Novel simulation-based statistical inference with applications to epidemic models

参数模型在统计建模中起着关键作用。参数模型假设有一个基础模型产生我们观察到的数据，这些数据取决于某些参数和随机量。例如，对于疾病的传播，模型参数决定了疾病的传染性，但谁被感染将取决于模型设置和随机性。在实践中，我们很少知道模型的参数，统计学的一个关键要素是获得参数的良好估计。在贝叶斯统计中，参数具有量化模型参数中的不确定性的后验分布。通过研究后验分布，我们可以计算我们感兴趣的参数的任何汇总统计量。然而，贝叶斯统计的一个主要缺点是后验分布很少以我们可以容易使用的形式提供。有许多方法可以从后验分布中获得样本，其中最常见的是MCMC。最近一系列的实际问题，在统计遗传学已经确定MCMC不能使用或它是特别困难的。一个解决方案已提供的ABC（近似贝叶斯计算）算法的形式。ABC算法使用来自模型的模拟，通过适当的机制（通常是先验分布）选择参数，以估计参数。(The先验分布表示我们对模型参数的先验信念。ABC算法形式化了我们从具有不同参数的模型进行模拟的想法，接受那些导致模拟数据与观测数据紧密一致的参数值。MCMC和ABC算法都是迭代算法，在每次迭代时从后验分布产生单个参数。最近，研究人员介绍了一种新的ABC算法，该算法在每次迭代时从后验分布中产生一组参数。这种新的ABC算法被证明是相当有效的比标准的ABC算法，并已直接应用于分析传染病的传播的流行病模型。拟议研究的目的是双重的。首先，开发更有效的MCMC和ABC算法，从后验分布中获得值集。这将导致更强大的参数估计与较低的不确定性参数估计的基础上，一个给定的大小的样本。其次，将新方法应用于一系列流行病模型，以更好地理解传染病的传播。特别是制定易于非专家使用和解释的程序。

Parametric models play a key role in statistical modelling. Parametric models assume that there is an underlying model giving rise to the data we observe with the data dependent upon certain parameters and random quantities. For example, for the spread of a disease, the model parameters dictate the infectiveness of the disease but who becomes infected will depend upon the model setup and randomness. In practice, we rarely know the parameters of the model and a key element of statistics is to obtain good estimates of the parameters. In Bayesian statistics the parameters have a posterior distribution which quantifies the uncertainty in the parameters of the model. By studying the posterior distribution we can calculate any summary statistics of the parameters we are interested in. However, a major drawback of Bayesian statistics is that the posterior distribution is rarely available in a form which we can easily use. There are a number of approaches for obtaining samples from the posterior distribution, the most common of which is MCMC. Recently a range of practical problems in statistical genetics have been identified where MCMC can either not be used or it is particularly difficult to do so. A solution has been provided in the form of the ABC (approximate Bayesian computation) algorithm. The ABC algorithm uses simulation from the model with parameters chosen via an appropriate mechanism, often the prior distribution, to estimate the parameters. (The prior distribution represents our prior beliefs about the model parameters.) The ABC algorithm formalises the idea that we simulate from the model with different parameters, accepting those parameter values which lead to simulated data in close agreement with the observed data.Both the MCMC and the ABC algorithms are iterative algorithms producing a single parameter from the posterior distribution at each iteration. Recently the investigator has introduced a new ABC algorithm which produces a set of parameters from the posterior distribution at each iteration. This new ABC algorithm is shown to be considerably more efficient than standard ABC algorithms, and has straightforwardly been applied to the analysis of epidemic models for the spread of infectious diseases. The aim of the proposed research is two-fold. Firstly, to develop more efficient MCMC and ABC algorithms which obtain sets of values from the posterior distribution. This should lead to more robust parameter estimation with lower uncertainty in parameter estimates based upon a sample of a given size. Secondly, to apply the new methods to a range of epidemic models to gain a better understanding of the spread of infectious diseases. In particular, the development of procedures which are easy to use and interpret by non-experts.