 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

The Kirkwood approximation does not generally produce a valid probability distribution (the normalization condition is violated).Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure.