Delta function in the context of Gaussian processes

I am learning gaussian processes from the book Bayesian Reasoning and Machine Learning, Chapter 19, page 400, section 19.1.2. A snapshot from the text,

I failed to understand the use of delta function here, why, the equation can't simply be written as, $$p(\mathbf{y}|\mathbf{x}) = \int_\mathbf{w} \mathbf{\Phi w} p(\mathbf{w})$$

Am I missing something trivial?

The integral is actually the following: $$p(\mathbf{y}|\mathbf{x})=\int p(\mathbf{y|\mathbf{x},\mathbf{w}})p(\mathbf{w})d\mathbf{w}$$ Here, $$p(\mathbf{y|\mathbf{x},\mathbf{w}})$$ is the distribution of $$\mathbf{y}$$ when we know $$\mathbf{x}$$ and $$\mathbf{w}$$, in which the relation is actually deterministic, i.e. $$\mathbf{y}=\mathbf{\Phi(x)}\mathbf{w}$$. So, $$\mathbf{y}$$ can only be single value, whose distribution is represented by dirac-delta functions to be able to do further analysis. This deterministic relation is automatically enforced because the delta function will ensure that no $$\mathbf{w}$$ value will be integrated if $$\mathbf{y}\neq \mathbf{\Phi(x)}\mathbf{w}$$, since $$\delta(\mathbf{y}-\mathbf{\Phi(x)}\mathbf{w})$$ is $$0$$ when the two are not equal. This is not the same as replacing $$p(\mathbf{y|\mathbf{x},\mathbf{w}})$$ with $$\mathbf{\Phi(x)w}$$.
• So in other words we are taking a subset of $\mathbf{w}$'s that satisfied the condition 𝐲=𝚽(𝐱)𝐰 , but is not it true that whatever the w is y will always be equal to 𝚽(𝐱)𝐰 ? That's how we wrote the equation, and created it deterministic. Sep 21, 2019 at 21:09