The Kirkwood superposition approximation was introduced in 1935 by John G. Kirkwood as a means of representing a discrete probability distribution. The Kirkwood approximation for a discrete probability density function P ( x 1 , x 2 , … , x n ) {\displaystyle P(x_{1},x_{2},\ldots ,x_{n})} is given by
P ′ ( x 1 , x 2 , … , x n ) = ∏ i = 1 n − 1 [ ∏ T i ⊆ V p ( T i ) ] ( − 1 ) n − 1 − i = ∏ T n − 1 ⊆ V p ( T n − 1 ) ∏ T n − 2 ⊆ V p ( T n − 2 ) ⋮ ∏ T 1 ⊆ V p ( T 1 ) {\displaystyle P^{\prime }(x_{1},x_{2},\ldots ,x_{n})=\prod _{i=1}^{n-1}\left[\prod _{{\mathcal {T}}_{i}\subseteq {\mathcal {V}}}p({\mathcal {T}}_{i})\right]^{(-1)^{n-1-i}}={\frac {\prod _{{\mathcal {T}}_{n-1}\subseteq {\mathcal {V}}}p({\mathcal {T}}_{n-1})}{\frac {\prod _{{\mathcal {T}}_{n-2}\subseteq {\mathcal {V}}}p({\mathcal {T}}_{n-2})}{\frac {\vdots }{\prod _{{\mathcal {T}}_{1}\subseteq {\mathcal {V}}}p({\mathcal {T}}_{1})}}}}}
where
∏ T i ⊆ V p ( T i ) {\displaystyle \prod _{{\mathcal {T}}_{i}\subseteq {\mathcal {V}}}p({\mathcal {T}}_{i})}
is the product of probabilities over all subsets of variables of size i in variable set V {\displaystyle \scriptstyle {\mathcal {V}}} . This kind of formula has been considered by Watanabe (1960) and, according to Watanabe, also by Robert Fano. For the three-variable case, it reduces to simply
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