Mercurial > hg > cip2012

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+** Information dynamics and temporal structure in music **
+
+
+It has often been observed that one of the more salient effects
+of listening to music to create expectations within the listener,
+and that part of the art of making music to create a dynamic interplay
+of uncertainty, expectation, fulfilment and surprise. It was not until
+the publication of Shannon's work on information theory, however, that
+the tools became available to quantify some of these concepts.
+
+In this talk, we will examine how a small number of
+\emph{time-varying} information measures, such as entropies and mutual
+informations, computed in the context
+of a dynamically evolving probabilistic model, can be used to characterise
+the temporal structue of a stimulus sequence, considered as a random process
+from the point of view of a Bayesian observer.
+
+One such measure is a novel predictive information rate, which we conjecture
+may provide an explanation for the `inverted-U' relationship often found between
+simple measures of randomness (\eg entropy rate) and
+judgements of aesthetic value [Berlyne 1971]. We explore these ideas in the context
+of Markov chains using both artificially generated sequences and
+two pieces of minimalist music by Philip Glass, showing that even an overly simple
+model (the Markov chain), when interpreted according to information dynamic
+principles, produces a structural analysis which largely agrees with that of an
+human expert listener and improves on those generated by rule-based methods.
+
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+../../figs
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+
+\title{Cognitive Music Modelling:\\An Information Dynamics Approach}
+\author{Samer Abdallah, Henrik Ekeus, Peter Foster,\\Andrew Robertson and Mark Plumbley}
+\institute{Centre for Digital Music\\Queen Mary, University of London}
+
+\date{\today}
+
+\def\X{\mathcal{X}}
+\def\Y{\mathcal{Y}}
+\def\Past{\mathrm{Past}}
+\def\Future{\mathrm{Future}}
+\def\Present{\mathrm{Present}}
+\def\param{\theta}
+\def\trans{a}
+\def\init{\pi^{\trans}}
+%\def\entrorate(#1){\mathcal{H}(#1)}
+%\def\entrorate(#1){\dot{\mathcal{H}}(#1)}
+\def\entrorate{h}
+\def\emcmarg(#1){b_#1}
+\def\mcmarg{\vec{b}}
+\def\domS{\mathcal{S}}
+\def\domA{\mathcal{A}}
+
+\def\Lxz(#1,#2){\mathcal{L}(#1|#2)}
+\def\LXz(#1){\overline{\mathcal{L}}(#1)}
+\def\LxZ(#1){\underline{\mathcal{L}}(#1)}
+\def\LXZ{\overline{\underline{\mathcal{L}}}}
+\def\Ixz(#1,#2){\mathcal{I}(#1|#2)}
+\def\IXz(#1){\overline{\mathcal{I}}(#1)}
+\def\IxZ(#1){\underline{\mathcal{I}}(#1)}
+\def\IXZ{\overline{\underline{\mathcal{I}}}}
+
+\def\ev(#1=#2){#1\!\!=\!#2}
+\def\sev(#1=#2){#1\!=#2}
+
+\def\FE{\mathcal{F}}
+
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+\newcommand\fut[1]{\overset{\rule{0pt}{0.1em}\smash{\rightarrow}}{#1}}
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+	\bl(#1)\&\bl(#2)\&\bl(#3)\&\bl(#4)\&\bl(#5)\&\bl(#6)\\};}}
+
+\begin{document}
+	\frame{\titlepage}
+	\section[Outline]{}
+	\frame{
+		\frametitle{Outline}
+		\tableofcontents
+	}
+
+
+
+\section{Expectation and surprise in music}
+\label{s:Intro}
+
+\begin{iframe}[`Unfoldingness']
+	Music is experienced as a
+	\uncover<2->{phenomenon}
+	\uncover<3->{that}
+	\uncover<4->{`unfolds'} \uncover<5->{in}\\
+	\only<6>{blancmange}%
+	\only<7>{(just kidding)}%
+	\uncover<8->{time,}
+	\uncover<9->{rather than being apprehended as a static object presented in its
+	entirety.}
+
+	\uncover<10->{[This is recognised in computation linguistics where the phenomenon is known as \emph{incrementality}, \eg in incremental parsing.]}
+
+	\uncover<11->{%
+	Meyer \cite{Meyer67} argued that musical experience depends on
+	how we change and revise our conceptions \emph{as events happen},
+	on how expectation and prediction interact with occurrence, and that, to a large
+	degree, the way to understand the effect of music is to focus on
+	this `kinetics' of expectation and surprise.%
+	}
+\end{iframe}
+
+\begin{iframe}[Expectation and suprise in music]
+
+	Music creates
+	\emph{expectations} of what is to come next, which may be fulfilled
+	immediately, after some delay, or not at all.
+	Suggested by music theorists, \eg
+	L. B. Meyer \cite{Meyer67} and Narmour \citep{Narmour77} but also
+	noted much earlier by Hanslick \cite{Hanslick1854} in the
+	 1850s:
+		\begin{quote}
+			\small
+			`The most important factor in the mental process which accompanies the
+			act of listening to music, and which converts it to a source of pleasure, is
+			\ldots
+%			frequently overlooked. We here refer to
+			the intellectual satisfaction which the
+			listener derives from continually following and anticipating the composer's
+			intentions---now, to see his expectations fulfilled, and now, to find himself
+			agreeably mistaken. It is a matter of course that this intellectual flux and
+			reflux, this perpetual giving and receiving takes place unconsciously, and with
+			the rapidity of lightning-flashes.'
+		\end{quote}
+\end{iframe}
+
+\begin{iframe}[Probabilistic reasoning]
+	\uncover<1->{%
+	Making predictions and assessing surprise is
+	essentially reasoning with degrees of belief and (arguably)
+	the best way to do this is using Bayesian probability theory \cite{Cox1946,Jaynes27}.%
+
+	[NB. this is \textbf{subjective} probability as advocated by \eg De Finetti and Jaynes.]
+	}
+
+%  Thus, we assume that musical schemata are encoded as probabilistic % \citep{Meyer56} models, and
+	\uncover<2->{%
+   We suppose that familiarity with different styles of music takes the form
+	of various probabilistic models, and that these models are adapted through listening.%
+	}
+%	various stylistic norms is encoded as
+%	using models that encode the statistics of music in general, the particular styles
+%	of music that seem best to fit the piece we happen to be listening to, and the emerging
+%	structures peculiar to the current piece.
+
+	\uncover<3->{%
+	Experimental evidence that humans are able to internalise statistical
+	knowledge about musical: \citep{SaffranJohnsonAslin1999,EerolaToiviainenKrumhansl2002}; and also
+	that statistical models are effective for computational analysis of music, \eg \cite{ConklinWitten95,Pearce2005}.%
+	}
+
+	% analysis of music, \eg \cite{ConklinWitten95,PonsfordWigginsMellish1999,Pearce2005}.
+%		\cite{Ferrand2002}. Dubnov and Assayag PSTs?
