Pious fraud is a term used to describe an act of dishonesty committed by someone who believes it will have beneficial moral results that outweigh the immorality of the act. For example, Christians have been caught placing bloody tears on icons of Mary to make it appear the statue is bleeding, and, when asked to explain their actions, they said they knew what they were doing was wrong, but they believed those who saw it would believe it to be a miracle and it would increase their faith and lead more people to Jesus.
Pious fraud is especially popular among people who have beliefs based on faith rather than objective evidence. Examples of pious fraud include stigmatics like Therese Neumann, authors of pseudepigraphal texts like the Testament of Adam and the Second Epistle of Peter, the creators of the Shroud of Turin, and so on. There are also examples of pious fraud in the health care industry where believers in alternative medicine forge test results in order to convince the medical community to put more research into their field.
Note that pious fraud shouldn't be confused with acts of dishonesty committed for other reasons such as personal gain. For example, discredited doctor, Andrew Wakefield, forged clinical results and lied about the safety of the MMR vaccine, probably not because he actually believed it was harmful (even his own research said it wasn't), but because he was working to patent a competing vaccine which would result in a lot of money. Such an act is not pious fraud, but just plain fraud.