Probability and statistics: a survival guide

Random variables

Probability measure

We start with a few elementary (and simplified) definitions from the theory of probability. Let us fix a sample space $\Omega = [0,1]$. A Borel set on $\Omega$ is a set that can be formed from open intervals of the form $(a,b), 0 \le a<b \le 1$, through the operations of countable union, countable intersection, and set difference. We will denote the collection of all Borel sets in $\Omega$ as $\Sigma$. It is pretty straightforward to show that $\Sigma$ contains the empty set, is closed under complement, and is closed under countable union. Such a set is known as $\sigma$-algebra and its elements (subsets of $\mathbb{R}$) are referred to as events.

A probability measure $P$ on $\Sigma$ is a function $P : \Sigma \rightarrow [0,1]$ satisfying $P(\emptyset) = 0$, $P(\mathbb{R}) = 1$ and additivity for every countable collection ${ E _n \in \Sigma }$,

\[P\left( \bigcup _n E _n \right) = \sum _{n} P(E _n).\]

Random variables

A random variable $\mathpzc{X}$ is a measurable map $\mathpzc{X} : \Omega \rightarrow \mathbb{R}$, i.e., a function such that for every $a$, ${ \mathpzc{X} \le a } = { \alpha : \mathpzc{X}(\alpha) \le a } \in \Sigma$. The map $\mathpzc{X}$ pushes forward the probability measure $P$; the pushforward measure $\mathpzc{X} _\ast P$ is given by

\[(\mathpzc{X} _\ast P)(A) = P(\mathpzc{X}^{-1}(A)),\]

where $\mathpzc{X}^{-1}(A) = { \alpha : X(\alpha) \in A }$ is the preimage of $A \subseteq \mathbb{R}$. (In short, we can write $\mathpzc{X} _\ast P = P\mathpzc{X}^{-1}$). This pushforward probability measure $\mathpzc{X} _\ast P$ is usually referred to as the probability distribution (or the law) of $\mathpzc{X}$.

When the range of $\mathpzc{X}$ is finite or countably infinite, the random variable is called discrete and its distribution can be described by the probability mass function (PMF):

\[f _{\mathpzc{X}}(x) = P(\mathpzc{X}=x),\]

which is a shorthand for $P( {\alpha : \mathpzc{X}(\alpha) = x } )$. Otherwise, $\mathpzc{X}$ is called a continuous random variable. Any random variable can be described by the cumulative distribution function (CDF)

\[F _{\mathpzc{X}}(x) = P({\mathpzc{X} \le x}),\]

which is a shorthand for $F _{\mathpzc{X}}(x) = P( {\alpha : \mathpzc{X}(\alpha) \le x } )$. If $X$ is absolutely continuous, the CDF can be described by the integral

\[F _{\mathpzc{X}}(x) = \int _{-\infty}^x f _{\mathpzc{X}}(x') dx',\]

where the integrand $f _{\mathpzc{X}}$ is known as the probability density function (PDF)¹.

Uniform distribution and uniformization

A random variable $\mathpzc{U}$ is said to be uniformly distributed on $[0,1]$ (denoted as $\mathpzc{U} \sim \mathcal{U}[0,1]$) if

\[P(\mathpzc{U} \in [a,b]) = b-a = \lambda([a,b]).\]

In other words, the map $\mathpzc{U}$ pushes forward the standard Lebesgue measure on $[0,1]$, $\mathpzc{U} _\ast P = \lambda$. The corresponding CDF is $F _\mathpzc{U}(u) = \max{ 0, \min{ 1, u } }$. Let $\mathpzc{X}$ be some other random variable characterized by the CDF $F _\mathpzc{X}$. We define $\mathpzc{U} = F _\mathpzc{X}(\mathpzc{X})$. Let us pick an arbitrary $x \in \mathbb{R}$ and let $u = F _\mathpzc{X}(x) \in [0,1]$. From monotonicity of the CDF, it follows that $\mathpzc{U} \le u$ if and only if $\mathpzc{X} \le x$. Hence, $F _\mathpzc{U}(u) = P(\mathpzc{U} \le u) = P(\mathpzc{X} \le x) = F _\mathpzc{X}(x) = u$. We conclude that by transforming a random variable with its own CDF uniformizes it on the interval $[0,1]$.

Applying the relation in inverse direction, let $\mathpzc{U} \sim \mathcal{U}[0,1]$ and let $F$ be a valid CDF. Then, the random variable $\mathpzc{X} = F^{-1}(\mathpzc{U})$ is distributed with the CDF $F _\mathpzc{U} = F$.

Expectation

The expected value (a.k.a. the expectation or mean) of a random variable $\mathpzc{X}$ is given by

\[\mathbb{E} \mathpzc{X} = \int _{\mathbb{R}} \mathrm{id}\, d(\mathpzc{X} _\ast P) = \int _{\Omega} \mathpzc{X}(\alpha) d\alpha,\]

where the integral is the Lebesgue integral w.r.t. the measure $P$; whenever a probability density function exists, the latter can be written as

\[\mathbb{E} \mathpzc{X} = \int _{\mathbb{R}} x f _{\mathpzc{X}}(x) dx.\]

Note that due to the linearity of integration, the expectation operator $\mathbb{E}$ is linear. Using the Lebesgue integral notation, we can write for $E \in \Sigma$

\[P(E) = \int _E dP = \int _\mathbb{R} \ind _E \, dP = \mathbb{E} \ind _E,\]

where

\[\ind _E(\alpha) = \left\{ \begin{array}{ccc} 1 & : & \alpha \in E \\ 0 & : & \mathrm{otherwise} \end{array} \right.\]

is the indicator function of $E$, which is by itself a random variable. This relates the expectation of the indicator of an event to its probability.

