According to one aspect, transaction information is received and applied to multiple fraud risk mathematical models that each produce a respective raw score, which are transformed with respective sigmoidal transform functions to produce optimized likelihood of fraud risk estimates to provide to a merchant. In one embodiment, the respective fraud risk estimates are combined using fusion proportions that are associated with the respective risk estimates, producing a single point risk estimate, which is transformed with a sigmoidal function to produce an optimized single point risk estimate for the transaction. The sigmoidal functions are derived to approximate a relationship between risk estimates produced by fraud risk detection models and a percentage of transactions associated with respective risk estimates, where the relationship is represented in terms of real-world distributions of fraudulent transaction and non-fraudulent transaction. One embodiment is directed to computing respective risk test penalties for multiple risk tests in one or more of the multiple fraud risk mathematical models used to estimate the likelihood of fraud, given a certain pattern of events represented by the transaction information, wherein the respective risk test penalties are computed as the inverse of the sum of one and a false positive ratio for the respective risk test.