frame-readoff.tex

% SPDX-License-Identifier: CC-BY-4.0
% Copyright 2018 Toni Dietze
\documentclass[beamer]{standalone}
\input{preamble.tex}
\begin{document}
%         1 2 3 4 5 6 7 8 9 0 1 2
\timeline{- + + + + + + + + + + -}{tlRuns}
\timeline{- - + - - - - - - - - -}{tlRuleSsAB}
\timeline{- - - + - - - - - - - -}{tlRuleAa}
\timeline{- - - - + - - - - - - -}{tlRuleBbB}
\timeline{- - - - - + - - - - - -}{tlRuleBbA}
\timeline{- - - - - - + - - - - -}{tlRootState}
\timeline{- - - - - - - + + + + +}{tlMLE}
\timeline{- - - - - - - - + + + +}{tlEM}
\timeline{- - - - - - - - - + + +}{tlWeights}
\timeline{- - - - - - - - - - + -}{tlWeightsInTrees}
\timeline{- - - - - - - - - - + +}{tlLklhd}
\timeline{- - - - - - - - - - - +}{tlOdd}
\begin{standaloneframe}{\jobname}
\centering
\begin{tikzpicture}
[ text height=0.75em
, text depth=0.25em
]
	\node at (2em, 3em) {corpus \(c\):};
	\node<\tlRuleSsAB-> at (15em, 3em) {automaton \(\oalt<\tlWeights>{ℳ}{\mathcal{A}}\):};
	\begin{scope}
	[ level distance=2.25em
	, sibling distance=1.5em
	, anchor=base
	, inner sep=0.1em
	, node distance=0.4em
	]
		\node[onslide={<\tlRuleSsAB>alert.bg}] (t1) at (0, 0) {\(σ\)}
			child { node[onslide={<\tlRuleAa>alert.bg}] {\(α\)}
			}
			child { node[onslide={<\tlRuleBbA>alert.bg}] {\(β\)}
				child { node[onslide={<\tlRuleAa>alert.bg}] {\(α\)}
				}
			};
		\node[onslide={<\tlRuleSsAB>alert.bg}] (t2) at (6em, 0) {\(σ\)}
			child { node[onslide={<\tlRuleAa>alert.bg}] {\(α\)}
			}
			child { node[onslide={<\tlRuleBbB>alert.bg}] {\(β\)}
				child { node[onslide={<\tlRuleBbB>alert.bg}] {\(β\)}
					child { node[onslide={<\tlRuleBbA>alert.bg}] {\(β\)}
						child { node[onslide={<\tlRuleAa>alert.bg}] {\(α\)}
						}
					}
				}
			};
		\node[base left = 1em of t1] {$t_1\colon$};
		%\node at (0, -9em) {$c(t_1) = 1$};
		\node[base left = 1em of t2] {$t_2\colon$};
		%\node[anchor=base] at (5em, -9em) {$c(t_2) = 1$};
	\end{scope}

	\matrix
	[ matrix anchor=base.base west
	, every node/.style={anchor=base west}
	, row sep=0.1em
	, ampersand replacement=\&
	] at (13em, 0) {
		\uncover<\tlRuleSsAB->{\node[onslide=<\tlRuleSsAB>alert.bg] (base) {$\nt{S} → σ(\nt{A}, \nt{B})$};}
	\&
		\uncover<\tlWeights>{\node {${} ↦ 1$};}
	\\
		\uncover<\tlRuleAa->{\node[onslide=<\tlRuleAa>alert.bg] {$\nt{A} → α$};}
	\&
		\uncover<\tlWeights>{\node {${} ↦ 1$};}
	\\
		\uncover<\tlRuleBbB->{\node[onslide=<\tlRuleBbB>alert.bg] {$\nt{B} → β(\nt{B})$};}
	\&
		\uncover<\tlWeights>{\node {${} ↦ 0.5$};}
	\\
		\uncover<\tlRuleBbA->{\node[onslide=<\tlRuleBbA>alert.bg] {$\nt{B} → β(\nt{A})$};}
	\&
		\uncover<\tlWeights>{\node {${} ↦ 0.5$};}
	\\[0.25em]
		\uncover<\tlRootState->{\node[onslide={<\tlRootState>alert.bg}] (iota) {\(\nt{S}\)};}
	\&
		\uncover<\tlWeights>{\node {${} ↦ 1$};}
	\\};

