Robustness Guarantees via Probabilistic Gaussian Process Models

The article discusses the calculation of the expectation of the positive part of a Gaussian random vector, which is an essential concept in probability theory and statistics. The authors explain that this expectation can be computed using the Karhunen-Loeve expansion, which is a mathematical tool used to represent a Gaussian random variable as a linear combination of its moments.
The authors begin by defining the positive part of a Gaussian random vector, denoted as $N(\gamma)$, where $\gamma$ is a scalar greater than zero. They explain that this quantity represents the amount of probability mass assigned to the positive tail of the distribution. The authors then introduce the Karhunen-Loeve expansion, which is a way of representing a Gaussian random variable as a linear combination of its moments.
The authors then show how to use the Karhunen-Loeve expansion to compute the expectation of the positive part of a Gaussian random vector. They begin by expressing $N(\gamma)$ in terms of its mean and variance, which are given by $\mu = \gamma – 1$ and $\sigma^2 = \gamma^2/4$, respectively. They then use the Karhunen-Loeve expansion to represent $N(\gamma)$ as a linear combination of its moments, which are denoted as $m_k$.
The authors then explain how to compute the expectation of the positive part using the moments. They show that the expected value of $N(\gamma)$ can be expressed as a sum of expectations of the moments, each multiplied by a complex coefficient. The authors provide an example of how to compute the first moment, which is denoted as $m_1$, and explain how to generalize this result to higher moments.
The authors then discuss some important properties of the expected positive part. They show that it is a Gaussian random variable with mean $\mu$ and variance $\sigma^2$, which agrees with the classical result for the expectation of a Gaussian distribution. They also show that the expected positive part has a skewed distribution when $\gamma$ is large, which can lead to non-Gaussian behavior in some applications.
Finally, the authors conclude by summarizing the main results of the article and highlighting their implications for probability theory and statistics. They note that the Karhunen-Loeve expansion provides a powerful tool for analyzing Gaussian random variables, and that the expected positive part is an important quantity in many applications.
In summary, the article provides a detailed explanation of how to compute the expectation of the positive part of a Gaussian random vector using the Karhunen-Loeve expansion. The authors demystify complex concepts by using everyday language and engaging metaphors or analogies, making the material accessible to a wide range of readers.