Gaussian Linear Experts

Gaussian linear experts in the protocol of online regression incorporate information about the Gaussian nature of the noise in the outcomes. Each expert ⚠ $\theta \in \mathbb{R}^n$ predicts the following probability distribution over the outcomes ⚠ $y\in\mathbb{R}$:

⚠ $\displaystyle{\xi^\theta(y) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(\theta'x_t-y)^2}{2\sigma^2}}, y\in \mathbb{R}}$

at each step, where the variance ⚠ $\sigma^2$ is the same for all the experts, and is assumed to be known. The natural loss function for these experts is the logarithmic loss function ⚠ $\lambda(\xi^\theta,y) = -\ln \xi^\theta(y)$.

Fedor Zhdanov and Vladimir Vovk. Competing with Gaussian linear experts. Technical report, arXiv:0910.4683 [cs.LG], arXiv.org e-Print archive, 2009.