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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

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.

kgwgk 6y ago
> the log of the density can't have an upper bound at 0 I don't think he claims that. He even shows charts where positive log-densities appear.
- tmalsburg2 6y ago
  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.
  - kgwgk 6y ago
    When x goes to minus infinity log(x) approaches zero.
    - tmalsburg2 6y ago
      No, the logarithm is only defined for positive values. Plus probability densities are strictly non-negative. https://en.m.wikipedia.org/wiki/Logarithm https://en.m.wikipedia.org/wiki/Logarithm
      - kgwgk 6y ago
        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.
  - moultano 6y ago
    The log function has a vertical asymptote as x approaches 0.
    - tmalsburg2 6y ago
      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.
      - kgwgk 6y ago
        If the author himself tells you so, it’s more than likely. :-)

kgwgk 6y ago

> the log of the density can't have an upper bound at 0 I don't think he claims that. He even shows charts where positive log-densities appear.

tmalsburg2 6y ago
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.
- kgwgk 6y ago
  When x goes to minus infinity log(x) approaches zero.
  - tmalsburg2 6y ago
    No, the logarithm is only defined for positive values. Plus probability densities are strictly non-negative. https://en.m.wikipedia.org/wiki/Logarithm https://en.m.wikipedia.org/wiki/Logarithm
    - kgwgk 6y ago
      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.
- moultano 6y ago
  The log function has a vertical asymptote as x approaches 0.
  - tmalsburg2 6y ago
    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.
    - kgwgk 6y ago
      If the author himself tells you so, it’s more than likely. :-)