+\end{iframe}
+
+\begin{iframe}[Music and information theory]
+	\uncover<1->{
+	With probabilistic models in hand we can apply quantitative information theory: we can compute entropies,
+	relative entropies, mutual information, and all that.
+	}
+
+	\uncover<2->{
+	Lots of interest in application of information theory to perception, music and aesthetics since the 50s,
+	\eg Moles \cite{Moles66}, Meyer \cite{Meyer67}, Cohen \cite{Cohen1962}, Berlyne \cite{Berlyne71}.
+	(See also Bense, Hiller)
+	}
+
+	\uncover<3->{
+	Idea is that subjective qualities and
+	states like uncertainty, surprise, complexity, tension, and interestingness
+	are determined by information-theoretic quantities.
+	}
+
+	\uncover<4->{
+	Berlyne \cite{Berlyne71} called such quantities `collative variables', since they are
+	to do with patterns of occurrence rather than medium-specific details.
+	\emph{Information aesthetics}.
+	}
+%	Listeners then experience greater or lesser levels of surprise
+%	in response to departures from these norms.
+%	By careful manipulation
+%	of the material, the composer can thus define, and induce within the
+%	listener, a temporal programme of varying
+%	levels of uncertainty, ambiguity and surprise.
+\end{iframe}
+
+\begin{iframe}[Probabilistic model-based observer hypothesis]
+	\begin{itemize}
+		\item<1->
+		As we listen, we maintain a probabilistic model that enables
+		us to make predictions.  As events unfold, we revise our probabilistic `belief state',
+		including predictions about the future.
+		\item<2->
+		Probability distributions and changes in distributions are characterised in terms
+		of information theoretic-measures such as entropy and relative entropy (KL divergence).
+		\item<3->
+		The dynamic evolution of these information measures captures significant structure,
+		\eg events that are surprising, informative, explanatory \etc
+	\end{itemize}
+
+\end{iframe}
+
+\begin{iframe}[Features of information dynamics]
+	\uncover<1->{
+	\textbf{Abstraction}: sensitive mainly to \emph{patterns} of occurence,
+	rather than details of which specific things occur or the sensory medium.
+%	it operates at a level of abstraction removed from the details of the sensory experience and
+%	the medium through which it was received, suggesting that the same
+%	approach could, in principle, be used to analyse and compare information
+%	flow in different temporal media regardless of whether they are auditory, visual or otherwise.
+	}
+
+	\uncover<2->{
+	\textbf{Generality}: applicable in principle to any probabilistic model, in particular,
+	models with time-dependent latent variables such as HMMs.
+	Many important musical concepts like key, harmony, and beat are essentially `hidden variables'.
+	}
+
+	\uncover<3->{
+	\textbf{Richness}: when applied to models with latent variables, can result in many-layered
+	analysis, capturing information flow about harmony, tempo, \etc
+	}
+
+	\uncover<4->{
+	\textbf{Subjectivity}: all probabilities are \emph{subjective} probabilities relative to \emph{observer's}
+	model, which can depend on observer's capabilities and prior experience.
+	}
+\end{iframe}
+
+\section{Surprise, entropy and information in random sequences}
+\label{s:InfoInRandomProcs}
+
+\begin{iframe}[Information theory primer\nicedot Entropy]
+		Let $X$ be a discrete-valued random (in the sense of \emph{subjective} probability) variable.
+		Entropy is a measure of \emph{uncertainty}. If observer expects to see $x$ with probability $p(x)$,
+		then
+		\begin{align*}
+			H(X) &= \sum_{x\in\X} - p(x) \log p(x) \\
+			&= \expect{[-\log p(X)]}.
+		\end{align*}
+		Consider $-\log p(x)$ as the `surprisingness' of $x$, then the entropy is the `expected surprisingness'.
+		High for spread out distributions and low for concentrated ones.
+\end{iframe}
+
+\begin{iframe}[Information theory primer\nicedot Relative entropy]
+		Relative entropy or Kullback-Leibler (KL) divergence quantifies difference between
+		probability distributions.
+		If observer receives data $\mathcal{D}$, divergence between (subjective) prior and
+		posterior distributions is the
+		amount of information in $\mathcal{D}$ \emph{about} $X$ for this observer:
+		\[
+			I(\mathcal{D}\to X) =
+			D(p_{X|\mathcal{D}} || p_X)
+				= \sum_{x\in\X} p(x|\mathcal{D}) \log \frac{p(x|\mathcal{D})}{p(x)}.
+		\]
+		If observing $\mathcal{D}$ causes a large change in belief about $X$, then $\mathcal{D}$
+		contained a lot of information about $X$.
+
+		Like Lindley's (1956) information (thanks Lars!).
+\end{iframe}
+
+\begin{iframe}[Information theory primer\nicedot Mutual information]
+		Mutual information between (MI) $X_1$ and $X_2$ is the expected amount of information about
+		$X_2$ in an observation of $X_1$. Can be written in several ways:
+		\begin{align*}
+			I(X_1;X_2) &= \sum_{x_1,x_2} p(x_1,x_2) \log \frac{p(x_1,x_2)}{p(x_1)p(x_2)} \\
+					&= H(X_1) + H(X_2) - H(X_1,X_2) \\
+					&= H(X_2) - H(X_2|X_1).
+		\end{align*}
+		(1) Expected information about $X_2$ in an observation of $X_1$;\\
+		(2) Expected reduction in uncertainty about $X_2$ after observing $X_1$;\\
+		(3) Symmetric: $I(X_1;X_2) = I(X_2;X_1)$.
+\end{iframe}
+
+\begin{iframe}[Information theory primer\nicedot Conditional MI]
+		Information in one variable about another given observations of some third variable.
+		Formulated analogously by adding conditioning variables to entropies:
+		\begin{align*}
+			I(X_1;X_2|X_3) &= H(X_1|X_3) - H(X_1|X_2,X_3).
+		\end{align*}
+		Makes explicit the dependence of information assessment on background knowledge,
+		represented by conditioning variables.
+\end{iframe}
+
+
+\begin{isframe}[Information theory primer\nicedot I-Diagrams]
+		\newcommand\rad{2.2em}%
+		\newcommand\circo{circle (3.4em)}%
+		\newcommand\labrad{4.3em}
+		\newcommand\bound{(-6em,-5em) rectangle (6em,6em)}
+		\newcommand\clipin[1]{\clip (#1) \circo;}%
+		\newcommand\clipout[1]{\clip \bound (#1) \circo;}%
+		\newcommand\cliptwo[3]{%
+			\begin{scope}
+				\clipin{#1};
+				\clipin{#2};
+				\clipout{#3};
+				\fill[black!30] \bound;
+			\end{scope}
+		}%
+		\newcommand\clipone[3]{%
+			\begin{scope}
+				\clipin{#1};
+				\clipout{#2};
+				\clipout{#3};
+				\fill[black!15] \bound;
+			\end{scope}
+		}%
+		Information diagrams are a Venn diagram-like represention of entropies and mutual
+		informations for a set of random variables.
