  To make decisions organisms often accumulate information across multiple
timescales. However, most experimental and modeling studies of decision-making
focus on sequences of independent trials. On the other hand, natural
environments are characterized by long temporal correlations, and evidence used
to make a present choice is often relevant to future decisions. To understand
decision-making under these conditions we analyze how a model ideal observer
accumulates evidence to freely make choices across a sequence of correlated
trials. We use principles of probabilistic inference to show that an ideal
observer incorporates information obtained on one trial as an initial bias on
the next. This bias decreases the time, but not the accuracy of the next
decision. Furthermore, in finite sequences of trials the rate of reward is
maximized when the observer deliberates longer for early decisions, but
responds more quickly towards the end of the sequence. Our model also explains
experimentally observed patterns in decision times and choices, thus providing
a mathematically principled foundation for evidence-accumulation models of
sequential decisions.
