Episode Details

Richard Ngo challenged me to set a time box and write down as many of the most important features of my formal epistemology as I can in one sitting. Here goes.

Where probability distributions fail...

...to express beliefs

• There is no probability distribution over that says "". That is a support condition about probability distributions, namely, ⨾⨾. Some s satisfy this condition and some do not; but there is no that expresses the range belief itself.
• There is no probability distribution over that says " and are independent". This is an equation condition about probability distributions, namely, . Some s satisfy this equation and some do not; there is no that expresses the independence belief itself.
• There is no probability distribution over that can express a conditional probability distribution , even though this is just as essential a part of a Bayesian reasoner's epistemic state as her prior. A conditional probability distribution is a family of probability distributions indexed by the condition variable, not a probability distribution. There is no single distribution that expresses it.

...to make safety tradeoffs

• Suppose there is an unfair coin, which you know to be unfair (but not exactly how much or [...]

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Outline:

(00:21) Where probability distributions fail...

(00:25) ...to express beliefs

(01:26) ...to make safety tradeoffs

(02:17) Beliefs, according to davidad

(03:06) All other known notions of belief fit in nicely

(03:11) Bayesian beliefs

(03:21) Bayesian updating

(05:05) Infra-Bayesian beliefs (Kosoy and Appel)

(06:01) MWER (Halpern and Leung)

(06:45) Probabilistic dependency graphs (Richardson and Halpern)

(07:25) Credal sets (Cozman)

(07:52) Previsions (Goubault-Larrecq)

(08:40) The monad (Mio, Sarkis, and Vignudelli)

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First published:
July 29th, 2026

Source:
https://www.lesswrong.com/posts/e7Pd4Q9TF7jFdmPgz/imprecise-beliefs-a-tiny-introduction

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Narrated by TYPE III AUDIO.