I am reading the paper titled "A Theory for Record Linkage" by Pellegi & Stunter, where they present the following definition:
$$ \gamma [ \alpha (a), \beta (b)]=\{ \gamma ^{1}[ \alpha (a), \beta (b)],\ .\ .\ .\ , \gamma^{k}[ \alpha (a), \beta (b)]\} $$
I understand $ f(x,y) $ which we're all familiar with, but what is $ f[x,y] $ ?
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By having a fast read of the linked paper i think $f[x,y]$ indicates a scalar function having as independent variables the two records $\alpha$ and $\beta$ that are functions of $a$ and $b$
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Immediately before the definition, the paper introduces it by: "Formally we define the comparison vector as a vector function of the records $\alpha(a), \beta(b)$:" Prior to that, $a$ is identified as a generic element of a set $A$ and likewise $b\in B$, and finally (but verbosely) we are told that $\gamma : A\times B \rightarrow \Gamma$, where $\Gamma$ is the set of all realisations of $\gamma$ and called the comparison space.
It is a little odd for a vector to be given using brace-notation (i.e. $\gamma = \{\cdots \}$) and later on the paper $\gamma$ is referred to as $\gamma = (\gamma^1, \gamma^2, \ldots \gamma^K)$ There are similar inconsistencies with probabilities, where we see both $P(a,b)$ and $P\{\gamma [\alpha(a), \beta(b)] \}$ written but it does appear as though the author is attempting to use parentheses/brackets/braces to distinguish between scalar and vector quantities. The paper dates from 1969 when notational standards were slightly different.
You may safely interpret $\gamma[\alpha(a), \beta(b)]$ as $\gamma (\alpha(a), \beta(b))$, though you might, if you're writing things down yourself, need sometimes to clarify that something is a vector operation, or is performed component-wise, rather than being a scalar operation.
Perhaps the author is trying to comply with a rule that states you should avoid nested parenthesis, so he just uses brackets insted of double parenthesis but the meaning is the same.