Moments

For any measurable function $g : \mathbb{R} \rightarrow \mathbb{R}$, $\mathpzc{Z} = g(\mathpzc{X})$ is also a random variable with the expectation

\[\mathbb{E} \mathpzc{Z} = \mathbb{E} g(\mathpzc{X}) = \int _{\mathbb{R}} g\, dP = \int _{\mathbb{R}} g(x) f _{\mathpzc{X}}(x) dx.\]

Such an expectation is called a moment of $\mathpzc{X}$. Particularly, the $k$-th order moment is obtained by setting $g(x) = x^k$,

\[\mu _{k}(\mathpzc{X}) = \mathbb{E} \mathpzc{X}^k.\]

The expected value itself is the first-order moment of $\mathpzc{X}$, which is often denoted simply as $\mu _\mathpzc{X} = \mu _{1}(\mathpzc{X})$. The central $k$-th order moment is obtained by setting $g(x) = (x - \mu _\mathpzc{X})^k$,

\[m _{k}(\mathpzc{X}) = \mathbb{E} ( \mathpzc{X} - \mathbb{E} \mathpzc{X})^k.\]

A particularly important central second-order moment is the variance

\[\sigma _\mathpzc{X}^2 = \mathrm{Var}\, \mathpzc{X} = m _2 = \mathbb{E} ( \mathpzc{X} - \mathbb{E} \mathpzc{X})^2 = \mu _2 ( \mathpzc{X} ) - \mu^2 _\mathpzc{X}.\]

Random vectors

Joint and marginal distributions

A vector $\mathpzcb{X} = (\mathpzc{X} _1, \dots, \mathpzc{X} _n)$ of random variables is called a random vector. Its probability distribution is defined as before as the pushforward measure $P = \mathpzcb{X} _\ast \lambda$ Its is customary to treat $\mathpzcb{X}$ as a collection of $n$ random variables and define their joint CDF as

\[F _{\mathpzcb{X}}(\bb{x}) = P({\mathpzcb{X} \le \bb{x}}) = P(\mathpzc{X} _1 \le x _1, \dots, \mathpzc{X} _n \le x _n) = P(\{ \mathpzc{X} _1 \le x _1 \} \times \dots \times \{ \mathpzc{X} _n \le x _n \}).\]

As before, whenever the following holds

\[F _{\mathpzcb{X}}(\bb{x}) = \int _{-\infty}^{x _1} \cdots \int _{-\infty}^{x _n} f _{\mathpzcb{X}}(x _1',\dots, x _n') dx' _1 \cdots dx' _n,\]

the integrand $f _{\mathpzcb{X}}$ is called the joint PDF of $\mathpzcb{X}$. The more rigorous definition as the Radon-Nikodym derivative

\[f _{\mathpzcb{X}} = \frac{d(\mathpzc{X} _\ast P) }{ d\lambda}\]

stays unaltered, only that now $\lambda$ is the $n$-dimensional Lebesgue measure.

Note that the joint CDF of the sub-vector $(\mathpzc{X} _2, \dots, \mathpzc{X} _n)$ is given by

\[\begin{aligned} F _{\mathpzc{X} _2, \cdots, \mathpzc{X} _n } (x _2,\dots,x _n) &=& P(\mathpzc{X} _2 \le x _2, \dots, \mathpzc{X} _n \le x _n) = P(\mathpzc{X} _1 \le \infty, \mathpzc{X} _2 \le x _2, \dots, \mathpzc{X} _n \le x _n) \\ &=& F _{\mathpzc{X} _1, \cdots, \mathpzc{X} _n } (\infty, x _2,\dots,x _n). \end{aligned}\]

Such a distribiution is called marginal w.r.t. $\mathpzc{X} _1$ and the process of obtaining it by substituting $x _1 = \infty$ into $F _{\mathpzcb{X}}$ is called marginalization. The corresponding action in terms of the PDF consists of integration over $x _1$,

\[\begin{aligned} f _{\mathpzc{X} _2, \cdots, \mathpzc{X} _n } (x _2,\dots,x _n) &=& \int _\mathbb{R} f _{\mathpzc{X} _1, \cdots, \mathpzc{X} _n } (x _1, x _2,\dots,x _n) dx _1. \end{aligned}\]

Statistical independence

A set $\mathpzc{X} _1, \dots, \mathpzc{X} _n$ of random variables is called statistically independent if their joint CDF is coordinate-separable, i.e., can be written as the following tensor product

\[F _{\mathpzc{X} _1, \cdots, \mathpzc{X} _n} = F _{\mathpzc{X} _1} \otimes \cdots \otimes F _{\mathpzc{X} _n}.\]

An alternative definion can be given in terms of the PDF (whenever it exists):

\[f _{\mathpzc{X} _1, \cdots, \mathpzc{X} _n} = f _{\mathpzc{X} _1} \otimes \cdots \otimes f _{\mathpzc{X} _n}.\]

We will see a few additional alternative definitions in the sequel. Let $\mathpzc{X}$ and $\mathpzc{Y}$ be statistically-independent random variables with a PDF and let $\mathpzc{Z} = \mathpzc{X}+\mathpzc{Y}$. Then,

\[\begin{aligned} F _\mathpzc{Z}(z) &=& P(\mathpzc{Z} \le z) = P(X+Y \le z) = \int _{\mathbb{R}} \int _{\infty}^{z-y} f _{\mathpzc{X}\mathpzc{Y}}(x,y) dxdy \\ &=& \int _{\mathbb{R}} \int _{\infty}^{z} f _{\mathpzc{X}\mathpzc{Y}}(x'-y,y) dx' dy, \end{aligned}\]

where we changed the variable $x$ to $x’ = x+y$. Differentiating w.r.t. $z$ yields

\[\begin{aligned} f _\mathpzc{Z}(z) &=& \frac{dF _\mathpzc{Z}(z)}{dz} = \int _{\mathbb{R}} \frac{\partial}{\partial z} \int _{\infty}^{z} f _{\mathpzc{X}\mathpzc{Y}}(x'-y,y) dx' dy = \int _{\mathbb{R}} f _{\mathpzc{X}\mathpzc{Y}}(z-y,y) dy.\end{aligned}\]

Since $\mathpzc{X}$ and $\mathpzc{Y}$ are statistically-independent, we can substitute $f _{\mathpzc{X}\mathpzc{Y}} = f _{\mathpzc{X}} \otimes f _{\mathpzc{Y}}$ yielding

\[\begin{aligned} f _\mathpzc{Z}(z) &=& \int _{\mathbb{R}} f _{\mathpzc{X}}(z-y) f _{\mathpzc{Y}}(y) dy = (f _{\mathpzc{X}} \ast f _{\mathpzc{Y}} )(z).\end{aligned}\]

This result is known as the convolution theorem.

