frame-best-merger.tex

% SPDX-License-Identifier: CC-BY-4.0
% Copyright 2018 Toni Dietze
\documentclass[beamer]{standalone}
\input{preamble.tex}
\title{\jobname}
\begin{document}
\begin{standaloneframe}{\jobname}
	\hfill
	{\small\tikz{\node[fill=HKS92K20, inner sep=0]{\begin{tabular}{l@{ \ldots{} }l}
		\(\mathscr{A}\) & bottom-up deterministic ta \((Q, Σ, I, Δ)\)
	\\	\(c\) & corpus over \(\trees{Σ}\)
	\\	\(Π\) & set of all bottom-up determinism preserving mergers for \(\mathscr{A}\)
	\end{tabular}};}}
	\begin{block}{maximize the likelihood (again)}
		\[
			\argmax_{π ∈ Π} \lklhd{c}{\semantics{\mle[π(\mathscr{A})]{c}}}
			\uncover<3->{
				=
				\argmax_{π ∈ Π}
					\frac
						{\lklhd{c}{\semantics{\mle[π(\mathscr{A})]{c}}}}
						{\lklhd{c}{\semantics{\mle[\mathscr{A}]{c}}}}
			}
		\]
	\end{block}
	\vspace*{-1em}
	\begin{overprint}
	\onslide<2-3>
		\begin{block}{combining likelihood and mle}
			\[
				\lklhd{c}{\semantics{\mle[\mathscr{A}]{c}}}
				= \frac
					{∏_{q ∈ Q} {c_{\mathscr{A}}^{\mathrm{I}}(q)}^{c_{\mathscr{A}}^{\mathrm{I}}(q)}}
					{\corpussize{c}}
				· \frac
					{∏_{τ ∈ Δ} {c_{\mathscr{A}}^{\mathrm{Δ}}(τ)}^{c_{\mathscr{A}}^{\mathrm{Δ}}(τ)}}
					{∏_{q ∈ Q} {c_{\mathscr{A}}^{\mathrm{Q}}(q)}^{c_{\mathscr{A}}^{\mathrm{Q}}(q)}}
			\]
		\end{block}
	\onslide<4>
		\begin{block}{}
			Assuming every merger
			\begin{itemize}
			\item does not collapse rules or root states, and
			\item preserves bottom-up determinism,
			\end{itemize}
			it is best to merge the two least frequent non-terminals (w.r.t.\@ \(c_{\mathscr{A}}^{\mathrm{Q}}\)).
		\end{block}
	\end{overprint}
\end{standaloneframe}
\end{document}