Supplementary Material for Jump-means: Small-variance Asymptotics for Markov Jump Processes

To obtain the SVA objective from the parametric MJP model, we begin by scaling the exponential distribution f(t;λ) = λ exp(−λt), which is an exponential family distribution with natural parameter η = −λ, log-partition function ψ(η) = − ln(−η), and base measure ν(dt) = 1 [1]. To scale the distribution, introduce the new natural parameter η̃ = βη and log-partition function ψ̃(η̃) = βψ(η̃/β). The new base measure ν̃(dt) is uniquely defined by the integral equation [see 1, Theorem 5] ∫ exp(η̃t)ν̃(dt) = exp(ψ̃(η̃)) = exp(−β ln(η̃/β)) = β β η̃β .