Sequential Monte Carlo: Towards Degeneracy-Free Methods

不完全观察到的进化系统出现在整个人类世界。天气预报、为股票价格建模、转录音乐或自动翻译人类语音只是少数几种情况，在这种情况下，我们可以获得的只是对随时间演变的系统的不完美观测，而我们感兴趣的是底层系统：考虑到卫星观测和稀疏的局部测量，我们希望准确地描述现在的天气并预测未来的天气；给出离散时间的音高测量，我们希望计算机能够对当时所说的话产生有意义的描述。令人惊讶的是，使用一个称为状态空间模型(或隐马尔可夫模型)的通用框架来模拟大量这样的问题是可能的。根据一系列观测推断未观测过程的可能值，因为这些观测在原则上是相当简单的，但它需要评估积分，这些积分不能用分析数学解决，而且太复杂，不能用简单的数值方法准确处理。已经开发了基于模拟的技术来解决这些问题，并且现在是在迄今收到的所有观测结果的情况下估计未观测过程的当前状态的最强大的工具集合。近年来，许多人致力于设计算法，以类似的方式有效地描述从观测序列开始到当前时间的未观察过程的可能路径。这个问题要难得多，因为我们收到的每一次观测都会告诉我们更多关于这个过程可能的历史的信息，而且以一种有效的方式不断更新这个越来越长的位置列表远非简单。这里提出的方法将试图将基于模拟的统计技术扩展到一个新的方向，特别适合描述未观察到的过程的整个路径，而不仅仅是它的终端值。将研究基于同一前提的两种不同策略--在相同的计算成本下，有时几个较小的模拟在特定意义上可以胜过单个较大的模拟。开发的技术将从理论和经验两方面进行研究。除了开发和分析新的计算技术外，该项目将提供软件库，简化这些方法在实际问题中的使用(希望在一定程度上，在特定应用领域的专家能够将这些技术直接应用于他们自己的问题)。如果：1/它导致了在状态空间模型中执行推理的新方法。2/这些方法可以通过较少的特定于应用的调整来实现，而现有方法需要更少的特定于应用的调整，或者这些方法提供了更有效的计算资源的使用。3/这些方法足够强大，以允许使用比当前实际更复杂的模型。4/这些方法是在可能有效使用这些技术的许多领域中，至少有一些领域的从业者采用了这些技术。长期好处可能包括对金融体系中的风险进行更现实的评估，对气象现象和改进的技术产品进行更可靠的跟踪和预测，只要有必要动态纳入现有的测量所产生的知识。在对未完全观察到的基本过程的完整路径感兴趣的情况下，将有特别的优势，但即使情况并非如此，仍有改进的余地。

Imperfectly observed evolving systems arise throughout the human world. Weather forecasting, modelling stock prices, transcribing music or interpreting human speech automatically are just a few of the situations in which imperfect observations of a system which evolves in time are all that is available whilst the underlying system is the thing in which we are interested: Given satellite observations and sparse localised measurements, we'd like to accurately characterise the weather now and predict future weather; given measurements of pitch at discrete times we'd like a computer to be able to produce a meaningful description of what was being said at the time.Surprisingly, it's possible to model a great number of these problems using a common framework, known as a state space model (or hidden Markov model). Inferring the likely value of the unobserved process based upon a sequence of observations, as those observations become available is in principle reasonably straightforward but it requires the evaluation of integrals which cannot be solved by analytical mathematics and which are too complex to deal with accurately via simple numerical methods. Simulation-based techniques have been developed to address these problems and are now the most powerful collection of tools for estimating the current state of the unobserved process given all of the observations received so far. Much effort has been dedicated in recent years to designing algorithms to efficiently describe the likely path of the unobserved process from the beginning of the observation sequence up to the current time in a similar way. This problem is much harder as each observation we receive tells us a little more about the likely history of the process and continually updating this ever-longer list of locations in an efficient way is far from simple.The methods proposed here will attempt to extend simulation-based statistical techniques in a new direction which is particularly well suited to characterisation of the whole path of the unobserved process and not just its terminal value. Two different strategies based around the same premise - that sometimes several smaller simulations can in a particular sense outperform a single larger simulation for the same computational cost - will be investigated. The techniques developed will be investigated both theoretically and empirically.In addition to developing and analysing new computational techniques, the project will provide software libraries which simplify the use of these methods in real problems (hopefully to the extent that scientists who are expert in particular application domains will be able to apply the techniques directly to their own problems).The research could be considered successful if:1/ It leads to new methods for performing inference in state space models.2/ These methods can be implemented with less application-specific tuning that existing methods require or these methods provide more efficient use of computational resources.3/ These methods are sufficiently powerful to allow the use of more complex models than are currently practical.4/ The methods are adopted by practitioners in at least some of the many areas in which these techniques might be usefully employed.The long term benefits could include more realistic assessment of risk in financial systems, more reliable tracking and prediction of meteorological phenomena and improved technological products wherever there is a need to dynamically incorporate knowledge arising from measurements as they become available. There will be particular advantages in settings in which the full path of the imperfectly observed underlying process is of interest but there is scope for improvement even when this is not the case.