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In this section, we return to the issue of the cost of inference scaling quadratically with the number of candidate jump times |W |. Recall that this is a consequence of the fact that for a general sMJP, the future behaviour depends not just on the current state si, but also on the duration for which the sMJP has been in that state. Thus, when running the forward-backward algorithm, at stage i, we need to explicitly represent probabilities corresponding to all possible values of li, the time since the sMJP entered state si. For sMJPs observed over long intervals, the set W can be large, making quadratic scaling intolerably expensive.

Discussion 112

There are a number of approaches to addressing this issue for discrete-time sMJPs, and we can adapt these to the forward-backward sampling stage of our MCMC algo- rithm. One simple approach is break the observation interval into a number of smaller segments, and sequentially update the trajectory in each segment conditioned on the rest of the trajectory. An alternative is to follow a slice-sampling approach (Dewar et al., 2012). Here, at the ith stage of the forward-backward algorithm, rather than allowing li to take all values {0, wi − wi−1, · · · , wi − w0}, we restrict its range from {0, · · · , wi− wci} for some random slice variable ci ∈ {0, · · · , i}. By allowing si to vary from one iteration to the next, we can construct an exact MCMC chain, while taking advantage of the fact that very long holding times are atypical.

The quadratic scaling of inference for sMJPs is a worst case result, and there exist hazard rates which still allow efficient inference over long observation intervals. Clearly, the MJP, with constant transition rates is one such example. In general, if the hazard function has a constant tail (so that the corresponding waiting-time density has an exponential tail), then the system has only a finite window of memory. Thus, suppose that for some time τwin, the hazard function has the form:

A(τ ) = ˜A(τ ) τ ≤ τwin

= Atail τ > τwin (6.38)

Then at time t during the forward filtering state, we need to account only for those values of ln that range from 0 to τwin; all larger values of ln can be summarized by a single state. Consequently, the cost of the resulting forward-backward dynamic pro- gramming algorithm now scales quadratically with the length of this memory window τ , and only linearly with the total observation interval.

Even if the actual hazard function is not of this form, we can approximate it with such a hazard function, and use the sampled paths as Metropolis-Hastings proposals for samples from the original system. Such an approximation would be suitable for the gamma hazard functions fromchapter 5for example; recall that these plateau out to a constant value as τ → ∞.

6.8

Discussion

In this chapter, we described a general framework for MCMC inference in semi-Markov processes. Our scheme is based on a procedure of dependent thinning that general- izes uniformization. Given the state of the system at any instant, we define a hazard function that dominates the true hazard function. Our scheme then proceeds by se- quentially sampling the time of the next candidate event given this function, and then updating the state of the system at this time. Our scheme now allows us to perform MCMC inference by alternately sampling thinned events given the current trajectory,

Discussion 113

and then a new trajectory given all candidate event times. At a high level, the first step can be viewed as sampling a random discretization of time. We showed how this can be done relatively easily exploiting properties of the Poisson process. Given this discretization of time, we can leverage available discrete-time MCMC schemes to up- date the sMJP trajectory. In general, it is straightforward to extend our approach here to piecewise-constant stochastic processes with a more complicated dependence on the past.

There are a number of possible avenues for further study. In our experiments in this chapter, we set the dominating rates to be twice the true event rates at any time. While this is convenient, it can result in poor mixing because of the requirement that the new and old instantaneous hazard functions resemble each other. Recall that for uniformization, the dominating hazard function was a constant Ω independent of the state of the system. By constructing more complicated dominating functions, it is possible to approximate such a constant rate, while avoiding the need for a very high dominating rate that results from say, rare events with very high leaving rates. Such an approach allows us to trade off computational costs and mixing rates more carefully. The reason we needed the old and new hazard functions to resemble each other was because they both had to explain the same set of candidate transition times W . We can thus also consider schemes that propose a new set of candidate times Wnew, allowing more global moves.

We saw that for general sMJPs, inference scales quadratically with W , the number of candidate jump times. We discussed a number of possible approaches to dealing with this problem in section 6.7, it is worth studying them in further detail. In our exper- iments, we studied sMJPs with fixed parameters. Likesubsection 3.5.2, it is possible to take a fully Bayesian approach, placing priors on these parameters as well. For in- stance,Berger and Sun(1993) discuss parameter inference for the Weibull distribution. With such a Bayesian approach, care needs to be taken with state-dependent bounding functions that attempt to approximate uniformization, since this will have to adapt to the varying parameter values.

Chapter 7

MJPs with unbounded rates

7.1

Introduction

Armed with ideas from the previous chapter, we return back to the problem of MCMC inference for Markov jump processes. We consider two limitations of the uniformization- based approach described inchapter 3: the need to truncate the state-space of systems with unbounded event rates, and the inefficiency resulting from using a single bound- ing rate for systems with combinations of very stable and very unstable states. Our approach based on dependent thinning from chapter 6 is directly applicable to such systems, and allows us to construct MCMC algorithms with fewer thinned events than the uniformization-based sampler. In this chapter, we provide an alternate (but equiv- alent) description of the approach outlined inchapter 6; this is based on a construction of the MJP from a family of Poisson processes. Besides helping us understand our algorithm better, this can also lead to extensions which can improve the performance of the sampler. As an application, we will consider a model from queuing theory, the M/M/c/c queue.