# MLE for two independent factor CIR

Following the maximun likelihood estimation as done in Klavidko I would like to generalize this to more independent factors . In first istance I would use the transition function at time t as a sum of the non chi squared conditional distributions for each factor: $$p_t = p_{1 t} +.. + p_{N t}$$ then take the log of the product of all t times used in the data and optimize: $$L = \sum_{t}log(p_t)$$ On the other hand in the paper multi they just optimize the sum of each log product $$L = \sum_{t,i}log(p_{i t})$$ Which one is correct? Thanks