Distinguishing the cause and effect from bivariate observational data is the
foundational problem that finds applications in many scientific disciplines.
One solution to this problem is assuming that cause and effect are generated
from a structural causal model, enabling identification of the causal direction
after estimating the model in each direction. The heteroscedastic noise model
is a type of structural causal model where the cause can contribute to both the
mean and variance of the noise. Current methods for estimating heteroscedastic
noise models choose the Gaussian likelihood as the optimization objective which
can be suboptimal and unstable when the data has a non-Gaussian distribution.
To address this limitation, we propose a novel approach to estimating this
model with Student's $t$-distribution, which is known for its robustness in
accounting for sampling variability with smaller sample sizes and extreme
values without significantly altering the overall distribution shape. This
adaptability is beneficial for capturing the parameters of the noise
distribution in heteroscedastic noise models. Our empirical evaluations
demonstrate that our estimators are more robust and achieve better overall
performance across synthetic and real benchmarks.

通过使用学生 t 分布提出了一种估计异方差噪声模型的新方法，以解决现有估计方法在数据分布非高斯时存在的亚优和不稳定问题，并成功地在合成和真实基准测试中表现出更好的鲁棒性和整体性能。

鲁棒估计因果异方差噪声模型

Robust Estimation of Causal Heteroscedastic Noise Models

The real life time series are usually nonstationary, bringing a difficult
question of model adaptation. Classical approaches like ARMA-ARCH assume
arbitrary type of dependence. To avoid such bias, we will focus on recently
proposed agnostic philosophy of moving estimator: in time $t$ finding
parameters optimizing e.g. $F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta
(x_\tau))$ moving log-likelihood, evolving in time. It allows for example to
estimate parameters using inexpensive exponential moving averages (EMA), like
absolute central moments $E[|x-\mu|^p]$ evolving for one or multiple powers
$p\in\mathbb{R}^+$ using $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$.
Application of such general adaptive methods of moments will be presented on
Student's t-distribution, popular especially in economical applications, here
applied to log-returns of DJIA companies. While standard ARMA-ARCH approaches
provide evolution of $\mu$ and $\sigma$, here we also get evolution of $\nu$
describing $\rho(x)\sim |x|^{-\nu-1}$ tail shape, probability of extreme events
- which might turn out catastrophic, destabilizing the market.

该研究讨论了如何使用移动估计器来适应非平稳的实际时间序列数据，并以学生 t - 分布为例进行演示，以获得在不同时间单位内的泰尔指数演变， 以概括市场的稳定性。

非平稳时间序列的矩方法移动估计自适应学生 t 分布

Adaptive Student's t-distribution with method of moments moving  estimator for nonstationary time series

Classification models may often suffer from "structure imbalance" between
training and testing data that may occur due to the deficient data collection
process. This imbalance can be represented by the learning using privileged
information (LUPI) paradigm. In this paper, we present a supervised
probabilistic classification approach that integrates LUPI into a hidden
conditional random field (HCRF) model. The proposed model is called LUPI-HCRF
and is able to cope with additional information that is only available during
training. Moreover, the proposed method employes Student's t-distribution to
provide robustness to outliers by modeling the conditional distribution of the
privileged information. Experimental results in three publicly available
datasets demonstrate the effectiveness of the proposed approach and improve the
state-of-the-art in the LUPI framework for recognizing human activities.

本论文提出了一种称为 LUPI-HCRF 的模型，它将 LUPI 集成到隐藏条件随机场模型中，并使用学生 t 分布来增强对离群值的建模。实验结果表明，该方法在识别人类活动方面具有较好的性能，优于现有的 LUPI 框架。