As machine learning systems become increasingly integrated into real-world
applications, accurately representing uncertainty is crucial for enhancing
their safety, robustness, and reliability. Training neural networks to fit
high-dimensional probability distributions via maximum likelihood has become an
effective method for uncertainty quantification. However, such models often
exhibit poor calibration, leading to overconfident predictions. Traditional
metrics like Expected Calibration Error (ECE) and Negative Log Likelihood (NLL)
have limitations, including biases and parametric assumptions. This paper
proposes a new approach using conditional kernel mean embeddings to measure
calibration discrepancies without these biases and assumptions. Preliminary
experiments on synthetic data demonstrate the method's potential, with future
work planned for more complex applications.

使用条件核平均嵌入测量标定差异，以提高机器学习模型的不确定性量化，并消除偏差和参数假设。初步实验基于合成数据展示了该方法的潜力，并计划用于更复杂的应用。

关于测量离散概率神经网络的校准

On Measuring Calibration of Discrete Probabilistic Neural Networks

Conditional kernel mean embeddings form an attractive nonparametric framework
for representing conditional means of functions, describing the observation
processes for many complex models. However, the recovery of the original
underlying function of interest whose conditional mean was observed is a
challenging inference task. We formalize deconditional kernel mean embeddings
as a solution to this inverse problem, and show that it can be naturally viewed
as a nonparametric Bayes' rule. Critically, we introduce the notion of task
transformed Gaussian processes and establish deconditional kernel means as
their posterior predictive mean. This connection provides Bayesian
interpretations and uncertainty estimates for deconditional kernel mean
embeddings, explains their regularization hyperparameters, and reveals a
marginal likelihood for kernel hyperparameter learning. These revelations
further enable practical applications such as likelihood-free inference and
learning sparse representations for big data.

本研究提出了非参数贝叶斯规则的去条件化核均值嵌入，并且展示了其与任务转换高斯过程的后验预测均值的联系。这提供了贝叶斯解释和不确定性估计，解释了它们的正则化超参数，揭示了内核超参数学习的边缘似然函数。这些发现进一步促进了无似然推断和大数据的稀疏表征等实际应用。

贝叶斯去条件核均值嵌入

Bayesian Deconditional Kernel Mean Embeddings

Conditional kernel mean embeddings are nonparametric models that encode
conditional expectations in a reproducing kernel Hilbert space. While they
provide a flexible and powerful framework for probabilistic inference, their
performance is highly dependent on the choice of kernel and regularization
hyperparameters. Nevertheless, current hyperparameter tuning methods
predominantly rely on expensive cross validation or heuristics that is not
optimized for the inference task. For conditional kernel mean embeddings with
categorical targets and arbitrary inputs, we propose a hyperparameter learning
framework based on Rademacher complexity bounds to prevent overfitting by
balancing data fit against model complexity. Our approach only requires batch
updates, allowing scalable kernel hyperparameter tuning without invoking kernel
approximations. Experiments demonstrate that our learning framework outperforms
competing methods, and can be further extended to incorporate and learn deep
neural network weights to improve generalization.

本研究提出了一种基于 Rademacher 复杂度界限的超参数学习框架，用于平衡数据拟合与模型复杂度，从而达到无需核逼近的可扩展核超参数调整方法，并证明本方法优于现行竞争者，可进一步扩展以融合有效应用深度神经网络的权重以提高泛化性能。