# What is the integral under the Gaussian distribution in the context of a-posteriori estimation?

According to my text book about Pattern Recognition (Bishop), the predictive distribution is defined as the following $$p(t_{n+1}|x_{n+1}, \vec{x}, \vec{t}) = \int p(t_{n+1}|x_{n+1}, \vec{w})\cdot p(\vec{w}| \vec{x}, \vec{t})\:\mathrm{d}\vec{w}$$ with $$dim(x)=dim(t)=n$$ so that $$n+1$$ denotes a newly received value while $$\vec{x}$$ is the vector of all former inputs and $$\vec{t}$$ is the vector of all former target outputs. The probability of $$t_{n+1}$$ is here expressed as relying on all known input-output/target combinations and the corresponding new input. $$\vec{w}$$ is the vector of all previously defined paramters describing the function that is the best known regression for all known input-output combinations (e. g. found through least squares method or similar).

Both functions under the integral are modelled as Gaussian distributions.

But the solutions of the equation above states $$p(t_{n+1}|x_{n+1}, \vec{x}, \vec{t})=\mathcal{N}(t_{n+1}|m(x_{n+1}), s^2(x_{n+1}))$$ with parameters $$m,s$$ that are constant for one choice of all parameters above and are not varying over time or similar. Thus, the latter equation says that the result is a Gaussian again. But since the integral under a Gaussian is an S-shaped curve, that does not make sense to me. Has it something to do with the multi-dimensionality?

Thanks!

• Could you explain what this "Euler function" is to which the title refers? – whuber Apr 24 at 14:32
• I should have called it: Gaussian distribution. Sorry. Is it now understandable? – Kutsubato Apr 25 at 7:55
• This is eq. 3.57 of Bishop PRML. He says it’s the convolution of two Gaussians so eq 2.115 applies and the result follows. Eq. 2.115 is derived in section 2.3.3. – Don Slowik May 6 at 12:39