# 1.12 The hilbert space of random variables

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Describes random variables in terms of Hilbert spaces, defining inner products, norms, and minimum mean square error estimation.

## Probability – notation primer

Definition 1 A random variable x is defined by a distribution function

$P\left(x\right)={F}_{X}\left(x\right)=\mathrm{Prob}\left(X\le x\right)$

The density function is given by

$\frac{\partial P\left(x\right)}{\partial x}={f}_{X}\left(x\right)=\frac{\partial \mathrm{Prob}\left(X\le x\right)}{\partial x}$

Definition 2 The expectation of a function $g\left(x\right)$ over the random variable $x$ is

${E}_{X}\left[g\left(x\right)\right]={\int }_{-\infty }^{\infty }g\left(x\right){f}_{X}\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$

Definition 3 Pairs of random variables $X,Y$ are defined by the joint distribution function

$P\left(x,y\right)={F}_{XY}\left(x,y\right)=\mathrm{Prob}\left(X\le x,Y\le y\right)$

The joint density function is given by

$\frac{{\partial }^{2}P\left(x,y\right)}{\partial x\partial y}={f}_{XY}\left(x,y\right)=\frac{{\partial }^{2}\mathrm{Prob}\left(X\le x,Y\le y\right)}{\partial x\partial y}$

The expectation of a function $g\left(x,y\right)$ is given by

${E}_{X,Y}\left[g\left(x,y\right)\right]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }g\left(x,y\right){f}_{XY}\left(x,y\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy$

## A hilbert space of random variables

Definition 4 Let $\left\{{Y}_{1},\cdots ,{Y}_{n}\right\}$ be a collection of zero-mean ( $E\left[{Y}_{i}\right]=0$ ) random variables. The space $H$ of all random variables that are linear combinations of those $n$ random variables $\left\{{y}_{1},\cdots ,{y}_{n}\right\}$ is a Hilbert space with inner product

$⟨X,Y⟩=E\left[x\overline{y}\right]={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }x\overline{y}{f}_{XY}\left(x,y\right)\phantom{\rule{0.166667em}{0ex}}\text{d}x\text{d}y.$

We can easily check that this is a valid inner product:

• $⟨x,x⟩=E\left[x\overline{x}\right]={\int }_{-\infty }^{\infty }{|x|}^{2}{f}_{x}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\text{d}x=E\left[{|x|}^{2}\right]\ge 0$ ;
• $⟨x,x⟩=0$ if and only if ${f}_{X}\left(x\right)=\delta \left(x\right)$ , i.e., if $X$ is a random variable that is deterministically zero (and this random variable is the “zero” of this Hilbert space);
• $⟨x,y⟩=\overline{⟨y,x⟩}$ ;
• $⟨x+y,z⟩=E\left[\left(x+y\right)\overline{z}\right]=E\left[x\overline{z}\right]+E\left[y\overline{z}\right]=⟨x,z⟩+⟨y,z⟩;$

Note in particular that orthogonality, i.e., $⟨x,y⟩=0$ , implies $E\left[x\overline{y}\right]=0$ , i.e., $x$ and $y$ are independent random variables. Additionally, the induced norm $\parallel X\parallel =\sqrt{⟨X,X⟩}=\sqrt{E\left[|x{|}^{2}\right]}$ is the standard deviation of the zero-mean random variable $X$ .

## A hilbert space of random vectors

One can define random vectors $X$ , $Y$ whose entries are random variables:

$X=\left[\begin{array}{c}{X}_{1}\\ ⋮\\ {X}_{N}\end{array}\right],Y=\left[\begin{array}{c}{Y}_{1}\\ ⋮\\ {Y}_{N}\end{array}\right].$

For these, the following inner product is an extension of that given above:

$⟨X,Y⟩=E\left[{y}^{H}x\right]=E\left[\sum _{i=1}^{n},\overline{{y}_{i}},{x}_{i}\right]=E\left[\mathrm{trace},\left[,x,{y}^{H},\right]\right].$

The induced norm is

$\parallel X\parallel =\sqrt{⟨X,X⟩}=E\left[\sqrt{{x}^{H}x}\right]=E\left[\sqrt{{\sum }_{i=1}^{N}{|{x}_{i}|}^{2}}\right],$

the expected norm of the vector $x$ .

## Minimum mean square error estimation

In an MMSE estimation problem, we consider $Y=AX+N$ , where $X,Y$ are two random vectors and $N$ is usually additive white Gaussian noise ( $Y$ is $m×1$ , $A$ is $m×n$ , X is $n×1$ , and $N\sim \mathcal{N}\left(0,{\sigma }^{2}I\right)$ is $m×1$ ). Due to this noise model, we want an estimate $\stackrel{^}{X}$ of $X$ that minimizes $E\left[\parallel X-,\stackrel{^}{X},{\parallel }^{2}\right]$ ; such an estimate has highest likelihood under an additive white Gaussian noise model. For computational simplicity, we often want to restrict the estimator to be linear, i.e.

