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In the Bishop Book on Machine Learning, when discussinf Variational Inference, he gives an example of the univariate Gaussian.

He defines in 10.1.3 the mean prior as:


What is the meaning of the conditional mean over an initial $\mu_0$ and what would be the analytical form for that kind of expression.

Thank You

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The expression $N(x \mid \mu_0, \sigma_0)$ is the notation of Bishop to denote the Gaussian distribution under the condition that the mean and standard deviation are given by $\mu_0$ and $\sigma_0$ respectively. So, your expression means: $\mu$ is normally distributed with mean $\mu_0$ and standard deviation $( \lambda_0 \tau )^{-1}$, which translates to the pdf $$ \frac{\lambda_0 \tau}{\sqrt{ 2 \pi }} e^{\frac{(\lambda_0 \tau)^2}{2} (\mu - \mu_0)^2} . $$

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