I propose a new application of profile Hidden Markov Models in the area of SNP discovery from resequencing data, to greatly reduce false SNP calls caused by misalignments around insertions and deletions (indels). The central concept is per-Base Alignment Quality, which accurately measures the probability of a read base being wrongly aligned. The effectiveness of BAQ has been positively confirmed on large datasets by the 1000 Genomes Project analysis subgroup.