Limit theorems

Given independent identically distributed (i.i.d.) variables $\mathpzc{X} _1, \dots, \mathpzc{X} _n$ with mean $\mu$ and variance $\sigma^2$, we define their sample average as

\[\mathpzc{S} _n = \frac{1}{n}( \mathpzc{X} _1 + \cdots + \mathpzc{X} _n ).\]

Note that $\mathpzc{S} _n$ is also a random variable with $\mu _{\mathpzc{S} _n} = \mu$ and $\displaystyle{\sigma^2 _{\mathpzc{S} _n} = \frac{\sigma^2}{n}}$. It is straightforward to see that the variance decays to zero in the limit $n \rightarrow \infty$, meaning that $\mathpzc{S} _n$ approaches a deterministic variable $\mathpzc{S} = \mu$. However, a much stronger result exists: the (strong) law of large numbers states that in the limit $n \rightarrow \infty$, the sample average converges in probability to the expected value, i.e.,

\[P\left( \lim _{n \rightarrow \infty} \mathpzc{S} _n = \mu \right) = 1.\]

This fact is often denoted as $\mathpzc{S} _n \mathop{\rightarrow}^P \mu$. Furthermore, defining the normalized deviation from the limit $\mathpzc{D} _n = \sqrt{n}(\mathpzc{S} _n - \mu)$, the central limit theorem states that $\mathpzc{D} _n$ converges in distribution to $\mathcal{N}(0,\sigma^2)$, that is, its CDF converges pointwise to that of the normal distribution. This is often denoted as $\mathpzc{D} _n \mathop{\rightarrow}^D \mathcal{N}(0,\sigma^2)$.

A slightly more general result is known as the delta method in statistics: if $g : \mathbb{R} \rightarrow \mathbb{R}$ is a $\mathcal{C}^1$ function with non-vanishing derivative, then by the Taylor theorem,

\[g(\mathpzc{S} _n) = g(\mu) + g'(\nu)(\mathpzc{S} _n-\mu) + \mathcal{O}(| \mathpzc{S} _n-\mu |^2),\]

where $\nu$ lies between $\mathpzc{S} _n$ and $\mu$. Since by the law of large numbers $\mathpzc{S} _n \mathop{\rightarrow}^P \mu$, we also have $\nu \mathop{\rightarrow}^P \mu$; since $g’$ is continuous, $g’(\nu) \mathop{\rightarrow}^P g’(\mu)$. Rearranging the terms and multiplying by $\sqrt{n}$ yields

\[\sqrt{n}( g(\mathpzc{S} _n) - g(\mu) ) = g'(\nu) \sqrt{n}( \mathpzc{S} _n) - \mu ) = g'(\nu) \mathpzc{D} _n,\]

from where (formally, by invoking the Slutsky theorem):

\[\sqrt{n}( g(\mathpzc{S} _n) - g(\mu) ) \mathop{\rightarrow}^D \mathcal{N}(0,g^{\prime} (\mu)^2 \sigma^2).\]

Joint moments

Given a measurable function $\bb{g} : \mathbb{R}^n \rightarrow \mathbb{R}^m$, a (joint) moment of a random vector $\mathpzcb{X} = (\mathpzc{X} _1, \dots, \mathpzc{X} _n)$ is

\[\mathbb{E} \bb{g}(\mathpzcb{X}) = \int \bb{g}(\bb{x}) dP = \left(\begin{array}{c} \int g _1(\bb{x}) dP \\ \vdots \\ \int g _m(\bb{x}) dP \end{array} \right) = \left(\begin{array}{c} \int _{\mathbb{R}^n} g _1(\bb{x}) f _{\mathpzcb{X}}(\bb{x}) d\bb{x} \\ \vdots \\ \int _{\mathbb{R}^n} g _m(\bb{x}) f _{\mathpzcb{X}}(\bb{x}) d\bb{x} \end{array} \right);\]

the last term migh be undefined if the PDF does not exist. The mean of a random vector is simply $\bb{\mu} _\mathpzcb{X} = \mathbb{E} \mathpzcb{X}$. Of particular importance are the second-order joint moments of pairs of random variables,

\[r _{\mathpzc{X}\mathpzc{Y}} = \mathbb{E} \mathpzc{X}\mathpzc{Y}\]

and its central version

\[\sigma^2 _{\mathpzc{X}\mathpzc{Y}} = \mathrm{Cov}(\mathpzc{X},\mathpzc{Y}) = \mathbb{E} \left( (\mathpzc{X} - \mathbb{E} \mathpzc{X} )(\mathpzc{Y} - \mathbb{E} \mathpzc{Y}) \right) = r _{\mathpzc{X}\mathpzc{Y}} - \mu _\mathpzc{X} \mu _\mathpzc{Y}.\]

The latter quantity is known as the covariance of $\mathpzc{X}$ and $\mathpzc{Y}$.

Two random variables $\mathpzc{X}$ and $\mathpzc{Y}$ with $r _{\mathpzc{X}\mathpzc{Y}} = 0$ are called orthogonal²

The variables with $\sigma^2 _{\mathpzc{X}\mathpzc{Y}} = 0$ are called uncorrelated. Note that for a statistically independent pair $(\mathpzc{X},\mathpzc{Y})$,

\[\begin{aligned} \sigma^2 _{\mathpzc{X}\mathpzc{Y}} &=& \int _{\mathbb{R}^2} (x-\mu _\mathpzc{X}) (y-\mu _\mathpzc{Y}) d((\mathpzc{X} \times \mathpzc{Y}) _\ast P) = \int _{\mathbb{R}} (x-\mu _\mathpzc{X}) d(\mathpzc{X} _\ast P) \, \int _{\mathbb{R}} (y-\mu _\mathpzc{Y}) d(\mathpzc{Y} _\ast P) \\ &=& \mathbb{E} (\mathpzc{X} - \mathbb{E} \mathpzc{X} ) \cdot \mathbb{E} (\mathpzc{Y} - \mathbb{E} \mathpzc{Y}) = 0. \end{aligned}\]

However, the converse is not true, i.e., lack of correlation does not generally imply statistical independence (with the notable exception of normal variables). If $\mathpzc{X}$ and $\mathpzc{Y}$ are uncorrelated and furthermore one of them is zero-mean, then they are also orthogonal (and the other way around).

