Interesting article but with some factual imprecisions: 1. A bit nit-picky but the Beta distribution has support [0, 1]. So the distribution in the graph is not a Beta. It's probably a shifted and scaled Beta. 2. The probability density can be arbitrarily high, which means that the log of the density can't have an upper bound at 0. Probability is bounded at 1 but probability density isn't. The fact that none of the example distributions passes zero is a mere coincident. I'm surprised to see this beginner's mistake in an otherwise insightful article.
It's possible that I misunderstood the text but what's the alternative interpretation of "The log function asymptotes at 0"? Also note that the text says log probabilities when it actually refers to log densities.
Ah, that's likely the interpretation that the author had in mind. But it didn't help that he wrote log probability when he actually talked about log probability density.
You're absolutely right! I meant it the other way around:
As x approaches 0, log(x) goes to minus infinity.
I think that's what he means (the next sentence about "visualiz[ing] very small values of a function by expanding their range" makes sense in that context), but I agree it is not very clear.
