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The Allen-Cahn equation is a differential equation used to model the phase separation of two, or more, alloys. This model may also be used to model cell motility, including chemotaxis and cell division. The numerical approximation, via a finite difference scheme, ultimately leads to a large system of linear equation. In this project, using numerical linear algebra techniques, we will develop a computational solver for the linear systems. We will then investigate the robustness of the proposed solver.

In conclusion, ABC represents a class of well-founded and powerful methods for Bayesian statistical inference. However, reliable application of ABC requires additional caution to be considered, due to the approximations and biases introduced at the different stages of the approach. In its current incarnation, the ABC toolkit as a whole is best suited for inference about parameters or predictive inferences about observables in the presence of a single or few candidate model(s). How to make ABC practically feasible for problems involving large sets of models and/or high-dimensional target parameter spaces is currently largely an open issue. Since the computation of the likelihood function is bypassed, it can be tempting to attack high-dimensional problems using ABC, but inevitably this comes bundled with new challenges that investigators need to be aware of at each step of their analyses.

Photo provided by Flickr

Photo provided by Flickr

As for all statistical methods, a number of assumptions and approximations are inherently required for the application of ABC-based methods to real modeling problems. For example, setting the tolerance parameter to zero ensures an exact result but typically makes computations prohibitively expensive. Thus, values of larger than zero are used in practice, which introduces a bias. Likewise, sufficient statistics are typically not available, and instead, other summary statistics are used, which introduces an additional bias due to the loss of information. Additional sources of bias—for example, in the context of model selection—may be more subtle , .

The first Approximate Bayesian computation (ABC)-related space and using it to approximate the likelihood by running several simulations for each grid point.

Automatic Sampler Discovery via Probabilistic Programming and Approximate Bayesian Computation Yura Perov and Frank Wood Department of Engineering Science, University.

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AB - Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data to summary statistics of the observed data. This thesis looks at two related methodological issues for ABC.

Firstly a method is proposed to construct appropriate summary statistics for

ABC in a semi-automatic manner. The aim is to produce summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that, in some sense, optimal summary statistics are the posterior means of the parameters. While these cannot be calculated analytically, an extra stage of simulation is used to estimate how the posterior means vary as a function of the data, and these estimates are then used as summary statistics within ABC. Empirical results show that this is a robust method for choosing summary statistics, that can result in substantially more accurate ABC analyses than previous approaches in the literature.

Secondly, ABC inference for multiple independent data sets is considered. If

there are many such data sets, it is hard to choose summary statistics which capture the available information and are appropriate for general ABC methods. An alternative sequential ABC approach is proposed in which simulated and observed data are compared for each data set and combined to give overall results. Several algorithms are proposed and their theoretical properties studied, showing that exploiting ideas from the semi-automatic ABC theory produces consistent parameter estimation. Implementation details are discussed, with several simulation examples illustrating these and application to substantive inference problems.

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In the mid 90s, the fields of analog and digital computing as separate approaches to modelintelligence, have begun to merge using the idea of Bayesian inference: One can generalize the logic of digital computationto a probabilistic calculus, embodied in a so-called graphical model.

N2 - Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data to summary statistics of the observed data. This thesis looks at two related methodological issues for ABC.

Firstly a method is proposed to construct appropriate summary statistics for

ABC in a semi-automatic manner. The aim is to produce summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that, in some sense, optimal summary statistics are the posterior means of the parameters. While these cannot be calculated analytically, an extra stage of simulation is used to estimate how the posterior means vary as a function of the data, and these estimates are then used as summary statistics within ABC. Empirical results show that this is a robust method for choosing summary statistics, that can result in substantially more accurate ABC analyses than previous approaches in the literature.

Secondly, ABC inference for multiple independent data sets is considered. If

there are many such data sets, it is hard to choose summary statistics which capture the available information and are appropriate for general ABC methods. An alternative sequential ABC approach is proposed in which simulated and observed data are compared for each data set and combined to give overall results. Several algorithms are proposed and their theoretical properties studied, showing that exploiting ideas from the semi-automatic ABC theory produces consistent parameter estimation. Implementation details are discussed, with several simulation examples illustrating these and application to substantive inference problems.

The Bayesian paradigmhas greatly helped to integrate different schools of thought in particular in the field of artificial intelligence andmachine learning but also provides a computational paradigm for neuroscience.

My research is dedicated to the design of efficient and novel computational methods for Bayesian inference and stochasticcontrol theory using ideas and methods from statistical physics.

Bayesian models are probability models and the typical computation, whether in the context of a complex data analysisproblem or in a stochastic neural network, is to compute an expectation value, which is referred to as Bayesian inference.

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