MOTIVATION: Mutual information (MI) is a quantity that measures the dependence between two arbitrary random variables and has been repeatedly used to solve a wide variety of bioinformatic problems. Recently, when attempting to quantify the effects of sampling variance on computed values of MI in proteins, we encountered striking differences among various novel estimates of MI. These differences revealed that estimating the 'true' value of MI is not a straightforward procedure, and minor variations of assumptions yielded remarkably different estimates. RESULTS: We describe four formally equivalent estimates of MI, three of which explicitly account for sampling variance, that yield non-equal values of MI given exact frequencies. These MI estimates are essentially non-predictive of each other, converging only in the limit of implausibly large datasets. Lastly, we show that all four estimates are biologically reasonable estimates of MI, despite their disparity, since each is actually the Kullback-Leibler divergence between random variables conditioned on equally plausible hypotheses. CONCLUSIONS: For sparse contingency tables of the type universally observed in protein coevolution studies, our results show that estimates of MI, and hence inferences about physical phenomena such as coevolution, are critically dependent on at least three prior assumptions. These assumptions are: (i) how observation counts relate to expected frequencies; (ii) the relationship between joint and marginal frequencies; and (iii) how non-observed categories are interpreted. In any biologically relevant data, these assumptions will affect the MI estimate as much or more-so than observed data, and are independent of uncertainty in frequency parameters.

Motivation: Mutual information (MI) is a quantity that measures the dependence between two arbitrary random variables and has been repeatedly used to solve a wide variety of bioinformatic problems. Recently, when attempting to quantify the effects of sampling variance on computed values of MI in proteins, we encountered striking differences among various novel estimates of MI. These differences revealed that estimating the ‘true’ value of MI is not a straightforward procedure, and minor variations of assumptions yielded remarkably different estimates. Results: We describe four formally equivalent estimates of MI, three of which explicitly account for sampling variance, that yield non-equal values of MI given exact frequencies. These MI estimates are essentially non-predictive of each other, converging only in the limit of implausibly large datasets. Lastly, we show that all four estimates are biologically reasonable estimates of MI, despite their disparity, since each is actually the Kullback–Leibler divergence between random variables conditioned on equally plausible hypotheses. Conclusions: For sparse contingency tables of the type universally observed in protein coevolution studies, our results show that estimates of MI, and hence inferences about physical phenomena such as coevolution, are critically dependent on at least three prior assumptions. These assumptions are: (i) how observation counts relate to expected frequencies; (ii) the relationship between joint and marginal frequencies; and (iii) how non-observed categories are interpreted. In any biologically relevant data, these assumptions will affect the MI estimate as much or more-so than observed data, and are independent of uncertainty in frequency parameters. Contact: [email protected] Supplementary information: Supplementary data are available at Bioinformatics online.

MOTIVATION: In a nucleotide or amino acid sequence, not all sites evolve at the same rate, due to differing selective constraints at each site. Currently in computational molecular evolution, models incorporating rate heterogeneity always share two assumptions. First, the rate of evolution at each site is assumed to be independent of every other site. Second, the values of these rates are assumed to be drawn from a known prior distribution. Although often assumed to be small, the actual effect of these assumptions has not been previously quantified in the literature. RESULTS: Herein we describe an algorithm to simultaneously infer the set of n-1 relative rates that parameterize the likelihood of an n-site alignment. Unlike previous work (a) these relative rates are completely identifiable and distinct from the branch-length parameters, and (b) a far more general class of rate priors can be used, and their effects quantified. Although described in a Bayesian framework, we discuss a future maximum likelihood extension. CONCLUSIONS: Using both synthetic data and alignments from the Myc, Max and p53 protein families, we find that inferring relative rather than absolute rates has several advantages. First, both empirical likelihoods and Bayes factors show strong preference for the relative-rate model, with a mean Delta ln P=-0.458 per alignment site. Second, the computed likelihoods and Bayes factors were essentially independent of the relative-rate prior, indicating that good estimates of the posterior rate distribution are not required a priori. Third, a novel finding is that rates can be accurately inferred even when up to approximately 4 substitutions per site have occurred. Thus biologically relevant putative hypervariable sites can be identified as easily as conserved sites. Lastly, our model treats rates and tree branch-lengths as completely identifiable, allowing for the first time coherent simultaneous inference of branch-lengths and site-specific evolutionary rates. AVAILABILITY: Source code for the utility described is available under a BSD-style license at http://www.fernandes.org/txp/article/9/site-specific-relative-evolutionary-rates.