paper-BagRelationalPDBsAreHard/analysis.tex

% -*- root: main.tex -*-
\section{Analysis}
\label{sec:analysis}
We begin the analysis by showing that with high probability an estimate is approximately $\numWorldsP$, where $p$ is the probability measure for a given TIPD.  Note that $$\numWorldsP = \numWorldsSum.$$

The first step is to show that the expectation of the estimate of a tuple t's membership across all worlds is $\numWorldsSum$.

\begin{align}
&\expect \big[\estimate\big]\\
=&\expect \big[\estExpOne\big]\\
=&\expect \big[\sum_{\substack{j \in [B],\\
			 \wVec \in \pw~|~ \sketchHash{i}[\wVec] = j,\\
			 \wVec[w']\in \pw~|~ \sketchHash{i}[\wVec[w']] = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]\big]\\
=&\expect \big[ \sum_{\substack{j \in [B],\\
				\wVec~|~\sketchHashParam{\wVec}= j,\\
				\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
				\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime} +  \nonumber \\
&\phantom{{}\wIndParam{\wVec}}\sum_{\substack{j \in [B], \\
				\wVec~|~\sketchHashParam{\wVec} = j,\\
				\wVecPrime ~|~ \sketchHashParam{\wVecPrime} = j,\\ \wVec \neq \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot\polarFunc{\wVecPrime}\big]\textit{(by linearity of expectation)}\\
=&\expect \big[ \sum_{\substack{j \in [B],\\
				\wVec~|~\sketchHashParam{\wVec}= j,\\
				\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
				\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime}\big] \nonumber \\
&\phantom{{}\big[}\textit{(by uniform distribution in the second summation)}\\
&=  \sum_{\substack{j \in [B],\\
				\wVec~|~\sketchHashParam{\wVec}= j,\\}} \wIndParam{\wVec}
\end{align}

For the next step, we show that the variance of an estimate is small.$$\var{\estimate}$$

\begin{align}
&=\var{\estExpOne}\\
&= \big(\estTwo\big)^2\\
&=\sum_{\substack{
		\wVec_1, \wVec_2,\\
		 \wVecPrime_1, \wVecPrime_2 \in \pw,\\
		 \sketchHashParam{\wVec_1} = \sketchHashParam{\wVecPrime_1},\\
		 \sketchHashParam{\wVec_2} = \sketchHashParam{\wVecPrime_2}
		 }}\wIndParam{\wVec_1} \cdot \wIndParam{\wVec_2}\cdot\polarFunc{\wVec_1}\cdot\polarFunc{\wVec_2}\cdot\polarFunc{\wVecPrime_1}\cdot\polarFunc{\wVecPrime_2}
\end{align}