Maximum Value of Kernel Function in ABC Are there cases where a kernel function, must have 1 as the maximum value ??
The definition of a Kernel can be found in the following link,
https://en.wikipedia.org/wiki/Kernel_(statistics)#In_non-parametric_statistics
For example, in Approximate Bayesian Computing (ABC), we approximate the true likelihood of the data with the following integral
$$\int K_{h}(x_{true}-x)p(x|\theta)dx$$
where $x_{true}$ is our observed data, and $p(x|\theta)$ the model from which we sample.
In this case, can we assume that $K_{h}(x_{true}-x)$ is a weight function, so the sum across all $x$ has to be 1, so we  can assume that the maximum value of $K_{h}(x_{true}-x)$ is 1?
Because it is known that a scaled kernel, in general, is no constrained to have a maximum value of 1.
But in the case of the ABC approximation, should it have 1 as each maximum value?
 A: The answer is no. You must have that $\int K_h(u)\mathrm{d}u = 1$, but that doesn't mean that $K_h(u) \leq 1$. The same applies to pdfs: a pdf $f$ can take values bigger than $1$, but it's integral $\int_a^b f(x)\mathrm{d}x = P(a \leq X \leq b)$ must be always less than $1$.
You can think of the uniform distribution centered at $0$ with support $(-\epsilon, \epsilon)$ for an example: the pdf (and kernel) is
$$ K_\epsilon(x) = \dfrac{1}{2\epsilon}\mathbb{I}(-\epsilon \leq x \leq \epsilon)$$
Just set $\epsilon$ to any small value ( e.g. $\varepsilon\approx 0.001$) and you get that $K_\epsilon$ takes only two values: either $0$ or $500 = 1/0.002$.
TLDR: A kernel or a pdf can take arbitrarily high $y$-values, but the higher the value, the smaller the support.
A: In the original ABC version (Tavaré et al., 1999), the (dichotomous) probability of acceptance of a simulation $x(\theta)$ from $p(\cdot|\theta)$ (and hence of the parameter value $\theta$) is
$$\mathbb I_{|x_\text{obs}-x(\theta)|<\epsilon}\in\{0,1\}$$
It is a natural generalisation (Fearnhead and Prangle, 2010) to consider a smoother function of the distance like
$$K(|x_\text{obs}-x(\theta)|/\epsilon)$$
provided of course that $K(|x_\text{obs}-x(\theta)|/\epsilon)\in(0,1)$. There is no constraint though that $K(\cdot/\epsilon)$ is a probability density and indeed it does not integrate to one (since the rescaled version of a true density $K$ would be $K(\cdot/\epsilon)/\epsilon$ instead). Nonetheless, the name kernel was adopted (or abused!) for $K(\cdot/\epsilon)$ by analogy with non-parametrics (Blum and François, 2010).
The associated ABC posterior then writes as
$$\pi(\theta)K(|x_\text{obs}-x(\theta)|/\epsilon)p(x|\theta)\Big/
\int_\Theta\int_{\mathfrak X} \pi(\theta)K(|x_\text{obs}-x(\theta)|/\epsilon)p(x|\theta)\text dx\text d\theta$$
by an accept reject argument,
which does not require $K(\cdot/\epsilon)$ to be a probability density.
