Audio note: this article contains 629 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Last week, ARC released a paper called Towards a Law of Iterated Expectations for Heuristic Estimators, which follows up on previous work on formalizing the presumption of independence. Most of the work described here was done in 2023.

A brief table of contents for this post:

• What is a heuristic estimator? (One example and three analogies.)
• How might heuristic estimators help with understanding neural networks? (Three potential applications.)
• Formalizing the principle of unpredictable errors for heuristic estimation (the technical meat of the paper).

In "Formalizing the Presumption of Independence", we defined a heuristic estimator to be a hypothetical algorithm that estimates the values of mathematical expression based on arguments. That is, a heuristic estimator is [...]

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Outline:

(04:56) What is a heuristic estimator?

(05:31) Example: Sum of sixth digits of square roots

(07:11) Analogy #1: Proof verification

(08:36) Analogy #2: Conditional expectation

(09:38) Analogy #3: Subjective probabilities and estimates

(11:43) How can heuristic estimation help us understand neural networks?

(14:16) Mechanistic anomaly detection

(16:33) Safe distillation

(19:28) Low probability estimation

(21:37) Formalizing the principle of unpredictable errors

(22:50) The subjective approach: Iterated estimation and error orthogonality

(26:37) Challenges with the subjective approach

(28:43) The objective approach: Accuracy

(35:28) Challenges with the objective approach

(36:14) Estimating the product of jointly normal random variables

(41:07) Estimating the permanent of a matrix

(44:21) Conclusion

The original text contained 19 footnotes which were omitted from this narration.

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First published:

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Narrated by TYPE III AUDIO.

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