Pooling INference and COmbining Distributions Exactly: A Bayesian approach (PINCODE)

虽然基于可能性的统计，特别是贝叶斯范式在许多情况下代表了理论基础和不确定性资格的黄金标准，但实现贝叶斯方法通常具有挑战性，特别是在数据存储在一组不相连的服务器上且无法组合的情况下。这种情况正变得越来越普遍，例如在大数据环境中，将所有数据存储在单个服务器上在技术上是不可行的，或者在隐私限制排除共享单个数据分片的情况下。一个特别激励我们工作的应用涉及对不同欧洲国家存在的罕见疾病的推断，这些国家由于保密原因不能共享数据，但希望根据所有国家的整个数据集进行推断。我们可以合理地假设，我们可以根据任何一台单独服务器的数据获得后验分布，并且可以很容易地获得该分布（我们称之为次后验分布）中的样本，例如通过某种MCMC方法。这样，问题就简化为从密度的乘积中生成样本，我们可以从密度的乘积中单独采样。这个项目将开发一种解决这个问题的新方法。与现有的技术不同，它不是基于近似或渐近证明。在Dai， Pollock和Roberts（2019年，2021年）的两篇概念证明论文中，我们开发了两种密切相关的方法：蒙特卡罗融合（MCF）和贝叶斯融合（BF）。这些方法在精度和对子后验不一致的鲁棒性方面都有很好的表现。然而，这些方法不能扩展到非常大的问题，并且仅限于所有子后验描述相同数据参数集上的分布的设置。此外，MCF和BF都不能直接应用于受隐私约束的数据。该项目的目标是开发一种健壮的、可扩展的、可访问的融合方法，适用于上述各种上下文中的分布式推理。在隐私方面，我们将结合使用同态秘密共享（HSS），这是一种简单的多方信息理论上安全的加密技术，可用于安全地执行来自各方的摘要的算术组合，每个人都不希望向其他方透露他们的摘要。虽然我们不能直接使用HSS来解决机密约束下的融合问题（即所谓的混淆问题），但我们打算在MCF和BF算法中以新颖的方式使用该技术来解决问题。PINCODE将首先基于合并马尔可夫过程的随机模拟开发融合方法的一般框架，该过程的共同合并值来自组合后验分布。大量的工作将投入到这些结构的计算效率上，着眼于优化数据大小的可扩展性、分布式服务器的数量、参数集的维度以及对次后验异质性的鲁棒性。我们还将考虑这些算法的原则近似，并为这些方法提供可证明的准确性保证。我们考虑的方法将广泛适用。其中一个目标应用将涉及与FAIRVASC计划（地平线2020项目，旨在借鉴不同欧洲国家罕见疾病分布式数据集之间的优势）的密切合作。进一步的方向将涉及融合方法学的结合，以提供精确的ABC方法。在我们的项目中，将非常强调软件开发的广泛和有效的传播。

While Likelihood-based Statistics and in particular the Bayesian paradigm represent the gold standards for theoretical underpinnings and uncertainty qualification in many contexts, implementing Bayesian methods can often be challenging, particularly in contexts where data is stored on a collection of disjoint servers and cannot be combined. This situation is becoming increasing common, for instance in the big data context where storing all data on a single server is not technical feasible, or where privacy constraints preclude the sharing of individual shards of data. One application which particularly motivates our work involves inference for rare diseases present in different European countries which cannot share data for confidentiality reasons, but where there is a desire to carry out inference based on the entire data set across all countries.It is reasonable to assume that we can obtain the posterior distribution based just on the data from any one individual server, and that samples from this distribution (which we term a sub-posterior distribution) can be readily obtained, for example through some kind of MCMC approach. Thus the problem reduces to that of generating samples from the product of densities from which we can individually sample. This project will develop a novel approach to this problem. Unlike existing techniques, it is not based on approximation or asymptotic justification. In two proof of concept papers, Dai, Pollock and Roberts (2019, 2021) we have developed two closely related approaches: Monte Carlo Fusion (MCF) and Bayesian Fusion (BF). These methods have many promising properties in terms of accuracy and robustness to inconsistency between sub posteriors. However these methods are not scalable to very large problems and are restricted to the setting where all sub-posteriors describe distributions on the same data parameter sets. In addition, neither MCF nor BF can be applied directly to data subject to privacy constraints. The aims of this project will be to develop a robust scalable and accessible fusion methodology suitable for distributed inference in the variety of contexts described above.Within the privacy context, we shall incorporate the use of homomorphic secret sharing (HSS), a simple multi-party information theoretically secure encryption technique which can be used to securely carry out arithmetic combinations of summaries from multiple parties which each individually do not wish to reveal their summary to the other parties. Whilst we cannot use HSS directly to solve the fusion problem under confidentiality constraints (the so called ConFusion problem), we intend to use the technique in novel ways within the MCF and BF algorithms to solve the problem.PINCODE will first develop a general framework for fusion methods based on the stochastic simulation of coalescing Markov processes with the property that their common coalesced value comes from the combined posterior distribution. Considerable effort will go into the computational efficiency of these constructions with an eye to optimising scalability in data size, the number of distributed servers, dimensionality of the parameter set as well as robustness to sub-posterior heterogeneity. We shall also consider principled approximations of these algorithms and provide provable accuracy guarantees for these methods. The methodology we consider will be widely applicable. One targeted application will involve close collaboration with the FAIRVASC initiative (a Horizon 2020 project to borrow strength between distributed data sets for rare diseases in different European countries). A further direction will involve the incorporation of fusion methodology to provide exact ABC methods. There will be a strong emphasis within our project on software development for wide-ranging and effective dissemination.