$\stackrel{^}{X}=KY=\left[\begin{array}{c}{K}_{1}^{H}\\ ⋮\\ {K}_{n}^{H}\end{array}\right]Y,$

where ${K}_{i}^{H}$ denotes the ${i}^{th}$ row of the estimation matrix $K$ and ${\stackrel{^}{X}}_{i}={K}_{i}^{H}Y$ . We use the definition of the ${\ell }_{2}$ norm to simplify the equation:

$\underset{K}{min}E\left[\parallel X-,\stackrel{^}{X},{\parallel }_{2}^{2}\right]=\underset{K}{min}E\left[{\parallel X-KY\parallel }_{2}^{2}\right]=\underset{K}{min}E\left[\sum _{i=1}^{n},{\left({X}_{i}-{K}_{i}^{H}Y\right)}^{2}\right]$

Since the terms involved in the sum are independent from each other and nonnegative, this minimization can be posed in terms of $n$ individual minimizations: for $i=1,2,...,n$ , we solve

$\underset{{K}_{i}}{min}E\left[{\left({X}_{i}-{K}_{i}^{H}Y\right)}^{2}\right]=\underset{{K}_{i}}{min}E\left[{\left({X}_{i}-\sum _{i=1}^{n}\overline{{K}_{ij}}{Y}_{j}\right)}^{2}\right]=\underset{{K}_{i}}{min}∥{X}_{i},-,\sum _{i=1}^{n},\overline{{K}_{ij}},{Y}_{j}∥,$

where the norm is the induced norm for the Hilbert space of random variables. Note at this point that the set of random variables ${\sum }_{i=1}^{n}\overline{{K}_{ij}}{Y}_{j}$ over the choices of ${K}_{i}$ can be written as $\mathrm{span}\left({\left\{{Y}_{j}\right\}}_{j=1}^{m}\right)$ . Thus, the optimal ${K}_{i}$ is given by the coefficients of the closest point in $\mathrm{span}\left({\left\{{Y}_{j}\right\}}_{j=1}^{m}\right)$ to the random variable ${X}_{i}$ according to the induced norm for the Hilbert space of random variables. Therefore, we solve for ${K}_{i}$ using results from the projection theorem with the corresponding inner product. Recall that given a basis ${Y}_{i}$ for the subspace of interest, we obtain the equation ${\beta }_{i}=G{\left({K}_{i}^{H}\right)}^{T}=G\overline{{K}_{i}}$ , where ${\beta }_{i,j}=〈{X}_{i},,,{Y}_{j}〉$ and $G$ is the Gramian matrix. More specifically, we have

$\underset{{\beta }_{i}}{\underbrace{\left[\begin{array}{c}〈{X}_{i},,,{Y}_{1}〉\\ 〈{X}_{i},,,{Y}_{2}〉\\ ⋮\\ 〈{X}_{i},,,{Y}_{m}〉\end{array}\right]}}=\underset{G}{\underbrace{\left[\begin{array}{cccc}〈{Y}_{1},,,{Y}_{1}〉& 〈{Y}_{2},,,{Y}_{1}〉& \cdots & 〈{Y}_{m},,,{Y}_{1}〉\\ 〈{Y}_{1},,,{Y}_{2}〉& 〈{Y}_{2},,,{Y}_{2}〉& \cdots & 〈{Y}_{m},,,{Y}_{2}〉\\ ⋮& ⋮& \ddots & ⋮\\ 〈{Y}_{1},,,{Y}_{m}〉& 〈{Y}_{2},,,{Y}_{m}〉& \cdots & 〈{Y}_{m},,,{Y}_{m}〉\end{array}\right]}}\underset{{K}_{i}}{\underbrace{\left[\begin{array}{c}\overline{{K}_{i1}}\\ \overline{{K}_{i2}}\\ ⋮\\ \overline{{K}_{im}}\end{array}\right]}}.$

Thus, one can solve for $\overline{{K}_{i}}={G}^{-1}{\beta }_{i}$ . In the Hilbert space of random variables, we have

$\begin{array}{cc}\hfill G& =\left[\begin{array}{cccc}E\left[{Y}_{1}{Y}_{1}\right]& E\left[{Y}_{2}{Y}_{1}\right]& \cdots & E\left[{Y}_{m}{Y}_{1}\right]\\ E\left[{Y}_{1}{Y}_{2}\right]& E\left[{Y}_{2}{Y}_{2}\right]& \cdots & E\left[{Y}_{m}{Y}_{2}\right]\\ ⋮& ⋮& \ddots & ⋮\\ E\left[{Y}_{1}{Y}_{m}\right]& E\left[{Y}_{2}{Y}_{m}\right]& \cdots & E\left[{Y}_{m}{Y}_{m}\right]\end{array}\right]={R}_{Y},\hfill \\ \hfill \beta & =\left[\begin{array}{c}E\left[{X}_{i}{Y}_{1}\right]\\ E\left[{X}_{i}{Y}_{2}\right]\\ ⋮\\ E\left[{X}_{i}{Y}_{m}\right]\end{array}\right]={\rho }_{{X}_{i}Y}.\hfill \end{array}$

Here ${R}_{Y}$ is the correlation matrix of the random vector $Y$ and ${\rho }_{{X}_{i}Y}$ is the cross-correlation vector of the random variable ${X}_{i}$ and vector $Y$ . Thus, we have $\overline{{K}_{i}}={G}^{-1}{\beta }_{i}={R}_{Y}^{-1}{\rho }_{{X}_{i}Y}$ , and so ${K}_{i}^{H}={\rho }_{{X}_{i}Y}^{T}{R}_{Y}^{-1}$ . Concatenating all the rows of $K$ together, we get $K={R}_{X,Y}{R}_{Y}^{-1}$ , where ${R}_{X,Y}$ is the cross-correlation matrix for the random vectors $X$ and $Y$ . We therefore obtain the optimal linear estimator $\stackrel{^}{X}=KY={R}_{X,Y}{R}_{Y}^{-1}Y$ .

At first, there may be some confusion on the difference between least squares and minimum mean-square error. To summarize:

• Least Squares are applied when the quantities observed are deterministic (i.e., a “single draw” of data or observations).
• Minimum Mean Square Error Estimation are applied when random variables are observed under Gaussian noise; one must know a distribution over inputs, and the error must be measured in expectation.

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