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nomenclature.tex
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\chapter*{Nomenclature}
\markboth{Nomenclature}{Nomenclature}
As much as possible, and unless otherwise stated, the following conventions are used throughout this thesis.
\noindent\textbf{Conventions}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\ba,\bb,\bc,\dots$ & Boldface lower case letters denote real vectors \\
$\bA,\bB,\bC,\dots$ & Boldface upper case letters denote real matrices \\
$\cA,\cB,\cC,\dots$ & Calligraphic upper case letters denote sets \\
$x'$ & Primes are used to distinguish elements (not indicate derivatives) \\
$\hat\theta$ & Hats are used to denote estimators of parameters \\
\end{longtable}
\noindent\textbf{Indexing}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\bA_{ij}$, $A_{ij}$, $a_{ij}$ & The $(i,j)$'th element of the matrix $\bA$ \\
$\bA_{i\bigcdot}$ & The $i$'th row of the matrix $\bA$ as a tall vector (transposed row vector) \\
$\bA_{\bigcdot j}$ & The $j$'th column vector of the matrix $\bA$ \\
\end{longtable}
\noindent\textbf{Symbols}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\bbN$ & The set of natural numbers (excluding zero) \\
$\bbZ$ & The set of integers \\
$\bbR$ & The set of real numbers \\
$\bbR_{>0}$ & The set of positive real numbers, $\{ x\in\bbR|x>0 \}$ \\
$\bbR_{\geq 0}$ & The set of non-negative real numbers, $\{ x\in\bbR|x\geq 0 \}$ \\
$\bbR^d$ & The $d$-dimensional Euclidean space \\
% $\pi$ & The constant defined as the ratio of a circle's circumference to its diameter; approximately $3.14159$ \\
% $\cF$ & A vector space of functions, typically an RKKS \\
% $\cX$ & The set of regression covariates \\
% $h$ & The reproducing kernel of an RKHS/RKKS of functions over $\cX$ \\
$\cA^c$ & The complement of a set $\cA$ \\
$\cP(\cA)$ & The power set of the set $\cA$ \\
$\{ \}, \emptyset$ & The empty set \\
% $\cup$ & The union (of sets) \\
% $\cap $ & The intersection (of sets) \\
% $\sum $ & Summation \\
% $\prod $ & Product \\
$\bzero$ & A vector of zeroes \\
$\bone_n$ & A length $n$ vector of ones \\
$\bI_n$ & The $n \times n$ identity matrix \\
$\exists$ & (short hand) There exists\\
$\forall$ & (short hand) For all \\
% $\notin$ & (short hand) Does not belongs to/is not an element of \\
$\lim_{n \to \infty}$ & The limit as $n$ tends to infinity \\
$\xrightarrow{\text{dist.}}$ & Convergence in distribution \\
% $\to$ & Denotes mapping between two sets \\
% $\mapsto$ & Denotes outcome of mappings \\
$O(n)$ & Computational complexity (time or storage) \\
$\Delta x$ & A quantity representing a change in $x$
\end{longtable}
\noindent\textbf{Relations}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$a \approx b$ & $a$ is approximately or almost equal to $b$ \\
$a \propto b$ & $a$ is equivalent to $b$ up to a constant of proportionality \\
$a \equiv b$ & $a$ is identical to $b$ \\
$A \Rightarrow B$ & The statement $B$ being true is predicated on $A$ being true \\
$A \Leftrightarrow B$ & The statement $A$ is true if and only if $B$ is true \\
$a \in \cA$ & $a$ is an element of the set $\cA$ \\
$\cA \subseteq \cB$ & $\cA$ is a subset of $\cB$ which may include itself \\
$\cA \subset \cB$ & $\cA$ is a subset of $\cB$ which does not include itself \\
$\cA \cong \cB$ & The space $\cA$ is isometrically isomorphic to the space $\cB$ \\
$a := b$, $a \gets b$ & $a$ is assigned the value $b$ \\
