[kalman_2] Update code and descriptions

[HumphreyYang]

This PR closes #904 and makes a few improvements beyond it.

Hi @longye-tian, this PR resolves #904, and thanks for raising those issues!

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Commit: 20e2923

📚 Changed Lectures

• kalman_2

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• Workflow: Build Project [using jupyter-book]

[Copilot]

Pull request overview

This PR updates the kalman_2 lecture to better align the math, narrative timing, and code implementation of the Kalman filter with the intended worker model, addressing clarifications raised in issue #904.

Changes:

• Aligns the observation equation with the text by setting the measurement matrix to exclude direct dependence on effort (G = [g, 0]) and clarifies independence/initial-belief variance vs. standard deviation notation.
• Reorders Kalman-filter bookkeeping to record beliefs before incorporating the current observation (consistent with conditioning on (y^{t-1})), and adjusts arrays/loop indexing accordingly.
• Improves figure metadata/captions and narrative explanations around simulation initial conditions and learning experiments.

[HumphreyYang]

Hi @longye-tian,

I think this is ready to be merged. Please let me know if there is anything else you would like to add!

[longye-tian]

Many thanks @HumphreyYang !

They all looks good to me. Just some minor comments see above.

Best,
Longye

[HumphreyYang]

Many thanks @longye-tian for the detailed check!

I left out those two cases intentionally.

Personally, I think it would be a bit unclear if we don’t use subplot titles for figures with multiple subplots.

For instance, if we remove the Time Step {t} title from subplots, it would be a bit hard for readers to tell what the plot is showing at first glance.

Please let me know what you think between those two cases! If this makes sense, I can update the style guide accordingly (CC @mmcky and @jstac)!

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Without subplot titles:

With subplot titles:

[jstac]

Many thanks @HumphreyYang @longye-tian !

@HumphreyYang , i agree, subplots should be the exception --- they often need titles. This particular case is a good example. It would be great if you could update the style guide -- at titles to subplots as required, for clarity.

[HumphreyYang]

Many thanks @jstac, I will open a PR in the manual repo!

I will merge this one for now.

Many thanks @longye-tian as well for the detailed review!