In Bayesian statistics, the marginal likelihood, also known as the evidence,
is used to evaluate model fit as it quantifies the joint probability of the
data under the prior. In contrast, non-Bayesian models are typically compared
using cross-validation on held-out data, either through $k$-fold partitioning
or leave-$p$-out subsampling. We show that the marginal likelihood is formally
equivalent to exhaustive leave-$p$-out cross-validation averaged over all
values of $p$ and all held-out test sets when using the log posterior
predictive probability as the scoring rule. Moreover, the log posterior
predictive is the only coherent scoring rule under data exchangeability. This
offers new insight into the marginal likelihood and cross-validation and
highlights the potential sensitivity of the marginal likelihood to the choice
of the prior. We suggest an alternative approach using cumulative
cross-validation following a preparatory training phase. Our work has
connections to prequential analysis and intrinsic Bayes factors but is
motivated through a different course.

本文针对 Bayesian 统计推断中模型拟合的评估方法，提出了一种新思路，即把边缘似然与采用信息熵对数作为评分规则的对于所有大小为 p 的测试数据集迭代得出的穷举式的 leave-p-out 交叉验证等效形式，并探讨了边缘似然对先验选择敏感性的问题。同时，作者提出了累积交叉验证的替代方案。

边际似然和交叉验证

On the marginal likelihood and cross-validation

Max-stable processes are increasingly widely used for modelling complex
extreme events, but existing fitting methods are computationally demanding,
limiting applications to a few dozen variables. $r$-Pareto processes are
mathematically simpler and have the potential advantage of incorporating all
relevant extreme events, by generalizing the notion of a univariate exceedance.
In this paper we investigate score matching for performing high-dimensional
peaks over threshold inference, focusing on extreme value processes associated
to log-Gaussian random functions and discuss the behaviour of the proposed
estimators for regularly-varying distributions with normalized marginals. Their
performance is assessed on grids with several hundred locations, simulating
from both the true model and from its domain of attraction. We illustrate the
potential and flexibility of our methods by modelling extreme rainfall on a
grid with $3600$ locations, based on risks for exceedances over local quantiles
and for large spatially accumulated rainfall, and briefly discuss diagnostics
of model fit. The differences between the two fitted models highlight the
importance of the choice of risk and its impact on the dependence structure.

本文研究了高维门限推断的得分匹配方法，以及基于局部分位数的风险和大规模空间积累降水的极值降雨建模方法，重点讨论了两种模型拟合方法之间的差异和风险选择的影响。

高维门限内峰值推断

High-dimensional peaks-over-threshold inference

Most generative models for clustering implicitly assume that the number of
data points in each cluster grows linearly with the total number of data
points. Finite mixture models, Dirichlet process mixture models, and
Pitman--Yor process mixture models make this assumption, as do all other
infinitely exchangeable clustering models. However, for some tasks, this
assumption is undesirable. For example, when performing entity resolution, the
size of each cluster is often unrelated to the size of the data set.
Consequently, each cluster contains a negligible fraction of the total number
of data points. Such tasks therefore require models that yield clusters whose
sizes grow sublinearly with the size of the data set. We address this
requirement by defining the \emph{microclustering property} and introducing a
new model that exhibits this property. We compare this model to several
commonly used clustering models by checking model fit using real and simulated
data sets.

本研究提出一种具有微聚类特性的新模型，可应用于需要聚类大小与数据集大小呈次线性增长的任务，通过比对实验数据和模拟数据的适配情况，验证该模型的有效性。