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Suppose we have a probability measure $\gamma$ defined on the $d$-dimensional lattice $\mathbb{N}^d$. Let us say $d = 5$ for simplicity. I will also write $\gamma_{1:5}$ for this measure and write say $\gamma_{25}$ for the marginal on coordinates $2$ and $5$, as so on.

Let $A := [n] = \{1,\dots,n\}$, $A^c := [n]^c$, $B:= [n+1]$, $B^c = [n+1]^c$ and $b = \{n+1\}$. All of these are subsets of $\mathbb{N} := \{1,2,\dots\}$.

We know the values of $\gamma_{1:5}$ for subsets of the form $A^\circ \times \cdots \times A^\circ$ where $^\circ$ denotes either nothing or a complement symbol $^c$. We want to extend our knowledge to subsets of the form $B^\circ \times \dots \times B^\circ$ where $^\circ$ has the same meaning and $B$ is defined as above. What allows us to do so, is this extra knowledge that say $ \frac{\gamma_{1:5}(A^c \times b \times A^c \times A^c \times b)}{\gamma_{1:5} (b \times b \times b \times b \times b)} = \frac{1}{\rho_1 \rho_3 \rho_4} $ and the like, for some positive number $\rho_i, i \in [d]$. Dropping extra symbols and abusing notation (and more), we know values of sequence of $A^c$ and $b$, in terms of sequences of pure $b$'s.

It seems that there are algebraic rules involved in doing this extension. I would like to know if these look familiar to anyone and if there is general picture behind the scene. Here is how one proceeds:

  • We extend our knowledge to a sequence of $A^c$'s, $b$'s and a single $A$.

  • We extend our knowledge to a sequence of $A^c$'s, $b$'s and arbitrary numbers of $A$'s. The rule turns out to be: the probability of such a sueqnce is the probability of a "sequence obtained by replacing all $b$s with $A^c$s", multiplied by $\rho_i$'s in places of $b$s. Example $ A^c b A b A = \rho_2 \rho_4 A^c A^c A A^c A $

  • Now we write $B^c = A -b$ and $B = A +b$. To compute say probability of $B^cBB$ (say d = 3 here for simplicity) We write $B^c B B = (A^c - b) (A+b)(A+b).$ Replace $b$ in the $i$th parenthesis with with $\rho_i A^c$: $B^c B B = (A^c -\rho_1 A^c)(A + \rho_2 A^c)(A+\rho_3A^c),$ and then follow the rules of a noncommutative associative algebra, to obtain $B^c BB = \bar{\rho}_1 A^c A A + \bar{\rho}_1 \rho_3 A^c A A^c + \bar{\rho}_1 \rho_2 A^c A^c A + \bar{\rho}_1 \rho_2 \rho_3 A^c A^c A^c,$ where $\bar{\rho}_i = 1- \rho_i$.

This works in general an produces the correct expansion for any sequence of $B$'s and $B^c$'s in terms of sequences of $A$'s and $A^c$s. Again, my humble question is, does this look familiar to anyone? Is there something more general (or better grounded) of which this is a special case?

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