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Laplace approximation and natural gradient for Gaussian process regression with heteroscedastic student-f model

机译：La Laplace近似与自然梯度与异源学生-F模型的高斯过程回归

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摘要

We propose the Laplace method to derive approximate inference for Gaussian process (GP) regression in the location and scale parameters of the student-t probabilistic model. This allows both mean and variance of data to vary as a function of covariates with the attractive feature that the student-t model has been widely used as a useful tool for robustifying data analysis. The challenge in the approximate inference for the model, lies in the analytical intractability of the posterior distribution and the lack of concavity of the log-likelihood function. We present the natural gradient adaptation for the estimation process which primarily relies on the property that the student-t model naturally has orthogonal parametrization. Due to this particular property of the model the Laplace approximation becomes significantly more robust than the traditional approach using Newton's methods. We also introduce an alternative Laplace approximation by using model's Fisher information matrix. According to experiments this alternative approximation provides very similar posterior approximations and predictive performance to the traditional Laplace approximation with model's Hessian matrix. However, the proposed Laplace-Fisher approximation is faster and more stable to calculate compared to the traditional Laplace approximation. We also compare both of these Laplace approximations with the Markov chain Monte Carlo (MCMC) method. We discuss how our approach can, in general, improve the inference algorithm in cases where the probabilistic model assumed for the data is not log-concave.
机译：我们提出了LAPAPLE方法，以导出学生-T概率模型的位置和比例参数的高斯过程（GP）回归的近似推断。这允许数据的均值和方差随着具有增强特征的协变量而变化，即学生-T模型已被广泛用作稳定数据分析的有用工具。对模型的近似推理中的挑战在于，在后部分布的分析难以和对数似然函数的缺失。我们介绍了估计过程的自然梯度调整，主要依赖于学生-T模型自然具有正交参数化的性质。由于模型的这种特殊性，拉普拉斯近似比使用牛顿方法的传统方法变得明显更强大。我们还通过使用模型的Fisher信息矩阵来介绍替代的LAPLACE近似。根据实验，这种替代近似为与模型的Hessian矩阵的传统拉普拉斯近似提供了非常相似的后近似和预测性能。然而，与传统的拉普拉斯近似相比，所提出的Laplace-Fisher近似值更快，更稳定。我们还使用Markov Chain Monte Carlo（MCMC）方法比较这两个LAPPLACE近似值。我们讨论了我们的方法，通常可以改善推理算法，在假设为数据的概率模型不是日志凹的情况下。

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