perplexity_train_test.tex

\documentclass{article}

\usepackage{amsmath}
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\begin{document}
\title{Explanation\_Train\_Test}
\author{Sumit Sidana}
\maketitle
\newdate{date1}{26}{02}{2016}
\date{\displaydate{date1}}
\section{Explanation: Train and Test}
Notations:
\begin{equation}
z = ailment
\end{equation}
\begin{equation}
t = time (month, week,... etc.)
\end{equation}


\textit{Perplexity of topic model} depends on its ability to predict the \textit{probability of future words}.\\
\textit{Probability of words} depend on \textit{probability of topic} with the following formula.
\begin{equation}\label{probWord}
P(w) = \sum_zP(w|z)P(z)=\sum_z\underbrace{\frac{n(z,w)}{n(z)}}_	{\mbox{Constant, w $\in$ second month, n(z,w) $\in$ train}}\times \underbrace{P(z)}_{\mbox{Varies with topic model}}
\end{equation}

So focussing on the only thing in equation \ref{probWord} which varies: $P(z)$\\
\begin{equation}
P(z|t) = \sum_{tweet\ p \in t}P(z|p)
\end{equation}

\begin{equation}
P(z|p) = 	\sum_{word\ w \in p}P(z|w)P(w|p)=\sum_w\frac{n(z,w)}{n(w)}P(w|p)
\end{equation}

P(z) needs to be calculated on the 1st month and:
\begin{itemize}
\item \textbf{atam}: Underlying assumption of atam is that topics stay static with respect to time. That is why $P(z)$ of 1st month needs to be used for $P(z)$ of the second month. Because P(z) stays static with time. This is what tm-lda did for lda
\item \textbf{tmatam}:  $P(z)$ of second month needs to be predicted using the $P(z)$ of first month using the transition matrix learnt during training period
\item \textbf{tatam}: $P(z)$ of second month needs to be computed directly on second month because model itself learnt ailments using the knowledge of time in-built in the model. Can ailments inferred using a time-aware model  actual representative of words tweeted about in the time of interest or it just learns noise?
\end{itemize}



\end{document}