The Inverse Problem
The forward model is easy: given inputs, compute the output. The inverse is the question that actually matters: given the output, recover the input. Most of finance and most of model calibration is the inverse direction of an equation that's trivial forward.
the skeleton
- 1 forward modelWrite the function that maps inputs to outputs the straightforward way.
- 2 known + unknownMark which side of the equation is observed and which is the question.
- 3 invertApply log, root, or numerical solve depending on what the unknown is.
instances · 3
finance · bitcoin-pizza
The Bitcoin Pizza
objective F = P(1+r)^t with known F, P; unknown t or r
stops when log inverts to find t; root inverts to find r — same forward, two inverses.
finance · present-value
What Is Future Money Worth Today?
objective future cash flow known; present value unknown
stops when Divide forward expression by (1+r)^t — discounting *is* the inverse of compounding.
ml / dl · model-calibration
Model Calibration
objective model outputs predicted probabilities; observed frequencies known; unknown temperature T
stops when Minimise log-loss over T to invert: which T makes the forward pass match reality?
leans on
walk the instances
Working Backward →
Three pages, one question — given the output, find the input.