Somatic mutations in cancer genomes are caused by mutational processes such as ultraviolet light and disrupted DNA repair mechanisms. Each mutational process causes a distinctive pattern of mutations (a “mutational signature”), which can be inferred from whole genome and whole exome sequencing data. However, as we show via simulation studies, the methods used to infer the mutational signatures are not robust. Even slight perturbations to the assumed model for how mutations are generated leads to substantially incorrect inferences. Since we know the model is incorrect, our results suggest current methods are likely inferring many spurious processes and missing bone fide processes. We offer an alternative, more reliable method for inferring mutational signatures using scalable Bayesian inference and an adaptive power likelihood method. We validate our approach in simulation studies and apply it to 4,645 whole-genome sequences. We discuss some possible biological implications of our findings.