+	\begin{center}
+		\begin{tabular}{c@{\ }c}
+			\scalebox{0.8}{%
+			\begin{tikzpicture}[baseline=0pt]
+				\coordinate (p1) at (90:\rad);
+				\coordinate (p2) at (210:\rad);
+				\coordinate (p3) at (-30:\rad);
+				\clipone{p1}{p2}{p3};
+				\clipone{p2}{p3}{p1};
+				\clipone{p3}{p1}{p2};
+				\cliptwo{p1}{p2}{p3};
+				\cliptwo{p2}{p3}{p1};
+				\cliptwo{p3}{p1}{p2};
+            \begin{scope}
+               \clip (p1) \circo;
+               \clip (p2) \circo;
+               \clip (p3) \circo;
+               \fill[black!45] \bound;
+            \end{scope}
+				\draw (p1) \circo;
+				\draw (p2) \circo;
+				\draw (p3) \circo;
+				\path
+					(barycentric cs:p3=1,p1=-0.2,p2=-0.1) +(0ex,0) node {$I_{3|12}$}
+					(barycentric cs:p1=1,p2=-0.2,p3=-0.1) +(0ex,0) node {$I_{1|23}$}
+					(barycentric cs:p2=1,p3=-0.2,p1=-0.1) +(0ex,0) node {$I_{2|13}$}
+					(barycentric cs:p3=1,p2=1,p1=-0.55) +(0ex,0) node {$I_{23|1}$}
+					(barycentric cs:p1=1,p3=1,p2=-0.55) +(0ex,0) node {$I_{13|2}$}
+					(barycentric cs:p2=1,p1=1,p3=-0.55) +(0ex,0) node {$I_{12|3}$}
+					(barycentric cs:p3=1,p2=1,p1=1) node {$I_{123}$}
+					;
+				\path
+					(p1) +(140:\labrad) node {$X_1$}
+					(p2) +(-140:\labrad) node {$X_2$}
+					(p3) +(-40:\labrad) node {$X_3$};
+			\end{tikzpicture}%
+			}
+			&
+			\parbox{0.5\linewidth}{
+				\small
+				\begin{align*}
+					I_{1|23} &= H(X_1|X_2,X_3) \\
+					I_{13|2} &= I(X_1;X_3|X_2) \\
+					I_{1|23} + I_{13|2} &= H(X_1|X_2) \\
+					I_{12|3} + I_{123} &= I(X_1;X_2)
+				\end{align*}
+			}
+		\end{tabular}
+	\end{center}
+The areas of
+		the three circles represent $H(X_1)$, $H(X_2)$ and $H(X_3)$ respectively.
+		The total shaded area is the joint entropy $H(X_1,X_2,X_3)$.
+		Each undivided region is an \emph{atom} of the I-diagram.
+\end{isframe}
+
+
+
+
+\begin{isframe}[Information theory in sequences]
+	\def\bx{1.6em}%
+	\def\cn(#1,#2) {\node[circle,draw,fill=white,inner sep=0.2em] at(#1) {$#2$};}%
+	\def\dn(#1){\node[circle,inner sep=0.2em] at(#1) {$\cdots$};}%
+	\def\en(#1){coordinate(#1)}%
+	\def\tb{++(3.8em,0)}%
+	\def\lb(#1)#2{\path (#1)+(0,\bx) node[anchor=south] {#2};}
+	\def\nr(#1,#2,#3){\draw[rounded corners,fill=#3] (#1) rectangle (#2);}%
+
+		Consider an observer receiving elements of a random sequence
+		$(\ldots, X_{-1}, X_0, X_1, X_2, \ldots)$, so that at any time $t$ there is
+		a `present' $X_t$, an observed pasti $\past{X}_t$, and an unobserved future
+		$\fut{X}_t$. Eg, at time $t=3$:
+
+		\begin{figure}
+				\begin{tikzpicture}%[baseline=-1em]
+					\path (0,0) \en(X0) \tb \en(X1) \tb \en(X2) \tb \en(X3) \tb \en(X4) \tb \en(X5) \tb \en(X6);
+					\path (X0)+(-\bx,-\bx) \en(p1) (X2)+(\bx,\bx) \en(p2)
+					      (X3)+(-\bx,-\bx) \en(p3) (X3)+(\bx,\bx) \en(p4)
+					      (X4)+(-\bx,-\bx) \en(p5) (X6)+(\bx,\bx) \en(p6);
+					\nr(p1,p2,un3) \nr(p3,p4,un4) \nr(p5,p6,un5)
+					\dn(X0) \cn(X1,X_1) \cn(X2,X_2) \cn(X3,X_3) \cn(X4,X_4) \cn(X5,X_5) \dn(X6)
+					\lb(X1){Past: $\past{X}_3$}
+					\lb(X5){Future $\fut{X}_3$}
+					\lb(X3){Present}
+				\end{tikzpicture}%}%
+		\end{figure}
+	Consider how the observer's belief state evolves when, having observed up to
+	$X_2$, it learns the value of $X_3$.
+\end{isframe}
+
+\begin{iframe}[`Surprise' based quantities]
+	To obtain first set of measures, we ignore the future $\fut{X}_t$
+	and consider the probability distribution for $X_t$ give the
+	observed past $\past{X}_t=\past{x}_t$.
+
+	\begin{enumerate}
+		\item<1->
+		\textbf{Surprisingness}: negative log-probability
+		$\ell_t = -\log p(x_t|\past{x}_t)$.
+
+		\item<2->
+		Expected surprisingness given context $\past{X}=\past{x}_t$ is the entropy of the predictive distribution,
+		$H(X_t|\ev(\past{X}_t=\past{x}_t))$: uncertainty about $X_t$ before the observation is made.
+
+		\item<3->
+		Expectation over all possible realisations of process is the conditional entropy
+		$H(X_t|\past{X}_t)$ according to the observer's model. For stationary process, is
+		\emph{entropy rate} $h_\mu$.
+	\end{enumerate}
+\end{iframe}
+
+\begin{iframe}[Predictive information]
+	Second set of measures based on amount of information the observation $\ev(X_t=x_t)$
+	carries \emph{about} about the unobserved future $\fut{X}_t$, \emph{given} that we already
+	know the past $\ev(\past{X}_t=\past{x}_t)$:
+	is
+	\begin{equation*}
+		\mathcal{I}_t = I(\ev(X_t=x_t)\to\fut{X}_t|\ev(\past{X}_t=\past{x}_t)).
+	\end{equation*}
+	Is KL divergence between beliefs about future $\fut{X}_t$ prior and posterior
+	to observation $\ev(X_t=x_t)$.
+	Hence, for continuous valued variables, invariant to invertible
+	transformations of the observation spaces.
+\end{iframe}
+
+\begin{iframe}[Predictive information based quantities]
+	\begin{enumerate}
+	\item<1->
+		\emph{Instantaneous predictive information} (IPI) is just $\mathcal{I}_t$.
