SIMULATION FROM ENDPOINT-CONDITIONED, CONTINUOUS-TIME MARKOV CHAINS ON A FINITE STATE SPACE, WITH APPLICATIONS TO...

Area of Science:

• Computational Statistics
• Stochastic Processes
• Applied Probability

Background:

• Serially-sampled data analysis often assumes discrete samples from latent continuous-time stochastic processes.
• Continuous-time Markov chains (CTMCs) are widely used generative models across diverse fields like finance, genetics, and genomics.
• A key challenge is simulating CTMC sample paths conditioned on discretely observed data.

Purpose of the Study:

• To provide a general solution for simulating sample paths of a CTMC on a discrete, finite state space.
• To generate complete sample paths, including intermediate states and transition times, for CTMCs with known start and end states over a defined interval T.
• To analytically compare the complexity and efficiency of predominant sampling methods.

Main Methods:

• Unified discussion of three predominant sampling approaches: modified rejection sampling, direct sampling, and uniformization.
• Analytical derivation of complexity and efficiency for each method based on the CTMC's instantaneous transition rate matrix (Q), start/end states, and sampling time (T).
• Comparison of the three methods across various model specifications.

Main Results:

• Demonstrated that no single sampling method (modified rejection, direct sampling, uniformization) universally dominates across all CTMC specifications.
• Provided explicit analytical results and proofs identifying the most efficient method for any given Q, T, and endpoints.
• Highlighted the practical implications and potential pitfalls of selecting an inefficient sampler through three distinct applications.

Conclusions:

• The choice of simulation method for CTMCs with discrete observations is critical and depends heavily on specific model parameters (Q, T, endpoints).
• Understanding the analytical trade-offs between different sampling techniques is essential for accurate and efficient data analysis in fields utilizing CTMCs.
• This work offers guidance for selecting the optimal CTMC sampling strategy, preventing potential errors in complex data modeling.