Population Monte Carlo for Ion Channel Restoration

Texte intégral

(1)Population Monte Carlo for ion channel restoration Olivier Cappe´ ´ CNRS / ENST Departement TSI, Paris. Arnaud Guillin Jean–Michel Marin Christian P. Robert CEREMADE, Universite´ Paris Dauphine, Paris Summary. The analysis by Hodgson (1999) of the ion channel model was based on a reversible jump simulation of the hidden Gamma process. We reanalyse in this paper the same model using a fixed dimension model. The simulation of the Gamma process is based on an importance sampling scheme, using a hidden Markov representation of the ion channel model. We study through this model the degeneracy phenomenon associated with repeated calls to the iterated particle system, also called population Monte Carlo. Keywords: Forward-backward formula, hidden semi-Markov model, importance sampling, particle system, reversible jump, variable dimension model. 1. Introduction

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(235) Cappe´ et al. µ1. µ1 2.190. µ2. −0.02. 0.00. 0.02. 0.04. 0.06. 2.180. 0.035. 0.08. 2.10. 2.15. 2.20. 2.25. (µ0 − µ1) σ. σ. 2.170. −0.04. 0.025. 0. 0. 5. 5. 10. 15. 15. 25. 20. µ0. 0.045. 14. 1000. 2000. 0.60. 0.62. −2.95. −2.90. −2.85. −2.75. −2.70. 8. 0. 6. 60. 2000. 3000. 4000. 5000. 3000. 4000. 5000. 5000. λ1. 0.566. 12. 100. −2.80. λ1. 1000. 0.024. 0.578. −3.00. λ0. 1000. 2000. 3000. 4000. 5000. 0. 1000. 2000. 4. λ2. s1. 0.04. 0.05. 0.05. 0.10. 0.15. 0.20. 0.25. 0.30. s1. 10. 1.2 0.10. 1.0. 20. 2.0. 30. 3.0. s0. 1.4. 0.03. 1.0. 0.02. 0.12. 0.01. 1.6. 0.14. 0. 0. 2. 20. 0. 0.020. 0.58. 5000. 0.016. 0.56. 4000. 0.570. 0.54. 3000. σ. 0.574. 0. 0. 2. 10. 4. 20. 6. 30. 8. 0. 1000. 2000. 3.0. 3. 4. 5. 6. 7. 8. 9. s1 λ1. 3000. 4000. 5000. 3000. 4000. 5000. 60. 70. 80. 90. 100. 25. 30. 35. 40. 2000. 3000. 4000. 0. 200. 400. 600. 800. 74 70. 2.5. 110. 1000. 72. 3.0. 0.10 0.00. 0.02 0.00. 50. 0. s2. 3.5. 0.04. 0.20. s 0 λ0. 2. 78. 2.5. 76. 2.0. 4.5. 1.5. 4.0. 0. 0.0. 0. 1.0. 0. 1000. 2000. Fig. 9. Details of the MCMC sample for the dataset of Figure 8: (left) histograms of the components of the MCMC sample and (right) cumulative averages for the parameters of the models and evolution of the number of switches.. µ1. σ2. 0.00. 0.05. 0.60. mcmc. 0.54 0.52. 0.10. 2.10. 2.15. 2.20. particles. particles. λ0. λ1. 2.25. 0.55. 0.60. 0.65. particles. 1.5. mcmc. 1.0. 0.03 0.01. 0.5. 0.02. mcmc. 0.04. 2.0. 0.05. 2.5. 0.06. −0.05. 0.58 0.56. mcmc. 2.15 2.05. −0.05. 2.10. 0.00. mcmc. 2.20. 0.05. 0.62. 2.25. 0.64. 0.10. 0.66. µ0. 0.01. 0.02. 0.03 particles. 0.04. 0.05. 0.5. 1.0. 1.5. 2.0. 2.5. particles. Fig. 10. QQ-plot comparing the distribution of the particle system with the distribution of the MCMC £7¢7¢<¢ sample obtained after iterations started at the particles.. 1000.