In general, the correlation matrix of a random vector $\mathpzcb{X} = (\mathpzc{X} _1, \dots, \mathpzc{X} _n)$ is given by

\[\bb{R} _{\mathpzcb{X}} = \mathbb{E} \mathpzcb{X} \mathpzcb{X}^\Tr;\]

its $(i,j)$-th element is $(\bb{R} _{\mathpzcb{X}}) _{ij} = \mathbb{E} \mathpzc{X} _i \mathpzc{X} _j$. Similarly, the covariance matrix is defined as the central counterpart of the above moment,

\[\bb{C} _{\mathpzcb{X}} = \mathbb{E} (\mathpzcb{X} - \bb{\mu} _\mathpzcb{X} ) (\mathpzcb{X} - \bb{\mu} _\mathpzcb{X} )^\Tr;\]

its $(i,j)$-th element is $(\bb{C} _{\mathpzcb{X}}) _{ij} =\mathrm{Cov}( \mathpzc{X} _i , \mathpzc{X} _j)$. Given another random vector $\mathpzcb{Y} = (\mathpzc{Y} _1, \dots, \mathpzc{Y} _m)$, the cross-correlation and cross-covariance matrices are defined as $\bb{R} _{\mathpzcb{X}\mathpzcb{Y}} = \mathbb{E} \mathpzcb{X} \mathpzcb{Y}^\Tr$ and $\bb{C} _{\mathpzcb{X}\mathpzcb{Y}} = \mathbb{E} (\mathpzcb{X} - \bb{\mu} _\mathpzcb{X} ) (\mathpzcb{Y} - \bb{\mu} _\mathpzcb{Y} )^\Tr$, respectively.

Linear transformations

Let $\mathpzcb{X} = (\mathpzc{X} _1, \dots, \mathpzc{X} _n)$ be an $n$-dimensional random vector, $\bb{A}$ and $m \times n$ deterministic matrix, and $\bb{b}$ and $m$-dimensional deterministic vector. We define a random vector $\mathpzcb{Y} = \bb{A} \mathpzcb{X} + \bb{b} $ as the affine transformation of $\mathpzcb{X}$. Using linearity of the expectation operator, it is straightforward to show that

\[\begin{aligned} \bb{\mu} _\mathpzcb{Y} &=& \mathbb{E}(\bb{A} \mathpzcb{X} + \bb{b}) = \bb{A} \bb{\mu} _\mathpzcb{X} + \bb{b} \\ \bb{C} _\mathpzcb{Y} &=& \mathbb{E}(\bb{A} \mathpzcb{X} - \bb{A} \bb{\mu} _\mathpzcb{X} ) (\bb{A} \mathpzcb{X} - \bb{A} \bb{\mu} _\mathpzcb{X} )^\Tr = \bb{A} \bb{C} _\mathpzcb{X} \bb{A}^\Tr \\ \bb{C} _{\mathpzcb{X} \mathpzcb{Y}} &=& \mathbb{E}(\mathpzcb{X} - \bb{\mu} _\mathpzcb{X} ) (\bb{A} \mathpzcb{X} - \bb{A} \bb{\mu} _\mathpzcb{X} )^\Tr = \bb{C} _\mathpzcb{X} \bb{A}^\Tr.\end{aligned}\]

Estimation

Let $\mathpzcb{X}$ be a latent $n$-dimensional random vector, and let $\mathpzcb{Y}$ be a statistically related $m$-dimensional observation (measurement). For example $\mathpzcb{Y}$ can be a linearly transformed version of $\mathpzcb{X}$ corrupted by additive random noise, $\mathpzcb{Y} = \bb{A}\mathpzcb{X} + \mathpzcb{N}$. We might attempt using the information $\mathpzcb{Y}$ contains about $\mathpzcb{X}$ in order to estimate $\mathpzcb{X}$. For that purpose, let us construct a deterministic function $\bb{h} : \RR^m \rightarrow \RR^n$ that we are going to call an estimator. Supplying a realization $\mathpzcb{Y} = \bb{y}$ to this estimator will produce a deterministic vector $\hat{\bb{x}} = \bb{h}(\bb{y})$, which is referred to as the estimate of $\mathpzcb{X}$ given the measurement $\bb{y}$. With some abuse of notation, we will henceforth denote $\bb{h}(\bb{y})$ as $\hat{\bb{x}}(\bb{y})$. Note that supplying the random observation vector $\mathpzcb{Y}$ to $\hat{\bb{x}}$ produces the random vector $\hat{\mathpzcb{X}} = \hat{\bb{x}}(\mathpzcb{Y})$; here the deterministic function $\hat{\bb{x}}$ acts as a random variable transformation.

Ideally, $\hat{\mathpzcb{X}}$ and $\mathpzcb{X}$ should coincide; however, unless the measurement is perfect, there will be a discrepancy $\mathpzcb{E} = \hat{\mathpzcb{X}} - \mathpzcb{X}$ which we will refer to as the error vector.

Maximum likelihood

For the sake of simplicity of exposition, let us focus on a very common estimation setting with a linear forward model and an additive statisticaly independent noise, i.e.,

\[\mathpzcb{Y} = \bb{A}\mathpzcb{X} + \mathpzcb{N},\]

where $\bb{A}$ is a deterministic $m \times n$ matrix and $\mathpzcb{N}$ is independent of $\mathpzcb{X}$. In this case, we can assert that the distribution of the measurement $\mathpzcb{Y}$ given the latent signal $\mathpzcb{X}$ is simply the distribution of $\mathpzcb{N}$ at $\mathpzcb{N} = \mathpzcb{Y}- \bb{A} \mathpzcb{X}$,

\[P _{\mathpzcb{Y} | \mathpzcb{X}}( \bb{y} | \bb{x} ) = P _{\mathpzcb{N}}( \bb{y} - \bb{A} \bb{x} ).\]

Assuming i.i.d. noise (i.e., that the $N _i$’s are distributed identically and independently of each other), the latter simplifies to a product of one-dimensional measures. Note that this is essentially a parametric family of distributions – each choice of $\bb{x}$ yields a distribution $P _{\mathpzcb{Y} | \mathpzcb{X} = \bb{x}}$ of $\mathpzcb{Y}$. For the time being, let us treat the notation $\mathpzcb{Y} | \mathpzcb{X}$ just as a funny way of writing.

