elab model section

Author: mhtess

writing/cogsci/negant-cogsci2018.tex

@@ -110,16 +110,27 @@ \section{Computational model}
 \citeA{Lassiter2013} provided a probabilistic pragmatics model that constrains the threshold $\theta$ relative to a context, formalized by a prior distribution over the value of the scalar degree $x$.  
 %
 \begin{align}
-L_{1}(x, \theta \mid u) &\propto S_{1}(u \mid x, \theta) \cdot P_(x) \cdot P(\theta) \label{eq:L1} \\
-S_{1}(u \mid x, \theta) &\propto \exp{(\alpha_1 \cdot \ln {L_{0}(x \mid u, \theta)})} \label{eq:S1}\\
-L_{0}(x \mid u, \theta) &\propto {\delta_{[\![u]\!](x, \theta)} \cdot P_{c}(x)} \label{eq:L0}
+L_{1}(x, \theta \mid u) &\propto S_{1}(u \mid x, \theta) \cdot P(x) \cdot P(\theta) \label{eq:L1} \\
+S_{1}(u \mid x, \theta) &\propto \exp{(\alpha_1 \cdot \ln {L_{0}(x \mid u, \theta)} - \text{cost}(u))} \label{eq:S1}\\
+L_{0}(x \mid u, \theta) &\propto {\delta_{[\![u]\!](x, \theta)} \cdot P(x)} \label{eq:L0}
 \end{align}
 %
-This is a Rational Speech Act (RSA) model, a recursive Bayesian model where speaker $S$ and listener $L$ coordinate on an intended meaning \cite<for a review, see >{Goodman2016:RSA}.
-In this framework, the pragmatic listener $L_1$ tries to resolve the state of the world $x$ (e.g., the height) from the utterance she heard $u$ (e.g., ``tall'').
-She imagines the utterance came from an approximately rational Bayesian speaker $S_1$ trying to inform a naive listener $L_0$, who in turn updates her prior beliefs $P_(x)$ via an utterance's literal meaning $[\![u]\!](x)$.
-$\theta$ comes from an uninformed prior and is resolved by the listener by reasoning about the likely states of the world $P_(x)$ (e.g., possible heights) and the likelihood that a speaker would say the adjective given a state and a threshold $S(u \mid x, \theta)$.
-
+Equations \ref{eq:L1}-\ref{eq:L0} comprise a Rational Speech Act (RSA) model, a recursive Bayesian model wherein a pragmatic listener $L_1$ tries to resolve the intended meaning of an utterance $u$ by combining their prior beliefs $P(x)$ with the generative process of the utterance, a speaker model $S_1$ \cite<for a review, see>{Goodman2016:RSA}.
+The speaker model $S_1$  is an approximately rational Bayesian agent trying 
+to inform a naive listener $L_0$ (Eq.~\ref{eq:L0}) about the state of the world $x$.
+In the context of adjective interpretation, the state of the world is given by the value of the scalar degree $x$ (e.g., the height of the referent).
+The literal listener updates their prior beliefs $P_(x)$ via an utterance's literal meaning $[\![u]\!](x)$, the threshold function described above.
+The pragmatic listener has uncertainty about $\theta$, which comes from an uninformed prior and is resolved by jointly reasoning about the likely states of the world $P_(x)$ (e.g., possible heights) and the likelihood that a speaker would say the adjective given a state and a threshold $S(u \mid x, \theta)$.
+
+The proportionality in Eq.~\ref{eq:S1} implies normalization over a set of alternative utterances. 
+Formal models of adjective interpretation (including an alternative model proposed by \citeNP{Qing2014:Adjectives}) have so far restricted the set of alternative utterances to be only an adjective (e.g., either ``tall'' or ``short'') and an information-less, null utterance. 
+Here, we introduce antonyms in the alternative set, as well as their logical negation glossed as modifier negation (e.g., ``not tall'' and ``not short''). 
+Following standard approaches in formal semantics, antonyms are given their own thresholds ($\theta_-$), so Eq.~\ref{eq:L1} becomes:
+%
+\begin{align}
+L_{1}(x, \theta_+, \theta_- \mid u) &\propto S_{1}(u \mid x, \theta_+, \theta_-)  P(x)  P(\theta_+)  P(\theta_-) \label{eq:L1a}
+\end{align}
+\subsection{Model behavior}
 
 
 \section{Experiment 1: Measuring the ordering}

writing/cogsci/negant.bib

@@ -127,6 +127,19 @@ @article{Goodman2016:RSA
   publisher={Elsevier}
 }
 
+@article{Qing2014:Adjectives,
+abstract = {This paper addresses two issues that arise in a degree-based approach to the semantics of positive forms of gradable adjectives such as tall in the sentence John is tall (e.g., Kennedy {\&} McNally 2005; Kennedy 2007): First, how the standard of comparison is contextually determined; Second, why gradable adjectives exhibit the relative-absolute distinction. Combining ideas of previous evolutionary and probabilistic approaches (e.g., Potts 2008; Franke 2012; Lassiter 2011; Lassiter {\&} Goodman 2013), we propose a new model that makes exact and empirically testable probabilistic predictions about speakers' use of gradable adjectives and that derives the relative-absolute distinction from considerations of optimal language use. Along the way, we distinguish between vagueness and loose use, and argue that, within our approach, vagueness can be understood as the result of uncertainty about the exact degree distribution within the comparison class.},
+author = {Qing, Ciyang and Franke, Michael},
+file = {:Users/mht/Documents/Mendeley Desktop/Qing, Franke/Qing, Franke{\_}2014(2).pdf:pdf},
+journal = {Proceedings of SALT},
+mendeley-groups = {vagueness/comparison-class,negated-antonyms},
+pages = {23--41},
+title = {{Gradable adjectives, vagueness, and optimal language use: A speaker-oriented model}},
+volume = {24},
+year = {2014}
+}
+
+
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