$X \sim p(X)$ & The random variable $X$ is distributed according to the pdf $p(X)$ \\
$X \sim D$ & The random variable $X$ is distributed according to the pdf specified by the distribution $D$, e.g. $D\equiv\N(0,1)$ \\
$X_1\!\;\!,\! . . .,\!X_n\!\!\iid\!\! D$ & Each random variable $X_i$, $i=1,\dots,n$ is independently and identically distributed according to the pdf specified by the distribution $D$ \\
$X | Y$ & The (random) variable $X$ given/conditional on $Y$ \\
\end{longtable}
\noindent\textbf{Functions}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\inf \cA$ & The infimum of a set $\cA$ \\
$\sup \cA$ & The supremum of a set $\cA$ \\
$\min \cA$ & The minimum value of a set $\cA$ \\
$\max \cA$ & The maximum value of a set $\cA$ \\
$\argmin_x f(x)$ & The value of $x$ which minimises the function $f(x)$ \\
$\argmax_x f(x)$ & The value of $x$ which maximises the function $f(x)$ \\
$\vert a \vert$ with $a\in\bbR$ & The absolute value of $a$; $\vert a \vert = a$ if $a$ is positive, and $-a$ if $a$ is negative, and $\vert 0 \vert = 0$ \\
$\delta_{xx'}$ & The Kronecker delta; $\delta_{xx'} = 1$ if $x = x'$, and 0 otherwise \\
$[A]$ & The Iverson bracket; $[A] = 1$ if the logical proposition $A$ is true, and 0 otherwise \\
$\ind_\cA(x)$ & The indicator function; $\ind_\cA(x) = 1$ if $x \in \cA$, and 0 otherwise \\
$e^x$, $\exp(x)$ & The natural exponential function \\
$\log(x)$ & The natural logarithmic function \\
$\frac{\d}{\d x} f(x)$, $\dot f(x)$ & The derivative of $f$ with respect to $x$ \\
$\frac{\d^2}{\d x^2} f(x)$, $\ddot f(x)$ & The second derivative of $f$ with respect to $x$ \\
$f \circ g$ & Composition of functions, i.e. $g$ following $f$ \\
\end{longtable}
\noindent\textbf{Abstract vector space operations and notations}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\cV^\bot$ & The orthogonal complement of the space $\cV$ \\
$\cV^\vee$ & The algebraic dual space of $\cV$ \\
$\cV^*$ & The continuous dual space of $\cV$ \\
$\overline\cV$ & The closure of the space $\cV$ \\
$\cB(\cV)$ & The Borel $\sigma$-algebra of $\cV$ \\
$\text{L}^p(\cX,\nu)$ & The set of $p$-integrable functions over the space $\cX$ with measure $\nu$ \\
$\text{L}(\cV;\cW)$ & The set of bounded, linear operators from $\cV$ to $\cW$ \\
$\dim(\cV)$ & The dimensions of the vector space $\cV$ \\
$\ip{x,y}_\cV$ & The inner product between $x$ and $y$ in the vector space $\cV$\\
$\norm{x}_\cV$ & The norm of $x$ in the vector space $\cV$ \\
$D(x,y)$ & The distance between $x$ and $y$ \\
$x \otimes y$ & The tensor product of $x$ and $y$ which are elements of a vector space \\
$\cF \otimes \cG$ & The tensor product space of two vector spaces \\
$\cF \oplus \cG$ & The direct sum (or tensor sum) of two vector spaces \\
$\d f(x)$ & The first Fréchet differential of $f$ at $x$ \\
$\d^2 f(x)$ & The second Fréchet differential of $f$ at $x$ \\
$\partial_v f(x)$ & The first Gâteaux differential of $f$ at $x$ in the direction $v$ \\
$\partial_v^2 f(x)$ & The second Gâteaux differential of $f$ at $x$ in the direction $v$ \\
$\nabla f(x)$ & The gradient of $f$ at $x$ ($f$ is a mapping between Hilbert spaces) \\
$\nabla^2 f(x)$ & The Hessian of $f$ at $x$ ($f$ is a mapping between Hilbert spaces) \\
\end{longtable}
\noindent\textbf{Matrix and vector operations}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\ba^\top$, $\bA^\top$ & The transpose of a vector $\ba$ or a matrix $\bA$ \\
