Does anyone know how you get the likelihood function for the waiting time statistics, p11 of CT4-05? It says use a similar model to the two-state model, does that mean I have to solve the forward equations?
heya you!! What are you studying this time? The MLE for the able\ill\dead model... its not easy to explain in text
Hey yourself - so not the forward equations at least. The result seems kind of simple, as if there ought to be an easier way to get there, but then a likelihood for a mixed distribution is a bit of an odd critter to start with. Ah well, my colleagues don't reckon it's likely to come up. Thanks anyway!
Note to self It's amazing what comes to you on dull train journeys. In case anyone else gets caught on the same bit (or indeed cares in the first place): Where I was going wrong is that you're supposed to calculate the likelihood of a particular path, not of a particular combination of v's and w's and so on. It just happens to be a function of the v's, w's etc. So find density for first transition time : (mu+sigma)exp[-(mu+sigma)t_1], multiply by probability that transition goes in the right direction (say able to ill) sigma/(mu+sigma), multiply by next transition time density etc... BTW, I really don't like the "probability function" they give for the two-state model p10 section 4. It's not a density function and it's not a mass function; the function's meaningless until you specify where it's which. You also need to specify the range of possible values (outside which the density/mass is zero). So I think the "simple form" is deceptive to say the least. Let alone the "joint probability function"... For example, say I have a mixed distribution on [0,1], with "probability function" f(x)=1/2, 0<=x<=1. What's the distribution?