Given an estimate $\hat{\bb{x}}$ of the true realization $\bb{x}$ of $\mathpzcb{X}$, we can measure its “quality” by measuring some distance $D$ from $P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}}}$ to the true distribution $P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}}$ that created $\mathpzcb{Y}$, and try to minimize it. Our estimator of $\bb{x}$ can therefore be written as

\[\hat{\bb{x}} = \mathrm{arg}\min _{\hat{\mathpzcb{X}}} D(P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} || P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}}} ).\]

Note that we treat the quantity to be estimated as a deterministic parameter rather than a stochastic quantity.

A standard way of measuring distance³ between distributions is the so-called Kullback-Leibler (KL) divergence. To define it, let $P$ and $Q$ be two probability measures (such that $P$ is absolutely continuous w.r.t. $Q$). Then, the KL divergence from Q to P is defined as

\[D(P || Q) = \int _{} \, \log \frac{dP}{dQ} \, dP.\]

In other words, it is the expectation of the logarithmic differences between the probabilities $P$ and $Q$ when the expectation is taken over $P$. The divergence can be thought of as an (asymmetric) distance between the two distributions.

Let us have a closer look at the minimization objective

\[D(P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} || P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}}} ) = \mathbb{E} _{ \mathpzcb{Y} \sim P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} } \log\left( \frac{P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} }{ P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}} } } \right) = \mathbb{E} _{ \mathpzcb{Y} \sim P _{\mathpzcb{Y} | \mathpzcb{X} = \bb{x}} } \log P _{\mathpzcb{Y} | \mathpzcb{X} = \bb{x}} -\mathbb{E} _{ \mathpzcb{Y} \sim P _{\mathpzcb{Y} | \mathpzcb{X} = \bb{x}} } \log P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}} }.\]

Note that the first term (that can be recognized as the entropy of $\log P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}}$) does not depend on the minimization variable; hence, we have

\[\hat{\bb{x}} = \mathrm{arg}\min _{\hat{\bb{x}}} \, \mathbb{E} _{ \mathpzcb{Y} \sim P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} } \left( - \log P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}}} \right).\]

Let us now assume that $N$ realization ${ \bb{y} _1, \dots, \bb{y} _N }$ of $\mathpzcb{Y}$ are observed. In this case, we can express the joint p.d.f of the observations as the product of $f _\mathpzcb{N} (\bb{y} _i - \bb{A} \bb{x})$ or, taking the negative logarithm,

\[-\frac{1}{N} \log f _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} (\bb{y} _1,\dots, \bb{y} _N ) = - \frac{1}{N} \sum _{i=1}^N \log f _\mathpzcb{N} (\bb{y} _i - \bb{A} \bb{x} ) = L(\bb{y} _1,\dots,\bb{y} _N | \bb{x}).\]

This function is known as the negative log likelihood function. By the law of large numbers, when $N$ approaches infinity,

\[L(\bb{y} _1,\dots,\bb{y} _N | \bb{x}) \rightarrow \mathbb{E} _{ \mathpzcb{Y} \sim P _{\mathpzcb{Y} | \mathpzcb{X}=\bb{x}} } \left( - \log P _{\mathpzcb{Y} | \mathpzcb{X}=\hat{\bb{x}}} \right).\]

Behold our minimization objective!

To recapitulate, recall that we started with minimizing the discrepancy between the latent parametric distribution that generated the observation and that associated with our estimator. However, a closer look at the objective revealed that it is the limit of the negative log likelihood when the sample size goes to infinity. The minimization of the Kullback-Leibler divergence is equivalent to maximization of the likelihood of the data coming from a specific parametric distribution,

\[\hat{\bb{x}} = \mathrm{arg}\max _{\hat{\bb{x}}} \, P( \mathpzcb{Y}=\bb{y} | \mathpzcb{X}=\bb{x} ).\]

For this reason, the former estimator is called maximum likelihood (ML).

Conditioning

Before treating maximum a posteriori estimation, we need to briefly introduce the important notion of conditioning and conditional distributions. Recall our construction of a probability space comprising the triplet $\Omega$ (the sample space), $\Sigma$ (the Borel sigma algebra), and $P$ (the probability measure). Let $X$ be a random variable and $B \subset \Sigma$ a sub sigma-algebra of $\Sigma$. We can then define the conditional expectation of $\mathpzc{X}$ given $B$ as a random variable $\mathpzc{Z} = \mathbb{E} \mathpzc{X} | B$ satisfying for every $E \in B$

\[\int _E \mathpzc{Z} dP = \int _E \mathpzc{X} dP.\]

(we are omitting some technical details such as, e.g., integrability that $\mathpzc{X}$ has to satisfy).

Given another random variable $\mathpzc{Y}$, we say that it generates a sigma algebra $\sigma(\mathpzc{Y})$ as the set of pre-images of all Borel sets in $\mathbb{R}$,

\[\sigma(\mathpzc{Y}) = \{ \mathpzc{Y}^{-1}(A) : A \in \mathbb{B}(\mathbb{R}) \}.\]

We can then use the previous definition to define the conditional expectation of $\mathpzc{X}$ given $\mathpzc{Y}$ as

\[\mathbb{E} \mathpzc{X} | \mathpzc{Y} = \mathbb{E} \mathpzc{X} | \sigma(\mathpzc{Y}).\]

Conditional distribution

Recall that expectation applied to indicator functions can be used to define probability measures. In fact, for every $E \in \Sigma$, we may construct the random variable $\ind _E$, leading to $P(E) = \mathbb{E} \ind _E$. We now repeat the same, this time replacing $\mathbb{E} $ with $\mathbb{E} \cdot | \mathpzc{Y}$. For every $E \in \Sigma$,

\[\varphi(\mathpzc{Y}) = \mathbb{E} \, E | \mathpzc{Y}\]

is a random variable that can be thought of as a transformation of the random variable $\mathpzc{Y}$ by the function $\varphi$. We denote this function as $P(E |\mathpzc{Y})$ and refer to it as the (regular) conditional probability of event $E$ given $\mathpzc{Y}$. It is easy to show that for every measurable set $B \subset \mathbb{R}$,

\[\int _B P(E | \mathpzc{Y}=y) (\mathpzc{Y} _\ast P)(dy) = P(E \cap \{ \mathpzc{Y} \in B \});\]

Substituting $E = { \mathpzc{X} \in B}$ yields the conditional distribution of random variable $X$ given $\mathpzc{Y}$,

\[P _{\mathpzc{X} | \mathpzc{Y}} ( B | \mathpzc{Y}=y) = P(\mathpzc{X} \in B | \mathpzc{Y}=y).\]

