Notes 1A special event x0 is sometimes given at time 0 to mark the beginning of the sequence; the model then generatestherestofthesequenceconditionedonx0

NHP is a thoughtfully designed framework that has been demonstrated effective on temporal data, but our method can also be used for other models with parametric intensityfunctions. In this section, we prove the claim in section 2.2 that argmaxθJLL(θ) = Θ When we take the expectation under p, each summand gets weighted by the probability that x[0,t) and x[t,t+dt) would take on the values in that summand. Therefore,wehaveG θ( t, x[0, t)) < 0since the distributions in equation (9) are distinct for the given history x[0, t). This lemma says: if θ and θ are meaningfully different in that they predict different intensities at time t for some history, then they actually do so for a set of histories of non-zero measure, making this difference visible in the objective functions like JLL(θ) (see above) and JNC(θ) (see Appendix B). We use d to denote the maximal difference between the intensities over (t0,t00), i.e., d If x[0,t) doesn't have any event, then its probability p( x[0,t)) = exp( Suppose that t1 has been shifted by R. Recall that we need order-(1dt)I many such histories.