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Path integrals (and generalizations) are sums over configurations. A logical extension of that process would be to sum not over configurations, but over theories (configurations are possible solutions of a single theory).

Renormalization already plays around the "space of theories" (with changing parameters), but AFAIK, this path is not pursued much.

My (very basic) thinking goes alongside the central limit theorem: summing many realizations of uncertain distributions gives something very definite (close to a Gaussian distribution). So could the summation over many realizations of uncertain distributions (all possible theories) give something definite (the theory we live in, given some measurements)?