Changed p_{t, \textbf{W}_t} to p_{t, j} for j = W_t.

introduction.tex

@@ -8,7 +8,7 @@ $\pdb = \inparen{\worlds, \bpd}$ is defined over a set of tuples $\tupset$ and a
 Any such world can be encoded as a vector (of length $\numvar=\abs{\tupset}$) from $\worlds$, such that the multiplicity of each $\tup \in \tupset$ is stored at a distinct index.
 A given world $\worldvec \in\worlds$ can be interpreted as follows: for each $\tup \in \tupset$, $\worldvec_{\tup}$ is the multiplicity of $\tup$ in $\worldvec$.
 We note that encoding a possible world as a vector, while non-standard, is equivalent to encoding it as a bag of tuples (\Cref{prop:expection-of-polynom} in \Cref{subsec:expectation-of-polynom-proof}).
-Given that tuple multiplicities are independent events, the  probability distribution $\bpd$ can be expressed compactly by assigning each tuple a (disjoint) probability distribution over $[0,\bound]$. Let $\prob_{\tup,j}$ denote the probability that tuple $\tup$ is assigned multiplicity $j$. The probability of a world $\worldvec$ is then $\prod_{\tup \in \tupset} \prob_{\tup,\worldvec_{\tup}}$.
+Given that tuple multiplicities are independent events, the  probability distribution $\bpd$ can be expressed compactly by assigning each tuple a (disjoint) probability distribution over $[0,\bound]$. Let $\prob_{\tup,j}$ denote the probability that tuple $\tup$ is assigned multiplicity $j$. The probability of a world $\worldvec$ is then $\prod_{\tup \in \tupset} \prob_{\tup,j}$ for $j = \worldvec_{\tup}$.
 %
 % Allowing for $\leq \bound$ multiplicities across all tuples gives rise to having $\leq \inparen{\bound+1}^\numvar$ possible worlds instead of the usual $2^\numvar$ possible worlds of a $1$-\abbrTIDB, which (assuming set query semantics), is the same as the traditional set \abbrTIDB.
 % In this work, since we are generally considering bag query input, we will only be considering bag query semantics.