A efficient approximation to the parameter posterior, a particle

A main goal of this thesis was to design and identify  an appropriate mathematical  model of stochastic biological system.  A  CTMC model  show its ability to provide a well description of biological system and its behaviour. We have demonstrated that how designed CTMC is used to describe two biological case studies: LV and repressilator.  In repressilator model, the simulated data produced from CTMC model is consistent with original observed data and show similar behaviour.

  The main contribution of this thesis made on the part of  Bayesian inference method. The problem of intractable likelihood prohibit the direct Bayesian inference, alternatively,  ABC and PMMH were employed as an approximated methods. A variety of an existing algorithms were reviewed and critically  discussed.   The most challenging problem can be faced when employing the Bayesian inference using the proposed approaches is how to design and tuning algorithm to achieve an efficient estimation, also to balance an accuracy and computational cost. There are some aspects of the model specification that mitigate as much as possible thenumber of modes in the likelihood The  effective of proposed scheme  demonstrated on  two biological systems. The designing ABC  algorithm based on rejection and generalised  rejection technique proof it ability to infer uncertainties in LV model.  Furthermore, ABC algorithm within sequential version demonstrated on LV and provide a better approximation to the model parameter posterior. In order to assess the performance of ABC algorithm, a PMMH is considered as the second  inference, yet, designing algorithm for PMMH more  complex and difficult  than ABC and require a special tuning.

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    A resulting  parameter inference of LV model obtained from employing ABC and PMMH algorithms given an experimental data that generated from artificially CTMC   model are compared.    An application of those proposed approaches also demonstrated  on more complex case study repressilator compared to the simple LV case study. In our application, we illustrated the important algorithmic parameter that can influence the efficiency of performance of algorithm.  In attempt to enable a best performance of PMMH algorithm, a nontrivial potential practical  issues with its possible solution were investigated. This particle issues associated with number of particles,  mixing and  stationary distribution are  mitigated by monitoring the convergence of several indecent Markov chains and  tuning a proposal distribution to be adaptive which   adjust it through the iteration to have a  shape that similar to to the current posterior approximation.    An ABC SMC was applied on repressilator model given both simulated and observed data, to obtain  more efficient approximation to the  parameter posterior, a particle diversity maintained by monitoring the the ESS.     The ABC SMC suffer from the exact problem, where the exact approximation obtained if the tolerance is equal to zero which in practice impossible.

To obtain a sufficient approximation we aim to obtain a smallest tolerance value which often require a very large number of simulation. In attempt to save computational time,  we carried out a pilot runs to draw an idea about the possible low tolerance can be achieved. This can able  us to stop algorithm.                              Despite the stochastic nature of repressilator model,  we notice  that different independent runs of the ABC SMC  algorithm with exact  algorithmic setting lead to similar  parameter approximation. Hence, we demonstrate  that ABC SMC algorithm converge to its stationary distributeion.

   The ABC SMC method seems   to be   more appropriate inference method which offer a reasonable trade-off between accuracy of estimation  and computational time. In addition, we have demonstrated    the applicability of the ABC SMC scheme   to  different cases with both simulate and observed  dataset without  requiring a special   tuning.