Relationship to Kullback–Leibler Divergence

If (and only if) the tree topology prior is Discrete Uniform over all distinct tree topologies (in which case is a constant for all ), then Lindley information is equivalent to the Kullback–Leibler () divergence measured from prior () to posterior ():

This KL interpretation is useful because, as we show later, the overall information content can be partitioned into additive clade-specific components that are themselves clade-specific KL divergences weighted by the posterior probability of the clade.

The KL interpretation also gives rise to a means of estimating Lindley information. Letting denote the data on which the posterior distribution is based,

where is the likelihood marginalized over all model parameters on a fixed tree topology , and is the likelihood marginalized over all model parameters (including tree topology). This approach would clearly require considerable computation, as the total marginal likelihood as well as all tree-specific marginal likelihoods for trees having nonnegligible posterior probabilities must be estimated. We next consider a more tractable approach using conditional clade distribution estimated from a single posterior sample of trees.