This paper presents a novel approach for approximate integration over the
uncertainty of noise and signal variances in Gaussian process (GP) regression.
Our efficient and straightforward approach can also be applied to integration
over input dependent noise variance (heteroscedasticity) and input dependent
signal variance (nonstationarity) by setting independent GP priors for the
noise and signal variances. We use expectation propagation (EP) for inference
and compare results to Markov chain Monte Carlo in two simulated data sets and
three empirical examples. The results show that EP produces comparable results
with less computational burden.

本文提出了一种用于高斯过程回归中噪声和信号方差的不精确积分的新方法，并使用期望传播进行推理，在两个模拟数据集和三个实证示例中与马尔可夫链蒙特卡罗方法进行比较。结果表明，期望传播能够产生可比较的结果，而计算量较小。

非平稳异方差高斯过程回归的近似推断

Approximate Inference for Nonstationary Heteroscedastic Gaussian process  Regression

While Gaussian probability densities are omnipresent in applied mathematics,
Gaussian cumulative probabilities are hard to calculate in any but the
univariate case. We study the utility of Expectation Propagation (EP) as an
approximate integration method for this problem. For rectangular integration
regions, the approximation is highly accurate. We also extend the derivations
to the more general case of polyhedral integration regions. However, we find
that in this polyhedral case, EP's answer, though often accurate, can be almost
arbitrarily wrong. We consider these unexpected results empirically and
theoretically, both for the problem of Gaussian probabilities and for EP more
generally. These results elucidate an interesting and non-obvious feature of EP
not yet studied in detail.

研究了期望传播算法在高斯积分问题解法中的应用，发现在矩形积分区域中，期望传播算法的近似值是非常精确的。虽然在多面体积分区域中它的答案通常是准确的，但我们发现在这种多面体情况下，期望传播算法的答案几乎完全是错误的，这些结果揭示出了期望传播算法的一个有趣且不明显的特征。