It can be easily shown that $P _{\mathpzc{X} | \mathpzc{Y}}$ is a valid probability measure on $\Sigma$ and for every pair of measurable sets $A$ and $B$,

\[\int _B P _{\mathpzc{X} | \mathpzc{Y}} (A | \mathpzc{Y}=y) (\mathpzc{Y} _\ast P)(dy) = P(\{ \mathpzc{X} \in A \} \cap \{ \mathpzc{Y} \in B \}).\]

If density exists, $P _{\mathpzc{X} | \mathpzc{Y}}$ can be described using the conditional p.d.f. $f _{\mathpzc{X} | \mathpzc{Y}}$ and the latter identity can be rewritten in the form

\[\int _A \left( \int _B f _{\mathpzc{X} | \mathpzc{Y}} (x | y) f _\mathpzc{Y}(y) dy \right) dx = P(\{ \mathpzc{X} \in A \} \cap \{ \mathpzc{Y} \in B \}) = \int _A \int _B f _{\mathpzc{XY} } (x, y) dxdy.\]

This essentially means that $f _{\mathpzc{XY} } (x, y) = f _{\mathpzc{X} | \mathpzc{Y}} (x | y) f _\mathpzc{Y}(y)$. Integrating w.r.t. $y$ yields the so-called total probability formula

\[f _{\mathpzc{X} } (x) = \int _\mathbb{R} f _{\mathpzc{XY} } (x, y) dy = \int _\mathbb{R} f _{\mathpzc{X|Y} } (x|y) f _\mathpzc{Y}(y) dy.\]

We can also immediately observe that if $\mathpzc{X}$ and $\mathpzc{Y}$ are statistically independent, we have

\[f _{\mathpzc{XY} } (x, y) = f _{\mathpzc{X} }(x) f _{\mathpzc{Y} } (y) = f _{\mathpzc{X} | \mathpzc{Y}} (x | y) f _\mathpzc{Y}(y),\]

from where $f _{\mathpzc{X} | \mathpzc{Y}} = f _{\mathpzc{X}}$. In this case, conditioning on $\mathpzc{Y}$ does not change our knowledge of $\mathpzc{X}$.

Bayes’ theorem

One of the most celebrate (and useful) results related to conditional distributions is the following theorem named after Thomas Bayes. Exchanging the roles of $\mathpzc{X}$ and $\mathpzc{Y}$, we have

\[f _{\mathpzc{XY} } = f _{\mathpzc{X} | \mathpzc{Y}} f _\mathpzc{Y} = f _{\mathpzc{Y} | \mathpzc{X}} f _\mathpzc{X};\]

re-arranging the terms, we have

\[f _{\mathpzc{Y} | \mathpzc{X}} = f _{\mathpzc{X} | \mathpzc{Y}} \, \frac{ f _\mathpzc{X} }{ f _\mathpzc{Y} };\]

in terms of probability measures, the equivalent form is

\[P _{\mathpzc{Y} | \mathpzc{X}} = P _{\mathpzc{X} | \mathpzc{Y}} \, \frac{ dP _\mathpzc{X} }{ dP _\mathpzc{Y} }.\]

Law of total expectation

Note that treating the conditional density $f _{\mathpzc{X}|\mathpzc{Y}}(x|y)$ just as a funnily-decorated p.d.f. with the argument $x$, we can write the following expectation integral

\[\mathbb{E} (\mathpzc{X}|\mathpzc{Y}=y) = \int _{\RR} x f _{\mathpzc{X}|\mathpzc{Y}}(x|y) dx.\]

With (a very accepted) abuse of notation, we denote it as “$\mathbb{E} (\mathpzc{X}|\mathpzc{Y}=y)$”. Note, however, that this is a very different object from $\mathbb{E} \, \mathpzc{X}|\mathpzc{Y}$ – while the former is a deterministic value, the latter is a random variable (a transformation of $\mathpzc{Y}$). In order to construct $\mathbb{E} \, \mathpzc{X}|\mathpzc{Y}$ out of $\mathbb{E} (\mathpzc{X}|\mathpzc{Y}=y)$, we define the map $\varphi : y \mapsto \mathbb{E} (\mathpzc{X}|\mathpzc{Y}=y)$ and apply it to the random variable $\mathpzc{Y}$, obtaining $\mathbb{E} \, \mathpzc{X}|\mathpzc{Y} = \varphi(Y)$. Again, with a slight abuse of notation, we can write this as

\[\varphi(Y) = \mathbb{E} \, \mathpzc{X}|\mathpzc{Y} = \int _{\RR} x f _{\mathpzc{X}|\mathpzc{Y}}(x|Y) dx.\]

Let us now take a regular expectation of the transformed variable $\varphi(Y)$, which can be viewed as a generalized moment of $\mathpzc{Y}$,

\[\mathbb{E}\, \varphi(\mathpzc{Y}) = \int _{\RR} \varphi(y) f _{\mathpzc{Y}} (y) dy = \int _{\RR} \left( \int _{\RR} x f _{\mathpzc{X}|\mathpzc{Y}}(x|y) dx \right) f _{\mathpzc{Y}} (y) dy.\]

Rearranging the integrands and using $f _{\mathpzc{XY} } = f _{\mathpzc{X} | \mathpzc{Y}} f _\mathpzc{Y}$, we obtain

\[\mathbb{E}\, \varphi(\mathpzc{Y}) = \int _{\RR^2} x f _{\mathpzc{XY} }(x,y) dx dy = \mathbb{E}\, X.\]

Stated differently,

\[\mathbb{E}\left( \mathbb{E}\, X|Y \right) = \mathbb{E}\, X.\]

This result is known as the smoothing theorem or the law of total expectation and can be thought of as an integral version of the law of total probability.

