# Gaussian Distribution with conditional mean

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:

$p(\mu|\tau)=N(\mu|\mu_0,(\lambda_0*\tau)^{-1})$

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 $$\frac{\lambda_0 \tau}{\sqrt{ 2 \pi }} e^{\frac{(\lambda_0 \tau)^2}{2} (\mu - \mu_0)^2} .$$