Is analogy between database and probability concepts legitimate? In Relational Databases: Tutorial for Statisticians Joe R. Hill casts probability view onto database theory. In Table 1 the author summarized the parallels between the two disciplines, describing relation structure together with three relational algebra operations in probabilistic terms.  

I seek clarification on 2 items:


What probability distribution corresponds to empty relation? An intermediate step in Dr.Hill's paper is indicator function -- which is standard trick of thinking about relations in functional terms. My understanding is that the author generalizes {0,1} valued indicator function to probability distribution. There, the 1 valued tuples are assigned not vanishing discrete probabilities (assuming discrete domain), while all the tuples not belonging to relation inherit their 0 indicator function value as probability. My objection is that this would work for nonempty sets of tuples only.
Missing from the parallels is Relational Algebra union operation -- what probability concept is it analogous to? If we take average, then it would fail to honor associativity law.


Edit: Resolved confusion with transformation constraint. Consider a relation 

x y
---
1 a
2 a
3 b


honoring FD x-&gt;y. One can match it with joint distribution like this

\ |  1   2   3  |
--+-------------+----
a | 1/4 1/4  0  | 1/2
b |  0   0  1/2 | 1/2
--+-------------+----
  | 1/4 1/4 1/2 |  1


Still the analogy is weak: there is well known interpretation of Functional dependencies in terms of entropy.
I would expect that the empty relation corresponds to the always-zero probability distribution.  Disjoint union corresponds to addition of probability distributions.  However, standard union would be more complicated.  I don't really know if there is some way to encode the standard union for probability distributions.  (I have heard people say that probability distributions form a boolean algebra.  So there must be some operation that serves as the union.)