Doob Finite Markov Bridge

Conditioning a discrete-time Markov chain (DTMC) on its endpoints transforms its transition probabilities. The Doob Finite Markov Bridge reweighs transitions to condition an entire Markov path on starting at state a and ending at state b after N steps.

Experiment with Actual Controls

You can interactively modify the variables and observe how the transition weights change over time to guarantee reaching the final state. Try it out at https://riano.app/module/doob-finite-markov-bridge.

1. Set Chain to "Lazy path".
2. Set Start to 0 and End to 2.
3. Set Length to 4 and Probe state to 1.
4. Observe the computed probabilities: The endpoint event probability is 0.21875, and both midpoint and first-step bridge probabilities to the probe state are ~0.571428.
5. Change the Chain to "Cycle". The endpoint event probability becomes 0 because state 2 cannot be reached from state 0 in exactly 4 steps. The bridge status becomes "Impossible endpoint event".
6. Change the Length to 2 with "Cycle" to see that the endpoint event probability becomes 1.0, and the bridge is defined.