Suppose one has an unknown univariate function f : \mathbb{R}^+ \to \mathbb{R} such that f is differentiable and strictly convex. Suppose further one has some data \{ X , Y = f(X) \}. The ultimate goal is to estimate some function g_\theta : \mathbb{R}^+ \to \mathbb{R} that approximates f, such that one can estimate the indexing parameters \theta and then fix them (to their posterior means, say) in a future Stan program.
I would appreciate insights and references to approaches that can accommodate what we know about f. I’m not very fluent on either Gaussian processes or splines, but I guess these would be good methods to check out. What I don’t know is how to enforce convexity, for instance.
Thanks in advance.
Edit: we also have an important constraint that f(0) = 0.