$\bA^{-1}$ & The inverse of a square matrix $\bA$ \\
$\Vert \ba \Vert^2$ & The squared 2-norm the vector $\ba$, equivalent to $\ba^\top\ba$ \\
$\vert \bA \vert$ & The determinant of a matrix $\bA$ \\
$\tr(\bA)$ & The trace of a square matrix $\bA$ \\
$\diag(\bA)$ & The diagonal elements of a square matrix $\bA$ \\
$\rank(\bA)$ & The rank of a matrix $\bA$ \\
$\vecc(\bA)$ & The column-wise vectorisation of a matrix $\bA$ \\
$\ba \otimes \bb$ & The outer product of two vectors $\ba$ and $\bb$ \\
$\bA \otimes \bB$ & The Kronecker product of matrix $\bA$ with matrix $\bB$ \\
$\bA \circ \bB$ & The Hadamard product two matrices $\bA$ and $\bB$ \\
\end{longtable}
\noindent\textbf{Statistical functions}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\Prob(A)$ & The probability of event $A$ occurring \\
$p(X|\theta)$ & The probability density function of $X$ given parameters $\theta$ \\
$L(\theta|X)$ & The log-likelihood of $\theta$ given data $X$, sometimes simply $L(\theta)$ \\
$\BF(M,M')$ & Bayes factor for comparing two models $M$ and $M'$ \\
$\cI(\theta)$ & The Fisher information for $\theta$ \\
$\E(X)$, $\E X$ & The expectation\footnote{\label{foot:exp}When there is ambiguity as to which random element the expectation or variance is taken under or what its distribution is, this is explicated by means of subscripting, e.g. $\E_{X\sim\N(0,1)}X$ to denote the expectation of a standard normal random variable.} of the random element $X$ \\
$\Var(X)$, $\Var X$ & The variance\footref{foot:exp}~of the random element $X$ \\
$\Cov(X,Y)$ & The covariance\footref{foot:exp}~between two random elements $X$ and $Y$ \\
$H(p)$ & The entropy of the distribution $p(X)$ \\
$\KL\big(q(x)\Vert p(x)\big)$ & The Kullback-Leibler divergence from $p(x)$ to $q(x)$, denoted also by $\KL(q\Vert p)$ for short \\
\end{longtable}
\pagebreak
\noindent\textbf{Statistical distributions}
\begin{longtable}{p{0.18\textwidth}p{0.79\textwidth}}
$\N(\mu,\sigma^2)$ & Univariate normal distribution with mean $\mu$ and variance $\sigma^2$ \\
$\N_d(\bmu,\bSigma)$ & $d$-dimensional multivariate normal distribution with mean vector $\bmu$ and covariance matrix $\bSigma$ \\
$\phi(z)$ & The standard normal pdf \\
$\Phi(z)$ & The standard normal cdf \\
$\phi(x|\mu,\sigma^2)$ & The pdf of $\N(\mu,\sigma^2)$ \\
$\phi(\bx|\bmu,\bSigma)$ & The pdf of $\N_d(\bmu,\bSigma)$\\
$\MN_{n,m}(\bmu,\bSigma,\bPsi)$ & Matrix normal distribution with mean $\bmu$ and row variances $\bSigma\in\bbR^{n\times n}$ and column variances $\bPsi\in\bbR^{m\times m}$ \\
$\tN(\mu,\sigma^2,a,b)$ & Truncated univariate normal distribution with mean $\mu$ and variance $\sigma^2$ restricted to the interval $(a,b)$ \\
$\N_+(0,1)$ & The half-normal distribution \\
$\N_+(0,\sigma^2)$ & The folded-normal distribution with variance $\sigma^2$ \\
$\tN_d(\bmu,\bSigma,\cA)$ & Truncated $d$-dimensional multivariate normal distribution with mean vector $\bmu$ and covariance matrix $\bSigma$ restricted to the set $\cA$ \\
$\Gamma(s,r)$ & Gamma distribution with shape $s$ and rate $r$ parameters \\
$\Gamma^{-1}(s,\sigma)$ & Inverse gamma distribution with shape $s$ and scale $\sigma$ parameters \\
$\chi_d^2$ & Chi-squared distribution with $d$ degrees of freedom \\
$\Bern(p)$ & Bernoulli distribution with probability of success $p$ \\
$\Cat(p_1,\dots,p_m)$ & Categorical distribution with $m$ categories, and each category has probability of success $p_j$ \\
\end{longtable}