Most medical statistics methods require the number of patients in a data set to be much larger than the number of measurements that are recorded for each patient. This requirement is increasingly problematic. We can now measure vastly more patient characteristics than in the past, but unless we have similarly large numbers of patients in our data sets (which is expensive, and for rare diseases  impossible), our data cannot be used fully.

Pooling data from different medical centers is often impossible due to privacy regulations, consent limitations, and logistic hurdles. Bayesian Federated Inference (BFI) is a novel statistical approach via which one can recover reliably from specific local analyses in separate centers what would have been found if their local data had been pooled. One can thereby harness the statistical power of large combined data sets without any need for data sharing.

Overfitting correction in
medical inference

When data sets contain insufficient numbers of patients compared to the number of measurements per patients statistical methods are known to start `overfitting’, i.e. to misinterpret noise as signal. This leads to unreliable predictions, and implies that many expensive data remain under-used (see above description of the BFI method).

We have shown, using mathematical tools from statistical physics (the replica methods), that it is possible for a large family of statistical models to predict the relation between the incorrect inferences of an overfitting model and what would have been reported in the absence of overfitting. Our results, illustrated by application to linear, logistic, and Cox regression, enable us to correct statistical inferences in the hitherto forbidden overfitting regime.