Unbiased Inference for Complex Models

复杂模型的无偏推理

• 批准号: EP/N000188/1
• 负责人: Sebastian Vollmer
• 金额: $ 12.6万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FN000188%2F1
• 关键词: Unbiased; Inference; Complex; Models

Large scale mathematical models play an essential role in many applications throughout science. A prime example of an application of a complex continuous model is the research on climate change. The climate is usually modelled as a non-linear mathematical-physical system based on the coupling of a model of the atmosphere and an ocean circulation model. Both of these models involve a big set of state variables, such as the temperature, humidity, the wind, flow speed and pressure, to name but a few. The state of the system is modelled as the collection of these parameters for every grid cell. The evolution of these parameters arises as a discretisation of the underlying continuous model. In order to make inference about the climate, the model needs to be run at a fixed discretisation. The accuracy of the inference depends on the number of times the model can be simulated under a finite computational budget which thus influences the level of the discretisation. This trade-off results in a widening gap between the models that we can simulate and the ones for which we can make sound statistical inference. Bayesian methods are ubiquitous in statistical modelling and machine learning for the analysis of data. The aim is to model the posterior distribution which is in most cases only available indirectly as an appropriate limit of a sequence of probability measures. The most prominent technique to access the posterior distribution is via MCMC algorithms. MCMC algorithms are fully flexible and generally applicable. However, they require simulations of the full complex model on each step which limits the number of steps that are possible for a fixed computational budget, thus requiring huge computational budgets if we want to keep a high level of accuracy. For this reason Monte Carlo methods are often neglected, even though they are the only methods targeting the correct posterior.For a wide range of applications, for example in weather prediction, nuclear waste management and quantitative finance, the quantity to be inferred is often only given indirectly as an appropriate limit of distributions. The conventional approach to such an indirect representation requires a truncation. In most cases, the choice is to either run the MCMC only for a sufficiently long time, to fix a finite discretisation or to incorporate only a certain number of terms in the series expansion. However, all of these truncations lead to a systematic bias in the modelling of the target distribution and the exact accuracy of this bias is highly application-specific and very difficult to analyse. The main motivation of this proposal is to establish, extend and improve schemes for direct unbiased inference and thus to remove systematic bias in Bayesian inference for complex models. In particular, it aims at improving established inference methods by an incorporation of a new class of unbiased estimators and to develop new inference algorithms that yield unbiased estimators for Bayesian inference. This approach naturally allows the distribution of computations across different discretisations which results in great computational gain which is beneficial for a wide range of applications. The BBC has, for example, recently reported on the possibility to use the mobile phone records in Western Africa in order to predict the spreading of Ebola. Mathematical models have also become a factor for the strategic planning of the Metropolitan police in London to predict crime and to identify the areas of high probabilities of certain crimes in order to increase the presence of the police accordingly. Many more of these very recent examples in the press can be found undermining the importance of the use of accurate models around us. In the medium term, a sound statistical method, allowing an unbiased evaluation of complex models would therefore optimise a wide range of production processes in industry, politics and medicine to name just a few.

大规模的数学模型在整个科学的许多应用中扮演着重要的角色。应用复杂连续模型的一个最好的例子是气候变化研究。气候通常被模拟为一个基于大气模型和海洋环流模型耦合的非线性数学-物理系统。这两个模型都涉及一大组状态变量，如温度、湿度、风、流速和压力，仅举几例。系统的状态被建模为每个网格单元的这些参数的集合。这些参数的演变是作为基础连续模型的离散化出现的。为了对气候做出推断，模型需要以固定的离散度运行。推论的准确性取决于在有限的计算预算下模型可以被模拟的次数，从而影响离散化的水平。这种权衡导致我们可以模拟的模型和我们可以对其进行合理统计推断的模型之间的差距越来越大。贝叶斯方法在统计建模和机器学习中普遍用于数据分析。其目的是建立后验分布的模型，在大多数情况下，后验分布只能间接地作为概率度量序列的适当极限。获取后验分布的最重要的技术是通过MCMC算法。MCMC算法具有很强的灵活性和通用性。然而，它们需要在每一步上模拟完整的复杂模型，这限制了固定计算预算可能的步数，因此如果我们想要保持高水平的精度，就需要巨大的计算预算。因此，蒙特卡罗方法经常被忽视，尽管它们是唯一针对正确的后验的方法。对于广泛的应用，例如在天气预报、核废物管理和定量金融中，被推论的量通常只是间接地给出作为适当的分布极限。这种间接表示的传统方法需要截断。在大多数情况下，选择要么只运行MCMC足够长的时间，要么修复有限离散化，要么在级数展开中只包含一定数量的项。然而，所有这些截断都导致了目标分布建模中的系统偏差，这种偏差的准确精度是高度特定于应用的，非常难以分析。这一建议的主要动机是建立、扩展和改进直接无偏推理方案，从而消除复杂模型贝叶斯推理中的系统性偏差。特别是，它的目的是通过引入一类新的无偏估计来改进已有的推理方法，并开发新的推理算法，为贝叶斯推理产生无偏估计。这种方法自然允许将计算分布在不同的离散化上，从而产生巨大的计算收益，这对广泛的应用程序是有益的。例如，BBC最近报道了利用西非的手机记录来预测埃博拉传播的可能性。数学模型也已成为伦敦大都会警察战略规划的一个因素，以预测犯罪并确定某些犯罪的高概率地区，以便相应地增加警察的存在。更多的这些最近在媒体上的例子可以发现破坏了我们周围使用准确模型的重要性。因此，从中期来看，一种可靠的统计方法，允许对复杂模型进行公正的评估，将优化工业、政治和医药等领域的广泛生产流程。

项目成果

Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models

• DOI: 10.3150/16-bej911
• 发表时间: 2014-11
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: S. Agapiou;Gareth O. Roberts;Sebastian J. Vollmer

The Bouncy Particle Sampler: A Nonreversible Rejection-Free Markov Chain Monte Carlo Method

• DOI: 10.1080/01621459.2017.1294075
• 发表时间: 2018-01-01
• 期刊: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• 影响因子: 3.7
• 作者: Bouchard-Cote, Alexandre;Vollmer, Sebastian J.;Doucet, Arnaud
• 通讯作者: Doucet, Arnaud

Piecewise Deterministic Markov Processes for Scalable Monte Carlo on Restricted Domains

受限域上可扩展蒙特卡罗的分段确定性马尔可夫过程

• DOI: 10.48550/arxiv.1701.04244
• 发表时间: 2017
• 期刊: arXiv e-prints
• 作者: Bierkens Joris

Variance reduction for discretised diffusions via regression

通过回归减少离散扩散的方差

• DOI: 10.48550/arxiv.1510.03141
• 发表时间: 2015
• 期刊: arXiv e-prints
• 作者: Belomestny Denis

Multilevel path simulation for weak approximation schemes with application to Lévy-driven SDEs

弱近似方案的多级路径仿真及其在 Lévy 驱动的 SDE 中的应用

• DOI: 10.3150/15-bej764
• 发表时间: 2017
• 期刊: Bernoulli
• 影响因子: 1.5
• 作者: Belomestny D