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.