I am currently solving my way through problems in Bishop's "Pattern Recognition and Machine Learning" and I am slightly confused by one statement. It is given in problem 1.20 on the page 63 and it goes like this:
$\hat{r}$ - optimal radius that maximises probability density in polar coordinates (radius of the "soap bubble")
$D$ - number of dimensions
$\sigma$ - standard deviation of normal distribution
We have already seen that $\sigma<<\hat{r}$ for large $D$, and so we see that most of the probability mass is concentrated in a thin shell at large radius. Finally, show that the probability density $p(x)$ is larger at the origin than at the radius $\hat{r}$ by a factor of $\exp(D/2)$. We therefore see that most of the probability mass in a high-dimensional Gaussian distribution is located at a different radius from the region of high probability density. This property of distributions in spaces of high dimensionality will have important consequences when we consider Bayesian inference of model parameters in later chapters
It used in context of high-dimensional Gaussian distribution with origin in $0$. It basically means that there is an "soap bubble" where the main probability mass is concentrated. But I can't understand why there is mode (highest probability density) of distribution located at origin. I just can't wrap my head around it and probably need a little intuition for this interesting feature.