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. Respective fraud risk estimates are combined using fusion proportions 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. The sigmoidal functions approximate a relationship between risk estimates produced by fraud risk detection models and a percentage of transactions associated with respective risk estimates; the relationship is represented in terms of real-world distributions of fraudulent and non-fraudulent transaction.