Audio note: this article contains 61 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Jaynes’ Widget Problem[1]: How Do We Update On An Expected Value?

Mr A manages a widget factory. The factory produces widgets of three colors - red, yellow, green - and part of Mr A's job is to decide how many widgets to paint each color. He wants to match today's color mix to the mix of orders the factory will receive today, so he needs to make predictions about how many of today's orders will be for red vs yellow vs green widgets.

The factory will receive some unknown number of orders for each color throughout the day - _N_r_ red, _N_y_ yellow, and _N_g_ green orders. For simplicity, we will assume that Mr A starts out with a prior distribution _P[N_r, N_y, N_g]_ under which:

• Number of orders for each color is independent of the other colors, i.e. _P[N_r, N_y, N_g] = P[N_r]P[N_y]P[N_g]_
• Number of orders for each color is uniform between 0 and 100: _P[N_i = n_i] = frac{1}{100} I[0 leq n_i < 100]_[2]

… and then [...]

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Outline:

(00:24 ) Jaynes' Widget Problem : How Do We Update On An Expected Value?

(03:20 ) Enter Maxent

(06:02 ) Some Special Cases To Check Our Intuition

(06:35 ) No Information

(07:27 ) Bayes Updates

(09:27 ) Relative Entropy and Priors

(13:20 ) Recap

The original text contained 2 footnotes which were omitted from this narration.

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First published:

November 4th, 2025

Source:

https://www.lesswrong.com/posts/qEWWrADpDR8oGzwpf/the-zen-of-maxent-as-a-generalization-of-bayes-updates

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Narrated by TYPE III AUDIO.