Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userApproximate Bayesian computation (ABC) gives exact results under the assumption of model error
238
0
0.0
(
0
)
Added by
Richard Wilkinson
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
Authors
Richard D. Wilkinson
Ask ChatGPT about the research
No Arabic abstract
Approximate Bayesian computation (ABC) or likelihood-free inference algorithms are used to find approximations to posterior distributions without making explicit use of the likelihood function, depending instead on simulation of sample data sets from the model. In this paper we show that under the assumption of the existence of a uniform additive model error term, ABC algorithms give exact results when sufficient summaries are used. This interpretation allows the approximation made in many previous application papers to be understood, and should guide the choice of metric and tolerance in future work. ABC algorithms can be generalized by replacing the 0-1 cut-off with an acceptance probability that varies with the distance of the simulated data from the observed data. The acceptance density gives the distribution of the error term, enabling the uniform error usually used to be replaced by a general distribution. This generalization can also be applied to approximate Markov chain Monte Carlo algorithms. In light of this work, ABC algorithms can be seen as calibration techniques for implicit stochastic models, inferring parameter values in light of the computer model, data, prior beliefs about the parameter values, and any measurement or model errors.
rate research
Read More
Approximate Bayesian computation (ABC) is computationally intensive for complex model simulators. To exploit expensive simulations, data-resampling via bootstrapping can be employed to obtain many artificial datasets at little cost. However, when using this approach within ABC, the posterior variance is inflated, thus resulting in biased posterior inference. Here we use stratified Monte Carlo to considerably reduce the bias induced by data resampling. We also show empirically that it is possible to obtain reliable inference using a larger than usual ABC threshold. Finally, we show that with stratified Monte Carlo we obtain a less variable ABC likelihood. Ultimately we show how our approach improves the computational efficiency of the ABC samplers. We construct several ABC samplers employing our methodology, such as rejection and importance ABC samplers, and ABC-MCMC samplers. We consider simulation studies for static (Gaussian, g-and-k distribution, Ising model, astronomical model) and dynamic models (Lotka-Volterra). We compare against state-of-art sequential Monte Carlo ABC samplers, synthetic likelihoods, and likelihood-free Bayesian optimization. For a computationally expensive Lotka-Volterra case study, we found that our strategy leads to a more than 10-fold computational saving, compared to a sampler that does not use our novel approach.
Approximate Bayesian computation methods are useful for generative models with intractable likelihoods. These methods are however sensitive to the dimension of the parameter space, requiring exponentially increasing resources as this dimension grows. To tackle this difficulty, we explore a Gibbs version of the ABC approach that runs component-wise approximate Bayesian computation steps aimed at the corresponding conditional posterior distributions, and based on summary statistics of reduced dimensions. While lacking the standard justifications for the Gibbs sampler, the resulting Markov chain is shown to converge in distribution under some partial independence conditions. The associated stationary distribution can further be shown to be close to the true posterior distribution and some hierarchic
We study the class of state-space models and perform maximum likelihood estimation for the model parameters. We consider a stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood function with the novelty of using approximate Bayesian computation (ABC) within SAEM. The task is to provide each iteration of SAEM with a filtered state of the system, and this is achieved using an ABC sampler for the hidden state, based on sequential Monte Carlo (SMC) methodology. It is shown that the resulting SAEM-ABC algorithm can be calibrated to return accurate inference, and in some situations it can outperform a version of SAEM incorporating the bootstrap filter. Two simulation studies are presented, first a nonlinear Gaussian state-space model then a state-space model having dynamics expressed by a stochastic differential equation. Comparisons with iterated filtering for maximum likelihood inference, and Gibbs sampling and particle marginal methods for Bayesian inference are presented.
In this paper, we show how a complete and exact Bayesian analysis of a parametric mixture model is possible in some cases when components of the mixture are taken from exponential families and when conjugate priors are used. This restricted set-up allows us to show the relevance of the Bayesian approach as well as to exhibit the limitations of a complete analysis, namely that it is impossible to conduct this analysis when the sample size is too large, when the data are not from an exponential family, or when priors that are more complex than conjugate priors are used.
In this tutorial we schematically illustrate four algorithms: (1) ABC rejection for parameter estimation (2) ABC SMC for parameter estimation (3) ABC rejection for model selection on the joint space (4) ABC SMC for model selection on the joint space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
