42 lines
2 KiB
TeX
42 lines
2 KiB
TeX
% -*- root: main.tex -*-
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\section{Analysis}
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\label{sec:analysis}
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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 $$\numWorldsP = \numWorldsSum.$$
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The first step is to show that the expectation of the estimate of a tuple t's membership across all worlds is $\numWorldsSum$.
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\begin{align}
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&\expect \big[\estimate\big]\\
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=&\expect \big[\estExpOne\big]\\
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=&\expect \big[\sum_{\substack{j \in [B],\\
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\wVec \in \pw~|~ \sketchHash{i}[\wVec] = j,\\
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\wVec[w']\in \pw~|~ \sketchHash{i}[\wVec[w']] = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]\big]\\
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=&\expect \big[ \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime} + \nonumber \\
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&\phantom{{}\wIndParam{\wVec}}\sum_{\substack{j \in [B], \\
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\wVec~|~\sketchHashParam{\wVec} = j,\\
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\wVecPrime ~|~ \sketchHashParam{\wVecPrime} = j,\\ \wVec \neq \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot\polarFunc{\wVecPrime}\big]\textit{(by linearity of expectation)}\\
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=&\expect \big[ \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\
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\wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\
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\wVec = \wVecPrime}} \wIndParam{\wVec} \cdot \polarFunc{\wVec} \cdot \polarFunc{\wVecPrime}\big] \nonumber \\
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&\phantom{{}\big[}\textit{(by uniform distribution in the second summation)}\\
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&= \sum_{\substack{j \in [B],\\
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\wVec~|~\sketchHashParam{\wVec}= j,\\}} \wIndParam{\wVec}
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\end{align}
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For the next step, we show that the variance of an estimate is small.$$\var{\estimate}$$
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\begin{align}
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&=\var{\estExpOne}\\
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&= \big(\estTwo\big)^2\\
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&=\sum_{\substack{
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\wVec_1, \wVec_2,\\
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\wVecPrime_1, \wVecPrime_2 \in \pw}}\wIndParam{\wVec_1} \cdot \wIndParam{\wVec_2}\cdot\polarFunc{\wVec_1}\cdot\polarFunc{\wVec_2}\cdot\polarFunc{\wVecPrime_1}\cdot\polarFunc{\wVecPrime_2}
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\end{align}
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