In the last decade scientists have developed an enormous set of experimental methods to investigate biological systems. Combining their effort with mathematics as well as physics lead to several tools for modeling dynamical systems. However, every model contains a certain number of parameters that can not be measured with the existing experimental tools. The knowledge about the value of these parameters in vivo is very important for several aspects, e. g. dynamical analysis, prediction algorithms or drug design. Parameter inference (also known as parameter estimation) methods try to obtain these values by using experimental data.
Parameter inference methods combine experimental data with the knowledge about the underlying structure of a dynamical system to be investigated. Unknown model parameter (here k1, k2, k3 and k4) can be estimated.
Several parameter estimation methods have been introduced, e.g. least squares parameter estimation. However, for large and complex systems parameter estimation is not trivial and the classical methods might fail. Bayesian frameworks for parameter estimation provides more reliable results for these systems. Especially for systems where the likelihood can not be computed approximate bayesian computation (ABC) provides a way to approximate the parameter distribution. The most simplest ABC method ist the ABC rejection sampler (Pritchard et. al., 1999). Computationally less expensive are ABC MCMC (Markov Chain Monte Carlo) and ABC SMC (Sequential Monte Carlo).
The cartoon describes the principal of the ABC SMC algorithm. The probability of the particles to represent the data best is updated during the algorithm via intermediate distributions. This leads to a final distribution, i.e. the approximation of the posterior distribution. This sequential Monte Carlo algorithm uses the principal of particle filtering.
ABC SMC is based on the simplest ABC rejection sampler and combines it with a sequential Monte Carlo approach. Instead of trying to obtain a single minimal tolerance (epsilon) it uses a decreasing tolerance schedule. Parameter values, so called “particles”, are first sampled from a defined prior distribution. The system will be simulated using the sampled particles to obtain the trajectories. Then the distance between these trajectories and a given data set is computed using a specific distance function. If the distance is smaller than the first tolerance the particle will be accepted and it will be included in the first intermediate distribution. If the distance is larger than the first tolerance the particle will be rejected. This process is repeated until a specific number of particles is accepted. These particles create the first population. In the next step new particles will be sampled from this population. However, after sampling a particle it will be perturbed by a perturbation kernel. The system will then be simulated with the perturbed particle and it will be computed whether this particle is accepted or rejected regarding the second tolerance. This process will continue for several populations until the final population, i.e. the population with the smallest possible tolerance, is reached. Therefore the algorithm describe a particle filtering procedure. The final population is an approximation of the posterior distribution.
The advantages of ABC SMC compared to point estimates lie in the fact that a whole parameter distribution is obtained. This distribution can be used to investigate the sensitivity of a system. Furthermore the ABC SMC is computationally less expensive compared to other Approximate Bayesian Computation methods.