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+\chapter{Posterior Regularization}
+Posterior regularization is an alternative way of clustering the phrases.
+Unlike a Baysian approach where intuitions about the data are expressed through the priors, posterior regularization imposes constraints on posterior distributions of the data.
+
+In this chapter , we will introduce a basic clustering model with EM 
+and look at shortcomings of the basic model. This will motivate us for
+more complicated posterior regularized models.
+\section{Phrase Clustering Model} 
+As a brief recap, the clustering problem we are working with
+is to label phrases with $K$ induced categories, where
+$K$ is picked manually.
+Phrases are obtained from bi-text data.
+We also look at context
+ words before and after
+phrases as cues for clustering.
+The relationship between phrases, contexts and categories are
+represented with a generative model shown in 
+Figure \ref{fig:EM}: a phrase picks a 
+category and then that category generates the contex for the phrase.
+
+\begin{figure}[h]
+  \centering
+  \includegraphics[width=3.5in]{EMdigram}
+  \caption{Basic Phrase Clustering Model}
+  \label{fig:EM}
+\end{figure}
+
+The joint probability of a category $z$ and a context $\bf{c}$ 
+given a phrase $\bf{p}$ is
+\[
+P(z,\bf{c}|\bf{p})=P(z|\bf{p})P(\bf{c}|z).
+\]
+$P(z|\bf{p})$ is distribution of categories given a phrase.
+This can be learned from data.
+$P(\bf{c}|z)$ is distribution of context given a category.
+Since a context usually contains multiple slots for words, we further
+decompose this distribution into independent distributions at
+each slot. For example, suppose a context consists of two positions 
+before and after the phrase. Denote these words as 
+$c_{-2},c_{-1},c_1,c_2$.
+Use $P_{-2},P_{-1},P_1,P_2$ to denote distributions of words at each 
+position, $P(\bf{c}|z)$ is decomposed as
+\[
+P(\bf{c}|z)=P_{-2}(c_{-2}|z)P_{-1}
+(c_{-1}|z)P_1(c_1|z)P_2(c_2|z).
+\]
+The posterior probability of a category given a phrase
+and a context can be computed by normalizing the joint probability:
+\[
+P(z|\bf{p},\bf{c})=\frac{P(z,\bf{c}|\bf{p})}
+{\sum_{i=1,K}P(i,\bf{c}|\bf{p})}.
+\]
+With the mechanisms to compute the posterior probabilities, we can 
+apply EM to learn all the probabilities.
+\section{Sparsity Constraints}
+A common linguistic intuition we have about the phrase 
+clustering problem is that a phrase should be put into very
+few categories, e.g. a verb phrase is unlikely to be used as 
+a noun phrase. In other words, the categorization of
+a phrase should be sparse.
+The generative model we proposed above with EM
+allows a phrase to be labelled with many tags. As we observed
+from the output, EM is using more categories than we wanted for
+each phrase.
+Posterior regularization
+provides a way to enforce sparsity \citep{ganchev:penn:2009}.
+The constraint we use here is called $l_1/ l_\infty$
+regularization.
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