+
+%	Expectations over $X|\ev(Z=z)$, $Z|\ev(X=x)$, and $(X,Z)$ give 3 more information measures:
+	\item<2->
+		Expectation of $\mathcal{I}_t$ before observation at time $t$ is
+		$I(X_t;\fut{X}_t | \ev(\past{X}_t=\past{x}_t))$: mutual information conditioned on
+		observed past. Is the amount of new information about the future expected from the next observation.
+		Useful for directing attention towards the next event even before it happens?
+
+%	This is different from Itti and Baldi's proposal that Bayesian
+%	\emph{surprise} attracts attention \cite{IttiBaldi2005}, as it is a mechanism which can
+%	operate \emph{before} the surprise occurs.
+
+
+	\item<3->
+	Expectation over all possible realisations is the conditional mutual information
+	$I(X_t;\fut{X}_t|\past{X}_t)$. For stationary process, this is the global
+	\emph{predictive information rate} (PIR), the average rate at which new information arrives about
+	the future. In terms of conditional entropies, has two forms:
+	$H(\fut{X}_t|\past{X}_t) - H(\fut{X}_t|X_t,\past{X}_t)$ or
+	$H(X_t|\past{X}_t) - H(X_t|\fut{X}_t,\past{X}_t)$.
+	\end{enumerate}
+
+\end{iframe}
+
+\begin{iframe}[Global measures for stationary processes]
+	For a stationary random process model, the average levels of suprise and information
+	are captured by the time-shift invariant process information measures:
+	\begin{align*}
+		\text{entropy rate} &:  & h_\mu  &= H(X_t | \past{X}_t) \\
+		\text{multi-information rate}  &: & \rho_\mu  &= I(\past{X}_t;X_t)  = H(X_t) - h_\mu \\
+		\text{residual entropy  rate}  &: & r_\mu &= H(X_t | \past{X}_t, \fut{X}_t) \\
+		\text{predictive information  rate} &:  & b_\mu  &= I(X_t;\fut{X}_t|\past{X}_t)  = h_\mu - r_\mu
+	\end{align*}
+	Residual entropy also known as \emph{erasure entropy} \cite{VerduWeissman2006}.
+\end{iframe}
+
+\begin{isframe}[Process I-diagrams]
+%		\newcommand\subfig[2]{\shortstack{#2\\[0.75em]#1}}
+		\newcommand\subfig[2]{#2}
+		\newcommand\rad{1.75em}%
+		\newcommand\ovoid[1]{%
+			++(-#1,\rad)
+			-- ++(2 * #1,0em) arc (90:-90:\rad)
+ 			-- ++(-2 * #1,0em) arc (270:90:\rad)
+		}%
+		\newcommand\axis{2.75em}%
+		\newcommand\olap{0.85em}%
+		\newcommand\offs{3.6em}
+		\newcommand\longblob{\ovoid{\axis}}
+		\newcommand\shortblob{\ovoid{1.75em}}
+		\begin{figure}
+				\begin{tikzpicture}%[baseline=-1em]
+					\newcommand\rc{\rad}
+					\newcommand\throw{2.5em}
+					\coordinate (p1) at (180:1.5em);
+					\coordinate (p2) at (0:0.3em);
+					\newcommand\bound{(-7em,-2.6em) rectangle (7em,3.0em)}
+					\newcommand\present{(p2) circle (\rc)}
+					\newcommand\thepast{(p1) ++(-\throw,0) \ovoid{\throw}}
+					\newcommand\fillclipped[2]{%
+						\begin{scope}[even odd rule]
+							\foreach \thing in {#2} {\clip \thing;}
+							\fill[black!#1] \bound;
+						\end{scope}%
+					}%
+					\fillclipped{30}{\present,\bound \thepast}
+					\fillclipped{15}{\present,\bound \thepast}
+					\fillclipped{45}{\present,\thepast}
+					\draw \thepast;
+					\draw \present;
+					\node at (barycentric cs:p2=1,p1=-0.3) {$h_\mu$};
+					\node at (barycentric cs:p2=1,p1=1) [shape=rectangle,fill=black!45,inner sep=1pt]{$\rho_\mu$};
+					\path (p2) +(90:3em) node {$X_0$};
+					\path (p1) +(-3em,0em) node  {\shortstack{infinite\\past}};
+					\path (p1) +(-4em,\rad) node [anchor=south] {$\ldots,X_{-1}$};
+				\end{tikzpicture}%
+			\\[0.25em]
+				\begin{tikzpicture}%[baseline=-1em]
+					\newcommand\rc{2.2em}
+					\newcommand\throw{2.5em}
+					\coordinate (p1) at (210:1.5em);
+					\coordinate (p2) at (90:0.8em);
+					\coordinate (p3) at (-30:1.5em);
+					\newcommand\bound{(-7em,-2.6em) rectangle (7em,3.0em)}
+					\newcommand\present{(p2) circle (\rc)}
+					\newcommand\thepast{(p1) ++(-\throw,0) \ovoid{\throw}}
+					\newcommand\future{(p3) ++(\throw,0) \ovoid{\throw}}
+					\newcommand\fillclipped[2]{%
+						\begin{scope}[even odd rule]
+							\foreach \thing in {#2} {\clip \thing;}
+							\fill[black!#1] \bound;
+						\end{scope}%
+					}%
+%					\fillclipped{80}{\future,\thepast}
+					\fillclipped{30}{\present,\future,\bound \thepast}
+					\fillclipped{15}{\present,\bound \future,\bound \thepast}
+					\draw \future;
+					\fillclipped{45}{\present,\thepast}
+					\draw \thepast;
+					\draw \present;
+					\node at (barycentric cs:p2=0.9,p1=-0.17,p3=-0.17) {$r_\mu$};
+					\node at (barycentric cs:p1=-0.5,p2=1.0,p3=1) {$b_\mu$};
+					\node at (barycentric cs:p3=0,p2=1,p1=1.2) [shape=rectangle,fill=black!45,inner sep=1pt]{$\rho_\mu$};
+					\path (p2) +(140:3.2em) node {$X_0$};
+	%            \node at (barycentric cs:p3=0,p2=1,p1=1) {$\rho_\mu$};
+					\path (p3) +(3em,0em) node  {\shortstack{infinite\\future}};
+					\path (p1) +(-3em,0em) node  {\shortstack{infinite\\past}};
+					\path (p1) +(-4em,\rad) node [anchor=south] {$\ldots,X_{-1}$};
+					\path (p3) +(4em,\rad) node [anchor=south] {$X_1,\ldots$};
+				\end{tikzpicture}%
+%				\\[0.25em]
+%		The small dark
+%		region  below $X_0$ is $\sigma_\mu$ and the excess entropy
+%		is $E = \rho_\mu + \sigma_\mu$.
+		\end{figure}
+		Marginal entropy of `present' $X_0$ is $H(X_0)=\rho_\mu+r_\mu+b_\mu$.\\
+		Entropy rate is $h_\mu = r_\mu+b_\mu$.