Maximum a posteriori

Recall that in maximum likelihood estimation we treated $\mathpzcb{X}$ as a deterministic parameter and tried to maximize the conditional probability $P(\mathpzcb{Y} | \mathpzcb{X})$. Let us now think of $\mathpzcb{X}$ as of a random vector and maximize its probability given the data,

\[\hat{\bb{x}}(\bb{y}) = \mathrm{arg}\max _{ \hat{\bb{x}} } P _{\mathpzcb{X} | \mathpzcb{Y} } ( \mathpzcb{X} = \hat{\bb{x}} | \mathpzcb{Y} = \bb{y}).\]

Invoking the Bayes theorem yields

\[P _{\mathpzcb{X} | \mathpzcb{Y}} = P _{\mathpzcb{Y} | \mathpzcb{X} } \, \frac{ dP _{\mathpzcb{X}} }{dP _{\mathpzcb{Y}} }\]

In the Bayesian jargon, $P _{\mathpzcb{X}}$ is called the prior probability, that is, our initial knowledge about $\mathpzcb{X}$ before any observation thereof was obtained; $P _{\mathpzcb{X} | \mathpzcb{Y}}$ is called the posterior probability having accounted for the measurement $\mathpzcb{Y}$. Note that the term $P _{\mathpzcb{Y} | \mathpzcb{X}}$ is our good old likelihood. Since we are maximizing the posterior probability, the former estimator is called maximum a posteriori (MAP).

Taking negative logarithm, we obtain

\[-\log P _{\mathpzcb{X} | \mathpzcb{Y}} = -\log P _{\mathpzcb{Y} | \mathpzcb{X} } -\log P _\mathpzcb{X} +\log P _{\mathpzcb{Y}} = L(\mathpzcb{Y} | \mathpzcb{X}) - \log P _{\mathpzcb{X}} + \mathrm{const}.\]

This yields the following expression for the MAP estimator

\[\bb{h}(\bb{Y}) = \mathrm{arg}\min _{ \hat{\bb{x}} } L(\mathpzcb{Y} | \hat{\bb{x}} ) - \log P _\mathpzcb{X} ( \hat{\bb{x}} ).\]

The minimization objective looks very similar to what we had in the ML case; the only difference is that now a prior term is added. In the absence of a good prior, a uniform prior is typically assumed, which reduces MAP estimation to ML estimation.

Minimum mean squared error

Another sound way of constructing the estimator function $\hat{\bb{x}}$ is by minimizing some error criterion related to the error vector $\mathcal{E}(\mathpzcb{E})$. A very common pragmatic choice is the mean squared error (MSE) criterion,

\[\mathcal{E}(\mathpzcb{E}) = \mathbb{E} \, \| \mathpzcb{E} \| _2^2,\]

leading to the following optimization problem:

\[\hat{\bb{x}}^{\mathrm{MMSE}} = \mathrm{arg} \min _{ \bb{h} : \RR^m \rightarrow \RR^n } \mathbb{E} \, \| \bb{h}( \mathpzcb{Y} ) - \mathpzcb{X} \| _2^2.\]

The resulting estimator is called minimum mean squared error (or MMSE) estimator. Since the squared norm is coordinate separable, we can effectively solve for each dimension of $\hat{\bb{x}}^{\mathrm{MMSE}}$ independently, finding the best (in the MSE sense) estimator of $X _i$ given $\mathpzcb{Y}$,

\[\hat{x} _i^{\mathrm{MMSE}} = \mathrm{arg} \min _{ h : \RR^m \rightarrow \RR } \mathbb{E} \, ( h( \mathpzcb{Y} ) - X _i )^2.\]

The minimization objective can be written explicitly as

\[\begin{aligned} \mathbb{E} \, ( h( \mathpzcb{Y} ) - X _i )^2 &=& \mathbb{E} \, \left( \mathbb{E}\, ( h( \mathpzcb{Y} ) - X _i )^2 | \mathpzcb{Y} \right) = \mathbb{E} \, \left( \mathbb{E}\, h^2 ( \mathpzcb{Y} ) | \mathpzcb{Y} - 2 \mathbb{E}\, h ( \mathpzcb{Y} ) X _i | \mathpzcb{Y} + \mathbb{E}\, X^2 _i | \mathpzcb{Y} \right) \\ &=& \mathbb{E} \, \left( h^2 ( \mathpzcb{Y} ) - 2 \mathbb{E}\, X _i | \mathpzcb{Y} \cdot h ( \mathpzcb{Y} ) + \mathbb{E}\, X^2 _i | \mathpzcb{Y} \right) \\ &=& \int _{\RR^m} ( h^2 ( \bb{y} ) - 2 \mathbb{E} (X _i | \mathpzcb{Y}=\bb{y}) \cdot h ( \mathpzcb{Y} ) + \mathbb{E} ( X^2 _i | \mathpzcb{Y}=\bb{y}) ) f _{\mathpzcb{Y}} (\bb{y}) d\bb{y}. \end{aligned}\]

The latter integral is minimized iff its non-negative integrand is minimized at every point $\bb{y}$. Let us fix $\bb{y}$ and define $a = h(\bb{y})$. The expression to minimize is

\[\varphi(a) = a^2 - 2a\, \mathbb{E} (\mathpzc{X} _i | \mathpzcb{Y}=\bb{y}) + \mathbb{E} ( \mathpzc{X}^2 _i | \mathpzcb{Y}=\bb{y});\]

note that this is a convex quadratic function with the minimizer given by

\[a^\ast = \mathbb{E} (\mathpzc{X} _i | \mathpzcb{Y}=\bb{y}).\]

From here we conclude that

\[h(\bb{Y}) = \mathbb{E} \mathpzc{X} _i | \mathpzcb{Y};\]

consequently, the MMSE estimator of $\mathpzcb{X}$ given $\mathpzcb{Y}$ is given by the conditional expectation

\[\hat{\mathpzcb{X}}^{\mathrm{MMSE}} = \mathbb{E} \, \mathpzcb{X} | \mathpzcb{Y}.\]

The error vector produced by the MMSE estimator is given by $\mathpzcb{E} = \mathbb{E} \, \mathpzcb{X} | \mathpzcb{Y} - \mathpzcb{X}$. Taking the expectation yields

\[\mathbb{E}\, \mathpzcb{E} = \mathbb{E} \left( \mathbb{E} \, \mathpzcb{X} | \mathpzcb{Y} \right) - \mathbb{E}\, \mathpzcb{X} = \mathbb{E}\, \mathpzcb{X} - \mathbb{E}\, \mathpzcb{X} = \bb{0}.\]

In other words, the estimation error is zero mean – a property often stated by saying that the MMSE estimator is unbiased.

