This chapter gives some simple, useful techniques for approximating the
-values of various types of optimal alignment scores. It starts with general techniques: if, e.g., a dynamic programming computation has probabilistically independent inputs, its successive states form a Markov chain. Thus, if the states are not too numerous, a “Markov computation” yields their distribution. The chapter reviews the three extreme-value distributions, which are relevant to approximating the distribution of random maxima, in the same way the normal distribution is relevant to approximating the distribution of random sums. In general, convergence to an extreme-value distribution is often painfully slow, so the Poisson approximation for counting rare and weakly dependent events can be a more flexible tool for approximating the distribution of maxima. In particular, the extreme-value and Poisson distributionsyield an approximate distribution for the optimal local alignment score of two random sequences, and a finite-size correction can increase the accuracy of statistical approximations if the sequences are relatively short. Moreover, the concept of “islands” permits many statistical approximation problems in local alignment to be transformed to combinatorial problems. Finally, the “Independent Diagonals Approximation” broadens the application of many of the previous methods, and an “Independent Alignments Approximation” converts many alignment variants into the combinatorial problem of determining an “effective length”.