A special issue of the journal "Cognitive neuropsychiatry", Vol. 15, Issue 1-3. Delusional beliefs have sometimes been considered as rational inferences from abnormal experiences. We explore this idea in more detail, making the following points. First, the abnormalities of cognition that initially prompt the entertaining of a delusional belief are not always conscious and since we prefer to restrict the term “experience” to consciousness we refer to “abnormal data” rather than “abnormal experience”. Second, we argue that in relation to many delusions (we consider seven) one can clearly identify what the abnormal cognitive data are which prompted the delusion and what the neuropsychological impairment is which is responsible for the occurrence of these data; but one can equally clearly point to cases where this impairment is present but delusion is not. So the impairment is not sufficient for delusion to occur: a second cognitive impairment, one that affects the ability to evaluate beliefs, must also be present. Third (and this is the main thrust of our paper), we consider in detail what the nature of the inference is that leads from the abnormal data to the belief. This is not deductive inference and it is not inference by enumerative induction; it is abductive inference. We offer a Bayesian account of abductive inference and apply it to the explanation of delusional belief.