# Gaussian Linear Experts

## Main.GaussianLinearExperts History

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Gaussian linear experts in the protocol of [[Competing With Prediction Rules | 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 $y\in\mathbb{R}$:

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

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, [[http://arxiv.org/abs/0910.4683 | arXiv:0910.4683]] [cs.LG], arXiv.org e-Print archive, 2009.

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

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, [[http://arxiv.org/abs/0910.4683 | arXiv:0910.4683]] [cs.LG], arXiv.org e-Print archive, 2009.