I begin with an intuitively plausible principle: if proposition H retrodicts or predicts that proposition D is true, then Pr(D|H) > ½. I then consider the difference between forward-directed and backward-directed conditional probabilities. Until the 1980s, population genetics theory was dominated by the former; the situation changed with the appearance of coalescent theory, which deploys the latter. A forward-directed probability model cannot retrodict, and a backward-directed model cannot predict. A Bayesian model will do both if it deploys both types of conditional probability. I then describe ideas from model selection theory in statistics that throw doubt on the idea that true models of a process will always be more accurate predictors or retrodictors than false models of that process. I then discuss theorems in information theory that show how the passage of time degrades information in a causal chain. The rate of information loss can be reduced and even cancelled by the branching process in phylogenetic trees. The separation of within-lineage from across-lineage assessments of information loss is an instance of Simpson’s paradox.