $\Omega_1$ and $\Omega_2$ are finite- or enumerable sets, it is possible to reconstruct $Q$ by just considering one-element sets in $\mathfrak{A}_2$ and the related probabilities \begin{displaymath} p_{ij}:= Q(i,\{j\}) \end{displaymath} called \emph{transition probabilities} encoding $Q$ assemble to a (perhaps countably infinite) matrix $M$ called \emph{transition matrix of $Q$ resp. of $\overline{Q}$}. Note that $p_{ij}$ is the probability of the transition of the \emph{state} (aka. \emph{elementary event} or \emph{one-element event}) $i$ to the \emph{event} $\{j\}$ (which in this case happens to have only one element, too). We have $\sum_i p_{ij}=1$ forall $i\in \Omega_1$. If $\rho:=(p_i)_{i\in \Omega_1}$ is a counting density on $\Omega_1$, then \begin{displaymath} pM=(\sum_{i\in \Omega} p_i p_{ij})_{j\in \Omega_2} \end{displaymath} is a counting density on $\Omega_2$. The conditional expectation plays a defining role in the theory of \emph{martingales} which are \emph{stochastic processes} such that the conditional expectation of the next value (provided the previous values) equals the present realized value. \hypertarget{stochastic_processes}{}\subsubsection*{{Stochastic processes}}\label{stochastic_processes} The terminology of \emph{stochastic processes} is a special interpretation of some aspects of [[infinitary combinatorics]] in terms of [[probability theory]]. Let $I$ be a [[total order]] (i.e. transitive, antisymmetric, and total). A \emph{stochastic process} is a diagram $X_I: I\to \mathcal{R}$ where $\mathcal{R}$ is the class of random variables such that $X_I(i)=:X_i:(\Omega_i, \mathfrak{F}_i, P_i)\to (S_i, \mathfrak{S}_i)$ is a random variable. Often one considers the case where all $(S_i, \mathfrak{S}_i)=(S, \mathfrak{S})$ are equal; in this case $S$ is called \emph{state space of the process $X_I$}. If all $\Omega_i=\Omega$ are equal and the class of $\sigma$-algebras $(\mathfrak{A}_i)_{i\in I}$ is \emph{filtered} i.e. \begin{displaymath} \mathfrak{F}_i\subseteq \mathfrak{F}_j\;;iff\;; i\le j \end{displaymath} and all $X_l$ are $\mathfrak{F}_l$ measurable, the process is called \emph{adapted process}. For example the \emph{natural filtration} where $\mathfrak{F}_i=\sigma(\{X^{-1}_l(A), l\le i, A\in \mathfrak{S}\})$ gives an adapted process. In terms of a diagram we have for $i\le j$ \begin{displaymath} \itexarray{ (\Omega_j,\mathfrak{A}_j,P_j)& \stackrel{f}{\to}& (\Omega_i,\mathfrak{A}_i,P_i) \\ \downarrow^{X_j}&& \swarrow^{\omega_i\mapsto Q(\omega_i,-)} \\ (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda) } \end{displaymath} and $\overline{Q}:(\Omega_i,\mathfrak{A}_i,P_i)\to(\Omega_j,\mathfrak{A}_j,P_j)$ where $Q:\Omega_i\times\mathfrak{A}_j\to [0,1]$ is the transition probability for the passage from state $i$ to state $j$. \hypertarget{martingale}{}\subsubsection*{{Martingale}}\label{martingale} An adapted stochastic process with the natural filtration in discrete time is called a \emph{martingale} if all $E[X_i]\lt \infty$ and $\forall i\le j, E[X_j|\mathfrak{A}_i]=X_i$. \begin{displaymath} \itexarray{ (\Omega_j,\mathfrak{A}_j,P_j)& \stackrel{f}{\to}& (\Omega_i,\mathfrak{A}_i,P_i) \\ \downarrow^{X_j}&& \swarrow^{E[X_j|\mathfrak{A}_i]=X_i} \\ (\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda) } \end{displaymath} [[martingale]] (\ldots{}) \hypertarget{markow_process}{}\subsubsection*{{Markow Process}}\label{markow_process} An adapted stochastic process satisfying \begin{displaymath} P(X_t|\mathfrak{A}_s)=P(X_t|X_s)\;;\forall s\le t \end{displaymath} is called a \emph{Markow process}. \hypertarget{chapmankolmogorow_equation}{}\subsection*{{Chapman-Kolmogorow Equation}}\label{chapmankolmogorow_equation} For a Markow process the Chapman-Kolmogorow equation encodes the statement that the transition probabilities of the process form a [[semigroup]]. If in the notation from above $(P_t:\Omega\times\mathfrak{A}\to [0,1])_t$ is a family of stochastic kernels $(\Omega,\mathfrak{A})\to(\Omega,\mathfrak{A})$ such that all $P_t(\omega,-):\mathfrak{A}\to [0,1]$ are probabilities, then $(P_t)_t$ is called \emph{transition semigroup} if \begin{displaymath} \overline P_t (P_s(\omega,A))=P_{s+t} (\omega, A) \end{displaymath} where \begin{displaymath} \overline P_t: P_s(\omega,-)\mapsto (A\mapsto\int_\Omega P_t (y,A) P_s(\omega,-)(d_y)) \end{displaymath} \hypertarget{InQuantumProbability}{}\subsection*{{In quantum probability theory}}\label{InQuantumProbability} In the dual algebraic formulation of [[probability theory]] known as \emph{[[noncommutative probability theory]]} or \emph{[[quantum probability theory]]}, where the concept of \emph{[[expectation value]]} is primitive (while that of the corresponding [[probability space]] (if it exists) is a derived concept), the concept of conditional expection appears as follows (e.g. \hyperlink{RedeiSummers06}{Redei-Summers 06, section 7.3}): Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$. This means that for $A \in \mathcal{A}$ any [[observable]], its \emph{[[expectation value]]} in the given [[state on a star-algebra|state]] is \begin{displaymath} \mathbb{E}(A) \;\coloneqq\; \langle A \rangle \in \mathbb{C} \,. \end{displaymath} More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]] \begin{equation} P^\ast = P \,, \phantom{AAA} P P = P \label{RealIdempotent}\end{equation} thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is \begin{equation} \mathbb{E}(A \vert P) \;\coloneqq\; \frac{ \left \langle P A P \right\rangle }{ \left\langle P \right\rangle } \,. \label{ConditionalExpectation}\end{equation} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Bayesian reasoning]] \item [[wavefunction collapse]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} See also \begin{itemize}% \item Wikipedia, \emph{\href{https://en.wikipedia.org /wiki/Conditional_expectation}{Conditional expectation}} \end{itemize} Discussion form the point of view of [[quantum probability]] is in \begin{itemize}% \item [[Miklos Redei]], [[Stephen Summers]], section 7.3 of \emph{Quantum Probability Theory} (\href{https://arxiv.org/abs/quant-ph/0601158}{arXiv:quant-ph/0601158}) \end{itemize} [[!redirects conditional probability]] [[!redirects conditional probabilities]] [[!redirects conditional expectation value]] [[!redirects conditional expectation values]] \end{document}