	\node<\tlRuleSsAB-> [left=0.4em of base, inner sep=0] {\(\alt<\tlWeights>{δ}{Δ}\colon\)};
	\node<\tlRootState->[left=0.4em of iota, inner sep=0] {\(\alt<\tlWeights>{ι}{I}\colon\)};

	\uncover<\tlLklhd>{
		\node[anchor=base west] at (13em, -9.5em) {$\lklhd{c}{⟦ℳ⟧} = 0.5^4 = 0.0625$};
	}
	\only<\tlRuns>{
		\begin{scope}[node distance = -0.3em and -0.6em, text depth = 0]
			\node[above right = of t1, onslide={<\tlRootState>alert.bg}] {\(\nt{S}\)};
			\node[above left  = of t1-1      ] {\(\nt{A}\)};
			\node[above right = of t1-2      ] {\(\nt{B}\)};
			\node[above right = of t1-2-1    ] {\(\nt{A}\)};

			\node[above right = of t2, onslide={<\tlRootState>alert.bg}] {\(\nt{S}\)};
			\node[above left  = of t2-1      ] {\(\nt{A}\)};
			\node[above right = of t2-2      ] {\(\nt{B}\)};
			\node[above right = of t2-2-1    ] {\(\nt{B}\)};
			\node[above right = of t2-2-1-1  ] {\(\nt{B}\)};
			\node[above right = of t2-2-1-1-1] {\(\nt{A}\)};
		\end{scope}
	}
	\only<\tlWeightsInTrees>{
		\begin{scope}[node distance=0em, every node/.style=alert.bg, inner sep=0.2em, text depth = 0]
			\scriptsize

			\node[base right=of t1] {1};
			\node[base left=of t1-1] {1};
			\node[base right=of t1-2] {0.5};
			\node[base right=of t1-2-1] {1};

			\node[base right=of t2] {1};
			\node[base left=of t2-1] {1};
			\node[base right=of t2-2] {0.5};
			\node[base right=of t2-2-1] {0.5};
			\node[base right=of t2-2-1-1] {0.5};
			\node[base right=of t2-2-1-1-1] {1};
		\end{scope}
	}

	\draw<\tlOdd>[decorate, decoration=brace] (t1-2.north east) -- node[right, alert.bg, inner sep=0.1em, xshift=0.3em] {odd} (t1-2.south east);
	\draw<\tlOdd>[decorate, decoration=brace] (t2-2.north east) -- node[right, alert.bg, inner sep=0.1em, xshift=0.3em] {odd} (t2-2-1-1.south east);
\end{tikzpicture}
\begin{columns}
\column<\tlMLE>{0.59\linewidth}
	\begin{block}{mle for pta\s with fixed ta \(\mathcal{A}\)}
		\centering
		\(\displaystyle
			%\mle[\mathbfit{M}]{c} =
			\argmax_{(ι, δ) ∈ Θ}\ \lklhd{c}{⟦(\mathcal{A}, ι, δ)⟧}
		\)
		\\[0.5em]
		\(Θ = \{(ι, δ) \mid \text{\((\mathcal{A}, ι, δ)\) is a probabilistic wta}\}\)
	\end{block}
\column<\tlEM>{0.41\linewidth}
	\flushright
	\(⟹\) \emph{EM algorithm} [\cite{1977DempsterLairdRubin}] can approximate local optimum
\end{columns}
\end{standaloneframe}
\end{document}