+\end{isframe}
+
+\section{Markov chains}
+\label{s:InfoInMC}
+
+
+\begin{iframe}[Markov chains\nicedot Definitions]
+
+%	Now we'll look at information dynamics in one of the simplest possible models, a Markov chain.
+%	To illustrate the how the measures defined in \secrf{InfoInRandomProcs} can be computed
+%	in practice, we will consider one of the simplest random processes, a
+%	first order Markov chain.
+%	In this case, the dynamic information measures can be computed in closed-form.
+%
+
+	Let $X$ be a Markov chain with state space
+	$\{1, \ldots, K\}$, \ie the $X_t$ take values from $1$ to $K$.
+	\begin{center}
+   \begin{tikzpicture}[->]
+      \matrix[column sep=2em,ampersand replacement=\&]{
+        \cn(X,1) \&  \cn(X,2) \& \cn(X,3) \&  \cn(X,4)  \& \dn(XT) \\};
+      \rl(X1,X2) \rl(X2,X3) \rl(X3,X4) \rl(X4,XT)
+    \end{tikzpicture}
+	\end{center}
+%	For the sake of brevity let us assume that $\domA$ is the set of integers from 1 to $K$.
+	Parameterised by transition matrix $\trans \in \reals^{K\times K}$,
+%	encoding the distribution of any element of the sequence given previous one,
+	\ie $p(\ev(X_{t+1}=i)|\ev(X_t=j))=\trans_{ij}$.
+	Assume irreducibility, ergodicity \etc to ensure uniqueness of
+	stationary distribution $\pi$ such that
+	$p(\ev(X_t=i))=\init_i$ independent of $t$. Entropy rate as a function of
+	$a$ is
+% $\entrorate:\reals^{K\times K} \to \reals$:
+	\[
+		\entrorate(\trans) = \sum_{j=1}^K \init_j \sum_{i=1}^K -\trans_{ij} \log \trans_{ij}.
+	\]
+\end{iframe}
+
+\begin{iframe}[Markov chains\nicedot  PIR]
+	Predictive information rate for first order chains comes out in terms of entropy rate
+	function as
+	\[
+		b_\mu = h(a^2) - h(a),
+	\]
+	where $a^2$ is \emph{two-step} transition matrix.
+
+	\uncover<2->{
+	Can be generalised to higher-order transition matrices
+	\[
+		b_\mu = h(\hat{a}^{N+1}) - Nh(\hat{a}),
+	\]
+	where $N$ is the order of the chain and $\hat{a}$ is a sparse
+	$K^N\times K^N$ transition matrix over product state space of $N$
+	consecutive observations (step size 1).
+	}
+\end{iframe}
+
+\begin{iframe}[Entropy rate and PIR in Markov chains]
+
+	\begin{fig}{artseq}
+		\hangbox{\colfig[0.40]{matbase/fig8515}}%
+		\quad
+		\hangbox{%
+			\begin{tabular}{cc}%
+				\colfig[0.18]{matbase/fig1356} &
+				\colfig[0.18]{matbase/fig45647} \\
+				\colfig[0.18]{matbase/fig49938} &
+				\colfig[0.18]{matbase/fig23355}%
+			\end{tabular}%
+		}%
+%			\end{hanging}\\
+	\end{fig}
+	For given $K$, entropy rate varies between 0 (deterministic sequence)
+	and $\log K$ when $\trans_{ij}=1/K$ for all $i,j$.
+	Space of transition matrices explored by generating
+	them at random and plotting entropy rate vs PIR. (Note inverted
+	`U' relationship). %Transmat (d) is almost uniform.
+\end{iframe}
+
+\begin{iframe}[Samples from processes with different PIR]
+	\begin{figure}
+		\colfig[0.75]{matbase/fig847}\\
+		\colfig[0.75]{matbase/fig61989}\\
+		\colfig[0.75]{matbase/fig43415}\\
+		\colfig[0.75]{matbase/fig50385}
+	\end{figure}
+	Sequence (a) is repetition
+	of state 4 (see transmat (a) on previous slide).
+	System (b) has the highest PIR.
+\end{iframe}
+
+%				\begin{tabular}{rl}
+%					(a) & \raisebox{-1em}{\colfig[0.58]{matbase/fig9048}}\\[1em]
+%					(b) & \raisebox{-1em}{\colfig[0.58]{matbase/fig58845}}\\[1em]
+%					(c) & \raisebox{-1em}{\colfig[0.58]{matbase/fig45019}}\\[1em]
+%					(d) & \raisebox{-1em}{\colfig[0.58]{matbase/fig1511}}
+%				\end{tabular}
+
+\section{Application: The Melody Triangle}
+\begin{iframe}[Complexity and interestingness: the Wundt Curve]
+	\label{s:Wundt}
+		Studies looking into the relationship between stochastic complexity
+		(usually measured as entropy or entropy rate) and aesthetic value, reveal
+		an inverted `U' shaped curve \citep{Berlyne71}. (Also, Wundt curve \cite{Wundt1897}).
+		Repeated exposure tends to move stimuli leftwards.
+
+		\hangbox{%
+			\only<1>{\colfig[0.5]{wundt}}%
+			\only<2>{\colfig[0.5]{wundt2}}%
+		}\hfill
+		\hangbox{\parbox{0.43\linewidth}{\raggedright
+		%Too deterministic $\rightarrow$ predictable, boring like a monotone;\\
+		%Too random $\rightarrow$ are boring like white noise: unstructured,
+		%featureless, uniform.
+		Explanations for this usually appeal to a need for a `balance'
+		between order and chaos, unity and diversity, and so on, in a generally
+		imprecise way.}}
+
+
+%		Hence, a sequence can be uninteresting in two opposite ways: by
+%		being utterly predictable \emph{or} by being utterly
+%		unpredictable.
+%		Meyer \cite{Meyer2004} suggests something similar:
+%		hints at the same thing while discussing
+%		the relation between the rate of information flow and aesthetic experience,
+%		suggesting that
+%%		`unless there is some degree of order, \ldots
+%%		there is nothing to be uncertain \emph{about} \ldots
+%		`If the amount of information [by which he means entropy and surprisingness]
+%		is inordinately increased, the result is a kind of cognitive white noise.'
+
+\end{iframe}
+
+\begin{iframe}[PIR as a measure of cognitive activity]
+
+		The predictive information rate incorporates a similar balance automatically:
+		is maximal for sequences which are neither deterministic nor
+		totally uncorrelated across time.
