lnBF

lnBF stands for the natural logarithm of the Bayes factor and is used in Bayesian model comparison to quantify the strength of evidence the data provide for two competing models or hypotheses, typically denoted M1 and M0. It is defined as lnBF = ln p(D|M1) − ln p(D|M0), where p(D|Mk) is the marginal likelihood of the data under model Mk. Positive values indicate the data favor M1, while negative values favor M0. The magnitude of lnBF reflects the strength of evidence: small absolute values suggest weak evidence, larger values indicate stronger support for one model over the other.

Interpretation of lnBF often follows scales analogous to those used for Bayes factors, though thresholds vary by field. Roughly, lnBF near zero signals little to no preference; values around 1–2.5 are often described as modest to substantial evidence; 2.5–5 as strong to very strong evidence; and values above about 5 as decisive in favor of M1. Negative lnBF values imply evidence in favor of M0.

Computation typically requires the marginal likelihood p(D|Mk), which involves integrating the likelihood over the model’s parameter priors. This can be done analytically for simple models or with numerical methods such as bridge sampling or nested sampling. A common quick approximation relates lnBF to the Bayesian information criterion (BIC): lnBF ≈ 0.5 × (BIC0 − BIC1), where BICk is the BIC of model k.

Relation to posterior model probabilities: assuming equal prior probabilities for the models, P(M1|D) = BF × P(M1) / [BF × P(M1) + P(M0)], and since BF = exp(lnBF), the posterior odds can be computed from lnBF directly.

Limitations include sensitivity to priors and model specification; lnBF should be interpreted within the broader context of the data and domain.