Adaptive Markov Chain Monte Carlo Simulation

3 Adaptive Markov Chain Monte Carlo Simulation 2.5.3.1 Metropolis-Hastings Algorithm 

Let p ( 0) be a chosen PDF, called the proposal PDF, which is a PDF for f that depends on 0. The role of p will become clear shortly. For convenience in notation, define the following function  

The value of q can be computed readily for a given 0, while the same is not tme for piO V, C) because evaluating the normalizing constant /cq  

The candidate state is then accepted with probability min l, Q] and rejected with the remaining probability 1 — min l, Q] = max 0, 1 — Q.  

If the candidate state is accepted, it will be taken as the next state of the Markov chain, i.e., 0n+i = Otherwise, the current state is remained as the next state, i.e., 0n+i = On- In other words, the next state is given by  

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Figure 1. Illustration of the SS procedure a) Conditional level 0 Standard Monte Carlo simulation b) Conditional level 0 adaptive selection of yi c) Conditional level 1 Markov Chain Monte Carlo simulation d) Conditional level 1 adaptive selection ofy2-...

Unlike the coin example, in real applications, it is often not obvious how to decompose the rare event into a sequence in Eq. 1 and how to compute aU conditional probabilities in Eq. 2. In Subset Simulation, the sequencing of the rare event is done adaptively as the algorithm proceeds. This is achieved by employing Markov chain Monte Carlo, an advanced simulation technique, which constitutes the second -technical - idea behind SS. Finally, all conditional probabilities are automatically obtained as a by-product of the adaptive sequencing. 

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