See the first paragraph of Section 3 of http://arxiv.org/pdf/1508.03606v1.pdf

1. A short note on soft-plus polynomials
Tomonari MASADA @ Nagasaki University
February 3, 2016
Consider a set V of two binary variables {x1, x2} and the function of the form
ϕ : {0, 1}V
→ R; x → log(1 + exp(w⊤
x + c)) . (1)
As is explained in [1], this corresponds to the free energy added by one hidden binary variable interacting
pairwise with each of the two visible binary variable x1 and x2.
All possible values obtained by ϕ are
log(1 + exp(c)) for (x1, x2) = (0, 0),
log(1 + exp(w1 + c)) for (x1, x2) = (1, 0),
log(1 + exp(w2 + c)) for (x1, x2) = (0, 1), and
log(1 + exp(w1 + w2 + c)) for (x1, x2) = (1, 1). (2)
Therefore, the function ϕ can be rewritten as follows based on the fact that x1 and x2 are binary:
ϕ(x1, x2) = log(1 + exp(c))(1 − x1)(1 − x2)
+ log(1 + exp(w1 + c))x1(1 − x2)
+ log(1 + exp(w2 + c))(1 − x1)x2
+ log(1 + exp(w1 + w2 + c))x1x2 . (3)
That is, ϕ is a polynomial. The coefficients of the monomials can be given explicitly as below:
ϕ(x1, x2) ={log(1 + exp(c)) − log(1 + exp(w1 + c)) − log(1 + exp(w2 + c)) + log(1 + exp(w1 + w2 + c))}x1x2
+ {− log(1 + exp(c)) + log(1 + exp(w1 + c))}x1
+ {− log(1 + exp(c)) + log(1 + exp(w2 + c))}x2
+ log(1 + exp(c)) . (4)
The following formula gives the coefficients:
KB(w, c) =
∑
C⊆B
(−1)|BC|
log
(
1 + exp
( ∑
i∈C
wi + c
))
, B ∈ 2V
. (5)
For example, when B = {x1, x2},
K{x1,x2}(w, c) =(−1)|{x1,x2}∅|
log
(
1 + exp
( ∑
i∈∅
wi + c
))
+ (−1)|{x1,x2}{x1}|
log
(
1 + exp
( ∑
i∈{x1}
wi + c
))
+ (−1)|{x1,x2}{x2}|
log
(
1 + exp
( ∑
i∈{x2}
wi + c
))
+ (−1)|{x1,x2}{x1,x2}|
log
(
1 + exp
( ∑
i∈{x1,x2}
wi + c
))
= log(1 + exp(c)) − log(1 + exp(w1 + c)) − log(1 + exp(w2 + c)) + log(1 + exp(w1 + w2 + c)) .
(6)
A similar discussion can be made for the case where we have more than two binary variables. See [1].
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2. References
[1] Guido Montufar, Johannes Rauh. Hierarchical Models as Marginals of Hierarchical Models. 2015.
arXiv:1508.03606.
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