+
+		\vspace{1em}
+		\begin{tabular}{rr}%
+			\raisebox{0.5em}{too predictable:} &
+			\only<1>{\noderow(black,un0,un0,un0,un1,un1)}%
+			\only<2>{\noderow(black,black,un0,un0,un0,un1)}%
+			\only<3>{\noderow(black,black,black,un0,un0,un0)}%
+			\only<4>{\noderow(black,black,black,black,un0,un0)}%
+		\\[1.2em]
+			\raisebox{0.5em}{intermediate:} &
+			\only<1>{\noderow(black,un1,un2,un3,un4,un5)}%
+			\only<2>{\noderow(black,black,un1,un2,un3,un4)}%
+			\only<3>{\noderow(black,black,black,un1,un2,un3)}%
+			\only<4>{\noderow(black,black,black,black,un1,un2)}%
+		\\[1.2em]
+			\raisebox{0.5em}{too random:} &
+			\only<1>{\noderow(black,un5,un5,un5,un5,un5)}%
+			\only<2>{\noderow(black,black,un5,un5,un5,un5)}%
+			\only<3>{\noderow(black,black,black,un5,un5,un5)}%
+			\only<4>{\noderow(black,black,black,black,un5,un5)}%
+		\end{tabular}
+		\vspace{1em}
+
+		(Black: \emph{observed}; red: \emph{unobserved}; paler: \emph{greater uncertainty}.)
+		Our interpretation:
+%		when each event appears to carry no new information about the unknown future,
+%		it is `meaningless' and not worth attending to.
+		Things are `interesting' or at least `salient' when each new part supplies new information about parts to come.
+
+%		Quantitative information dynamics will enable us to test this experimentally with human
+%		subjects.
+\end{iframe}
+
+\begin{iframe}[The Melody Triangle\nicedot Information space]
+ \begin{figure}
+	\colfig[0.75]{mtriscat}
+	\end{figure}
+	Population of transition matrices in 3D space of $h_\mu$, $\rho_\mu$ and $b_\mu$.
+%	Concentrations of points along redundancy axis correspond to roughly periodic patterns.
+	Colour of each point
+	represents PIR.
+	%---highest values found at intermediate entropy and redundancy.
+	Shape is mostly (not completely) hollow inside: forming roughly
+	a curved triangular sheet.
+\end{iframe}
+
+\begin{iframe}[The Melody Triangle\nicedot User interface]
+ \begin{figure}
+	\colfig[0.55]{TheTriangle.pdf}
+	\end{figure}
+	Allows user to place tokens in the triangle
+	to cause sonification of a Markov chain with corresponding information
+	`coordinate'.
+\end{iframe}
+
+\begin{iframe}[Subjective information]
+	So far we've assumed that sequence is actually sampled from
+	from a stationary Markov chain with a transition matrix known
+	to the observer.
+	This means time averages of IPI and surprise should equal
+	expectations.
+
+	\uncover<2->{
+	What if sequence is sampled from some other Markov chain,
+	or is produced by some unknown process?
+	}
+
+	\begin{itemize}
+		\item<3->
+		In general, it may be impossible to identify any `true' model. There
+		are no `objective' probabilities; only subjective ones, as
+		argued by de Finetti \cite{deFinetti}.
+
+
+		\item<4->
+		If sequence \emph{is} sampled from some Markov chain, we can
+		compute (time) averages of observer's average subjective surprise
+		and PI and also track what happens if observer gradually learns
+		the transition  matrix from the data.
+	\end{itemize}
+\end{iframe}
+
+
+\begin{iframe}[Effect of learning on information dynamics]
+	\begin{figure}
+%		\colfig{matbase/fig42687} % too small text
+%		\colfig{matbase/fig60379} % 9*19 too tall
+%		\colfig{matbase/fig52515} % 9*20 ok, perhaps text still too small
+		\colfig[0.9]{matbase/fig30461} % 8*19  ok
+%		\colfig{matbase/fig66022} % 8.5*19  ok
+	\end{figure}
+%	Upper row shows actual stochastic learning,
+%	lower shows the idealised deterministic learning.
+	\textbf{(a/b/e/f)}: multiple runs starting from same
+	initial condition but using different generative transition matrices.
+	\textbf{(c/d/g/h)}: multiple runs starting from different
+	initial conditions and converging on transition matrices
+		with (c/g) high and (d/h) low PIR.
+\end{iframe}
+
+
+\section{More process models}
+\begin{iframe}[Exchangeable sequences and parametric models]
+	De Finetti's theorem says that an exchangeable random process can be represented
+	as a sequence variables which are iid \emph{given} some hidden probability
+	distribution, which we can think of as a parameterised model:
+	\begin{tabular}{lp{0.45\linewidth}}
+		\hangbox{\begin{tikzpicture}
+			[>=stealth',var/.style={circle,draw,inner sep=1pt,text height=10pt,text depth=4pt}]
+			\matrix[ampersand replacement=\&,matrix of math nodes,row sep=2em,column sep=1.8em,minimum size=17pt] {
+				\& |(theta) [var]| \Theta \\
+				|(x1) [var]| X_1 \& |(x2) [var]| X_2 \& |(x3) [var]| X_3 \&
+				|(etc) [outer sep=2pt]| \dots \\
+			};
+			\foreach \n in {x1,x2,x3,etc} \draw[->] (theta)--(\n);
+		\end{tikzpicture}}
+		&
+			\raggedright
+			\uncover<2->{Observer's belief state at time $t$ includes probability distribution
+			over the parameters $p(\ev(\Theta=\theta)|\ev(\past{X}_t=\past{x}_t))$.}
+	\end{tabular}\\[1em]
+	\uncover<3->{
+	Each observation causes revision of belief state
+	and hence supplies information
+	$
+		I(\ev(X_t=x_t)\to\Theta|\ev(\past{X}_t=\past{x}_t))
+	%		= D( p_{\Theta|\ev(X_t=x_t),\ev(\past{X}_t=\past{x}_t)} || p_{\Theta|\ev(\past{X}_t=\past{x}_t)} ).
+	$ about $\Theta$:
+	In previous work we called this the `model information rate'.
+	}
+	\uncover<4->{(Same as Haussler and Opper's \cite{HausslerOpper1995} IIG or
+	Itti and Baldi's \cite{IttiBaldi2005} Bayesian surprise.)}
+\end{iframe}
+
+		\def\circ{circle (9)}%
+		\def\bs(#1,#2,#3){(barycentric cs:p1=#1,p2=#2,p3=#3)}%
+\begin{iframe}[IIG equals IPI in (some) XRPs]
+	\begin{tabular}{@{}lc}
+		\parbox[c]{0.5\linewidth}{\raggedright
+		Mild assumptions yield a relationship between IIG (instantaneous information gain) and IPI.
