关于寻找任意过程族的预报子
On Finding Predictors for Arbitrary Families of Processes
摘要：
问题是在以下设置中的序列预测。离散值观测序列$x_1,...,x_n,...$是根据某个未知的概率律（测度）$μ$生成的。在观察每个结果之后，需要给出下一个观察的条件概率。测度$μ$属于随机过程测度的任意但已知的类$C$。我们感兴趣的预测$ρ$的条件概率收敛（在某种意义上）的“真正的”$μ$ -条件概率，如果任何$μ\in C$被选中生成的序列。这项工作的贡献是在家庭$C$的特点，这种预测存在，并在提供一个具体的和简单的形式，寻找一个解决方案。我们表明，如果任何预测的作品，那么存在一个贝叶斯预测，其先验是离散的，这也工作。我们还发现了几个充分和必要条件的预测存在的，在拓扑特征的家庭$C$，以及在本地行为的措施，在$C$，这在某些情况下导致程序，用于构建这样的预测。应该强调的是，该框架是完全通用的：所考虑的随机过程不需要是独立同分布的，固定的，或属于任何参数或可数的家庭。
英文摘要：
The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $μ$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $μ$ belongs to an arbitrary but known class $C$ of stochastic process measures. We are interested in predictors $ρ$ whose conditional probabilities converge (in some sense) to the "true" $μ$-conditional probabilities if any $μ\in C$ is chosen to generate the sequence. The contribution of this work is in characterizing the families $C$ for which such predictors exist, and in providing a specific and simple form in which to look for a solution. We show that if any predictor works, then there exists a Bayesian predictor, whose prior is discrete, and which works too. We also find several sufficient and necessary conditions for the existence of a predictor, in terms of topological characterizations of the family $C$, as well as in terms of local behaviour of the measures in $C$, which in some cases lead to procedures for constructing such predictors. It should be emphasized that the framework is completely general: the stochastic processes considered are not required to be i.i.d., stationary, or to belong to any parametric or countable family.
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