Abstract
The Double Chain Markov Model (DCMM) is used to model an observable process \(Y = \{Y_{t}\}_{t=1}^{T}\) as a Markov chain with transition matrix, \(P_{x_{t}}\), dependent on the value of an unobservable (hidden) Markov chain \(\{X_{t}\}_{t=1}^{T}\). We present and justify an efficient algorithm for sampling from the posterior distribution associated with the DCMM, when the observable process Y consists of independent vectors of (possibly) different lengths. Convergence of the Gibbs sampler, used to simulate the posterior density, is improved by adding a random permutation step. Simulation studies are included to illustrate the method. The problem that motivated our model is presented at the end. It is an application to real data, consisting of the credit rating dynamics of a portfolio of financial companies where the (unobserved) hidden process is the state of the broader economy.
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Appendix: Proof of Theorem 1
Appendix: Proof of Theorem 1
Proof
The joint mass function of the hidden states, given the parameters and the observed data as vectors at each time point is
The “typical term” can be written as
by Lemma 1. Now, \(\frac{\mbox{P}(\boldsymbol {x}^{t+2}, \boldsymbol {y}^{t+1} \vert \boldsymbol {y}_{,t},x_{t}, x_{t+1}, \theta)}{\mbox{P}(\boldsymbol {x}^{t+1},\boldsymbol {y}^{t+1}\vert \boldsymbol {y}_{,t}, \theta)}\) depends only on x t+1, and is therefore independent of x t and thus can become the normalizing constant. That is,
We continue, in more detail, to show
By the law of total probability and Lemma 1, we have
and consequently from (10) and (11),
This is initialized at t=u 0 by setting \(\mbox{P}(x_{0} \vert \boldsymbol {y}_{,M}, \theta) = \mbox{P}(x_{u_{0}} \vert r)\) to be the same as the Dirichlet prior on D(α 01,…,α 0a ). □
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Fitzpatrick, M., Marchev, D. Efficient Bayesian estimation of the multivariate Double Chain Markov Model. Stat Comput 23, 467–480 (2013). https://doi.org/10.1007/s11222-012-9323-y
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DOI: https://doi.org/10.1007/s11222-012-9323-y