+		(Everything here implicitly conditioned on $\past{X}_t$).}
+	&
+		\pgfsetxvec{\pgfpoint{1mm}{0mm}}%
+		\pgfsetyvec{\pgfpoint{0mm}{1mm}}%
+		\begin{tikzpicture}[baseline=0pt]
+			\coordinate (p1) at (90:6);
+			\coordinate (p2) at (210:6);
+			\coordinate (p3) at (330:6);
+			\only<4->{%
+				\begin{scope}
+					\foreach \p in {p1,p2,p3} \clip (\p) \circ;
+					\fill[lightgray] (-10,-10) rectangle (10,10);
+				\end{scope}
+				\path	(0,0) node {$\mathcal{I}_t$};}
+			\foreach \p in {p1,p2,p3} \draw (\p) \circ;
+			\path (p2) +(210:13) node {$X_t$}
+						(p3) +(330:13) node {$\fut{X}_t$}
+					(p1) +(140:12) node {$\Theta$};
+			\only<2->{\path	\bs(-0.25,0.5,0.5) node {$0$};}
+			\only<3->{\path	\bs(0.5,0.5,-0.25) node {$0$};}
+		\end{tikzpicture}
+	\end{tabular}\\
+	\begin{enumerate}
+			\uncover<2->{\item	$X_t \perp \fut{X}_t | \Theta$: observations iid given $\Theta$ for XRPs;}
+			\uncover<3->{\item $\Theta \perp X_t | \fut{X}_t$:
+%		$I(X_t;\fut{X}_t|\Theta_t)=0$ due to the conditional independence of
+%		observables given the parameters $\Theta_t$, and
+%		$I(\Theta_t;X_t|\fut{X}_t)=0$
+		assumption that $X_t$ adds no new information about $\Theta$
+		given infinitely long sequence $\fut{X}_t =X_{t+1:\infty}$.}
+\end{enumerate}
+\uncover<4->{Hence, $I(X_t;\Theta_t|\past{X}_t)=I(X_t;\fut{X}_t|\past{X}_t) = \mathcal{I}_t$.\\}
+\uncover<5->{Can drop assumption 1 and still get $I(X_t;\Theta_t|\past{X}_t)$ as an additive component (lower bound) of $\mathcal{I}_t$.}
+\end{iframe}
+
+\def\fid#1{#1}
+\def\specint#1{\frac{1}{2\pi}\int_{-\pi}^\pi #1{S(\omega)} \dd \omega}
+\begin{iframe}[Discrete-time Gaussian processes]
+	Information-theoretic quantities used earlier have analogues for continuous-valued
+	random variables.  For stationary Gaussian processes, we can obtain results in
+	terms of the power spectral density $S(\omega)$, (which for discrete time is periodic
+	in $\omega$ with period $2\pi$). Standard methods give
+	\begin{align*}
+		H(X_t) &= \frac{1}{2}\left( \log 2\pi e + \log \specint{}\right), \\
+		h_\mu &= \frac{1}{2} \left( \log 2\pi e  + \specint{\log} \right), \\
+		\rho_\mu &= \frac{1}{2} \left( \log \specint{\fid} - \specint{\log}\right).
+	\end{align*}
+	Entropy rate is also known as Kolmogorov-Sinai entropy.
+%	$H(X_t)$ is a function of marginal variance which is just the total power in the spectrum.
+\end{iframe}
+
+\begin{iframe}[PIR/Multi-information duality]
+	Analysis yeilds PIR:
+	\[
+		b_\mu = \frac{1}{2} \left( \log \specint{\frac{1}} - \specint{\log\frac{1}} \right).
+	\]
+	Yields simple expression for finite-order autogregressive processes, but beware: can diverge
+	for moving average processes!
+
+	\uncover<2->{
+	Compare with multi-information rate:
+	\[
+		\rho_\mu = \frac{1}{2} \left( \log \specint{\fid} - \specint{\log}\right).
+	\]
+	Yields simple expression for finite-order moving-average processes, but can diverge
+	for marginally stable autogregressive processes.
+	}
+
+	\uncover<3->{
+		Infinities are troublesome and point to problem with notion of infinitely
+		precise observation of continuous-valued variables.
+	}
+\end{iframe}
+
+%		Information gained about model parameters (measured as the KL divergence
+%		between prior and posterior distributions) is equivalent
+%		to \textbf{Itti and Baldi's `Bayesian surprise'} \cite{IttiBaldi2005}.
+
+
+	\section{Application: Analysis of minimalist music}
+	\label{s:Experiments}
+
+\begin{iframe}[Material and Methods]
+
+%		Returning to our original goal of modelling the perception of temporal structure
+%		in music, we computed dynamic information measures for
+		We took two pieces of minimalist
+		music by Philip Glass, \emph{Two Pages} (1969) and \emph{Gradus} (1968).
+		Both monophonic and isochronous, so representable very simply as
+		a sequence of symbols (notes), one symbol per beat,
+		yet remain ecologically valid examples of `real' music.
+
+		We use an elaboration of the Markov chain model---not necessarily
+		a good model \latin{per se}, but that wasn't the point of the experiment.
+		Markov chain model was chosen as it is tractable from and information
+		dynamics point of view while not being completely trivial.
+\end{iframe}
+
+\begin{iframe}[Time-varying transition matrix model]
+		We allow transition matrix to vary slowly with time to track
+		changes in the sequence structure.
+		Hence, observer's belief state includes a probabilitiy
+		distribution over transition matrices; we choose a product of
+		Dirichlet distributions:
+		\[
+			\textstyle
+			p(\trans|\param) = \prod_{j=1}^K p_\mathrm{Dir}(\trans_{:j}|\param_{:j}),
+		\]
+		where $\trans_{:j}$ is \nth{j} column of $\trans$ and $\param$ is an
+		$K \times K$ parameter matrix.
+%		(Dirichlet, being conjugate to discrete/multinomial distribution,
+%		makes processing of observations particularly simple.)
+%		such that $\param_{:j}$ is the
+%		parameter tuple for the $K$-component Dirichlet distribution $p_\mathrm{Dir}$.
+%		\begin{equation}
+%			\textstyle
+%			p(\trans|\param) = \prod_{j=1}^K p_\mathrm{Dir}(\trans_{:j}|\param_{:j})
+%			   = \prod_{j=1}^K (\prod_{i=1}^K \trans_{ij}^{\param_{ij}-1}) / B(\param_{:j}),
+%		\end{equation}
+%		where $\trans_{:j}$ is the \nth{j} column of $\trans$ and $\param$ is an
+%		$K \times K$ matrix of parameters.
+
+		At each time step, distribution first \emph{spreads} under mapping
+		\[
+			\param_{ij} \mapsto \frac{\beta\param_{ij}}{(\beta + \param_{ij})}
+		\]
+		to model possibility that transition matrix
+		has changed ($\beta=2500$ in our experiments). Then it \emph{contracts}
+		due to new observation providing fresh evidence about transition matrix.
+%
+%		Each observed symbol % provides fresh evidence about current transition matrix,
+%		enables observer to update its belief state.
+\end{iframe}
+
+
+\begin{iframe}[Two Pages\nicedot Results]
+
+%		\begin{fig}{twopages}
+			\begin{tabular}{c@{\hspace{1.5ex}}l}%
+%			\hspace*{-1.5em}
+%				\hangbox{\colfig[0.5]{matbase/fig20304}} % 3 plots
+%				\hangbox{\colfig[0.52]{matbase/fig39528}} % 4 plots with means
+%				\hangbox{\colfig[0.52]{matbase/fig63538}} % two pages, 5 plots
+%				\hangbox{\colfig[0.52]{matbase/fig53706}} % two pages, 5 plots
+				\hangbox{\colfig[0.72]{matbase/fig33309}} % two pages, 5 plots
+			&
+				\hangbox{%
+					\parbox{0.28\linewidth}{
+						\raggedright
+						\textbf{Thick lines:} part boundaries as indicated
+						by Glass; \textbf{grey lines (top four panels):} changes in the melodic `figures';
+					%	of which the piece is constructed.