Since the MSE is equivalent (isomorphic) to Euclidean length, MMSE estimation can be viewed as the minimization of the length of the vector $\mathpzcb{E}$ over the subspace of vectors of the form $\hat{\mathpzcb{X}} = \bb{h}( \mathpzcb{Y} )$ with $\bb{h} : \RR^m \rightarrow \RR^m$. We known from Euclidean geometry that the minimum length is obtained by the orthogonal projection of $\mathpzcb{X}$ onto the said subspace, meaning that $\hat{\mathpzcb{X}}$ is an MMSE estimator iff its error vector $\mathpzcb{E}$ is orthogonal to every $\bb{h}( \mathpzcb{Y} )$, that is,

\[\mathbb{E}\, \left( (\hat{\mathpzcb{X}} - \mathpzcb{X} ) \bb{h}^\Tr ( \mathpzcb{Y} ) \right) = \bb{0}\]

for every $\bb{h} : \RR^m \rightarrow \RR^m$.

Best linear estimator

Sometimes the functional dependence of $\hat{\mathpzcb{X}}^{\mathrm{MMSE}} $ on $\mathpzcb{Y}$ might be too complicated to compute. In that case, it is convenient to restrict the family of functions to some simple class such as that of linear (more precisely, affine) functions of the form $\bb{h}(\bb{y}) = \bb{A} \bb{y} + \bb{b}$. The MMSE estimator restricted to such a subspace of functions is known as the best linear estimator (BLE), and its optimal parameters $\bb{A}$ and $\bb{b}$ are found by minimizing

\[\min _{ \bb{A}, \bb{b} } \mathbb{E} \, \| \bb{A} \mathpzcb{Y} + \bb{b} - \mathpzcb{X} \| _2^2.\]

Note that since $\mathbb{E} \mathpzcb{E} = \bb{A} \mathbb{E}\, \mathpzcb{Y} + \bb{b} - \mathbb{E}\, \mathpzcb{X}$, we can always zero the estimator bias by setting $\bb{b} = \bb{\mu} _{\mathpzcb{X}} - \bb{A}\bb{\mu} _{\mathpzcb{Y}}$. With this choice, the problem reduces to

\[\min _{ \bb{A} } \mathbb{E} \, \| \bb{A} (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} ) - ( \mathpzcb{X} - \bb{\mu} _{\mathpzcb{X}} ) \| _2^2\]

or, equivalently,

\[\min _{ \bb{A} } \, \mathbb{E} \, \mathrm{tr} \left( (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} )^\Tr \bb{A}^\Tr \bb{A} (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} ) \right) - 2 \mathbb{E} \,\mathrm{tr} \left( (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} )^\Tr \bb{A}^\Tr (\mathpzcb{X} - \bb{\mu} _{\mathpzcb{X}} ) \right).\]

Manipulating the order of multiplication under the trace, exchaging its order with that of the expectation operator, and moving the constants outside the expectation yields the following minimization objective:

\[\begin{aligned} \varphi(\bb{A}) &=& \mathrm{tr} \left( \bb{A} \mathbb{E} (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} ) (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} )^\Tr \bb{A}^\Tr -2 \bb{A} \mathbb{E} (\mathpzcb{X} - \bb{\mu} _{\mathpzcb{X}} ) (\mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} )^\Tr \right) \\ &=& \mathrm{tr} \left( \bb{A} \bb{C} _{\mathpzcb{Y}} \bb{A}^\Tr -2 \bb{A} \bb{C} _{\mathpzcb{X} \mathpzcb{Y}} \right). \end{aligned}\]

Note that this is a convex (since $\bb{C} _{\mathpzcb{Y}} \succ 0$) quadratic function. In order to find its minimizer, we differentiate w.r.t. the parameter $\bb{A}$ and equate the gradient to zero:

\[0 = \nabla \varphi(\bb{A}) = 2\bb{A} \bb{C} _{\mathpzcb{Y}} - 2\bb{C} _{\mathpzcb{X} \mathpzcb{Y}}.\]

The optimal parameter is obtained as $\bb{A} = \bb{C} _{\mathpzcb{X} \mathpzcb{Y}}\bb{C} _{\mathpzcb{Y}}^{-1}$.

Combining this result with the expression for $\bb{b}$, the best linear estimator is

\[\hat{\mathpzcb{X}}^{\mathrm{BLE}} = \bb{C} _{\mathpzcb{X} \mathpzcb{Y}}\bb{C} _{\mathpzcb{Y}}^{-1} ( \mathpzcb{Y} - \bb{\mu} _{\mathpzcb{Y}} ) + \bb{\mu} _{\mathpzcb{X}} .\]

As the more general MMSE estimator, BLE is also unbiased and enjoys the orthogonality property, meaning that $\hat{\mathpzcb{X}}$ is an MMSE estimator iff its error vector $\mathpzcb{E}$ is orthogonal to every affine function of $\mathpzcb{Y} )$, that is,

\[\mathbb{E}\, \left( (\hat{\mathpzcb{X}} - \mathpzcb{X} ) ( \bb{A}\mathpzcb{Y} + \bb{b} )^\Tr \right) = \bb{0}\]

for every $\bb{A} \in \RR^{m \times n}$ and $\bb{b} \in \RR^m$.

1. To be completely rigorous, the proper way to define the PDF is by first equipping the image of the map $\mathpzc{X}$ with the Lebesgue measure $\lambda$ that assigns to every interval $[a,b]$ its length $b-a$. Then, we invoke the Radon-Nikodym theorem saying that if $\mathpzc{X}$ is absolutely continuous w.r.t. $\lambda$, there exists a measurable function $f : \mathbb{R} \rightarrow [0,\infty)$ such that for every measurable $A \subset \mathbb{R}$, $\displaystyle{(\mathpzc{X} _\ast P)(A) =P(\mathpzc{X}^{-1}(A)) = \int _A f d\lambda}$. $f$ is called the Radon-Nikodym derivative and denoted by $\displaystyle{f = \frac{d(\mathpzc{X} _\ast P)}{d\lambda}} $. It is exactly our PDF. ↩

2. In fact, $r _{\mathpzc{X}\mathpzc{Y}}$ can be viewed as an inner product on the space of random variables. This creates a geometry isomorphic to the standard Euclidean metric in $\mathbb{R}^n$. Using this construction, the Cauchy-Schwarz inequality immediately follows: $| r _{\mathpzc{X}\mathpzc{Y}} | \le \sigma _\mathpzc{X} \sigma _\mathpzc{Y}$. ↩

3. Actually, not a true metric (which what the term distance implies, but rather an asymmetric form thereof, formally termed a divergence. ↩