+						\textbf{grey lines (bottom panel):}
+						six most surprising moments chosen by expert listenter.
+					}
+				}
+			\end{tabular}
+%		\end{fig}
+\end{iframe}
+
+\begin{iframe}[Two Pages\nicedot Rule based analysis]
+	\begin{figure}
+		\colfig[0.98]{matbase/fig13377}
+%		\hangbox{\colfig[0.98]{matbase/fig13377}}
+	\end{figure}
+	Analysis of \emph{Two Pages} using (top) Cambouropoulos'
+	Local Boundary Detection Model (LBDM) and
+	(bottom) Lerdahl and Jackendoff's
+	grouping preference rule 3a (GPR3a), which is a function of pitch proximity.
+	Both analyses indicate `boundary strength'.
+\end{iframe}
+
+\begin{iframe}[Two Pages\nicedot Discussion]
+		Correspondence between the information
+		measures and the structure of the piece is quite close.
+		Good agreement between the six `most surprising
+		moments' chosen by expert listener and  model information signal.
+
+		What appears to be an error in the detection of
+		the major part boundary (between events 5000 and 6000) actually
+		raises a known anomaly in the score, where Glass places the boundary several events
+		before there is any change in the pattern of notes. Alternative analyses of \emph{Two Pages}
+		place the boundary in agreement with peak in our surprisingness signal.
+\end{iframe}
+
+\comment{
+\begin{iframe}[Gradus\nicedot Results]
+
+%		\begin{fig}{gradus}
+			\begin{tabular}{c@{\hspace{1.5ex}}l}
+%				&
+%				\hangbox{\colfig[0.4]{matbase/fig81812}}
+%				\hangbox{\colfig[0.52]{matbase/fig23177}} % two pages, 5 plots
+%				\hangbox{\colfig[0.495]{matbase/fig50709}} % Fudged segmentation
+%				\hangbox{\colfig[0.495]{matbase/fig3124}} % Geraint's segmentation
+				\hangbox{\colfig[0.715]{matbase/fig11808}} % Geraint's segmentation, corrected
+			&
+%				\hangbox{\colfig[0.5]{matbase/fig39914}}
+				\hangbox{%
+					\parbox{0.28\linewidth}{
+						\raggedright
+						\textbf{Thick lines:} part boundaries as indicated
+						by the composer.
+						\textbf{Grey lines:} segmentation by expert listener.
+
+						Note: traces smoothed with Gaussian
+						window about 16 events wide.
+					}
+				}
+			\end{tabular}
+%		\end{fig}
+\end{iframe}
+
+\begin{iframe}[Gradus\nicedot Rule based analysis]
+	\begin{figure}
+		\colfig[0.98]{matbase/fig58691}
+	\end{figure}
+	Boundary strength analysis of \emph{Gradus} using (top) Cambouropoulos'
+	\cite{CambouropoulosPhD} Local Boundary Detection Model  and
+	(bottom) Lerdahl and Jackendoff's \cite{LerdahlJackendoff83}
+	grouping preference rule 3a.
+\end{iframe}
+}
+\begin{iframe}[Gradus\nicedot Metrical analysis]
+	\begin{figure}
+		\begin{tabular}{cc}
+			\colfig[0.40]{matbase/fig56807} & \colfig[0.41]{matbase/fig27144} \\
+			\colfig[0.40]{matbase/fig87574} & \colfig[0.41]{matbase/fig13651} \\
+			\hspace*{1ex}\colfig[0.39]{matbase/fig19913} & \hspace*{1ex}\colfig[0.40]{matbase/fig66144}
+		\end{tabular}
+	\end{figure}
+\end{iframe}
+
+\comment{
+\begin{iframe}[Gradus\nicedot Discussion]
+
+		\emph{Gradus} is much less systematically structured than \emph{Two Pages}, and
+		relies more on the conventions of tonal music, which are not represented the model.
+
+		For example initial transition matrix is uniform, which does not correctly represent
+		prior knowledge about tonal music.
+
+		Information dynamic analysis does not give such a
+		clear picture of the structure; but some of the fine structure can be related
+		to specific events in the music (see Pearce and Wiggins 2006).
+%		nonetheless, there are some points of correspondence between the analysis and
+%		segmentation given by Keith Potter.
+
+\end{iframe}
+}
+
+	\section{Application: Beat tracking and rhythm}
+
+	\begin{iframe}[Bayesian beat tracker]
+		\uncover<1->{
+			Works by maintaining probabilistic belief state about time of next
+			beat and current tempo.
+
+			\begin{figure}
+				\colfig{beat_prior}
+			\end{figure}
+			}
+
+		\uncover<2->{
+			Receives categorised drum events (kick or snare) from audio analysis front-end.
+			}
+
+	\end{iframe}
+
+	\begin{iframe}[Information gain in the beat tracker]
+		\begin{tabular}{ll}
+			\parbox[t]{0.43\linewidth}{\raggedright
+			\uncover<1->{
+				Each event triggers a change in belief state, so we can compute
+				information gain about beat parameters.}\\[1em]
+
+			\uncover<2->{
+				Relationship between IIG and IPI
+				means we treat it as a proxy for IPI.}
+				}
+			&
+			\hangbox{\colfig[0.55]{beat_info}}
+		\end{tabular}
+	\end{iframe}
+
+	\begin{iframe}[Analysis of drum patterns]
+		We analysed 17 recordings of drummers, both playing solo or with a band.
+		All patterns in were in 4/4.
+		\begin{itemize}
+			\item
+			\uncover<1->{
+				Information tends to arrive at beat times: consequence of structure of model.
+			}
+			\item
+			\uncover<2->{
+				Lots of information seems to arrive after drum fills and breaks
+				as the drummer reestablishes the beat.
+			}
+			\item
+			\uncover<3->{
+				No consistent pattern of information arrival in relation to metrical
+				structure, so no obvious metrical structure in micro-timing of events.
+				However, still possible that metrical structure might emerge from predictive
+				analysis of drum pattern.
+				}
+		\end{itemize}
+	\end{iframe}
+
+	\section{Summary and conclusions}
+	\label{s:Conclusions}
+
+	\begin{iframe}[Summary]
+
+		\begin{itemize}
+		\item Dynamic, observer-centric information theory.
+		\item Applicable to any dynamic probabilistic model.
+		\item PIR potentially a measure of complexity.
+		\item Simple analysis for Markov chains and Gaussian processes.
+		\item Applications in music analysis and composition.
+		\item Search for neural correlates is ongoing (that's another talk\ldots).
+		\end{itemize}
+		Thanks!
+	\end{iframe}
+
+	\begin{bframe}[Bibliography]
+		\bibliographystyle{alpha}
+		{\small \bibliography{all,c4dm,compsci}}
+	\end{bframe}
+\end{document}