[kalman] update RV notation

[longye-tian]

Hi @mmcky

This PR updates the RV notation following #898 .

In particular, this extends the new uppercase/lowercase convention to the Implementation section and Exercises:

• use $X_t$ and $Y_t$ for random state and signal variables;
• keep lowercase notation for realizations, density arguments, filter estimates, and code variables;
• clarify that the Exercise 3 plotted errors use simulated realizations.

Best,
Longye

[mmcky]

Reviewed — the uppercase/lowercase RV convention is applied correctly and consistently, and the new x[:, t] is the realized value of $X_t$ note nicely bridges the math and the Ex3 code.

I pushed a small follow-up commit (dc7ca23) updating the two # Constant value of state x_t code comments to X_t to match. LGTM once CI is green.

[github-actions]

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Commit: 23a0563

📚 Changed Lectures

• kalman

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[Copilot]

Pull request overview

This PR updates the Kalman filter lecture (lectures/kalman.md) to consistently apply the “uppercase = random variable / lowercase = realization” convention across the Implementation section and Exercises, aligning with the notation introduced earlier in the lecture and requested in #898.

Changes:

• Updated the LinearStateSpace model equations in the Implementation section to use $X_t$ / $Y_t$ for random state and signal variables.
• Revised Exercises 1–4 text and code comments to use $X_t$ for random states while keeping lowercase for realizations/estimates/code variables.
• Clarified in Exercise 3 that the plotted squared errors are based on simulated realizations (and added an explicit note tying x[:, t] to the realized $X_t$).

[longye-tian]

Many thanks @mmcky for the update and I've addressed the comments from Copilot as well.

Best,
Longye

[mmcky]

Hi @longye-tian — I've resolved the merge conflict on this branch (merged latest main in via 23a0563), but I'd like you to double-check the resolution since it involved harmonising with another notation change.

The conflict arose because #908 ("[kalman] Fix errors, modernize code, and unify notation") landed on main after this branch was created, and it did a global notation overhaul of the same file: \hat x → \mu (e.g. \hat x_t → \mu_t, code variable x_hat → μ) and the state shock w_t → W_t. That collided with your RV-uppercase changes in three places.

I resolved each by keeping your changes and adopting main's now-established notation where they overlapped, so the file stays internally consistent. Specifically:

Location	How I resolved it
Exercise 1 task	Kept your grammar fix ("exercise is to") and "for $X_t$"; used main's N(\mu_t, \Sigma_t) instead of N(\hat x_t, \Sigma_t)
Exercise 3 intro	Kept your uppercase $X_t$ / $X_{t-1}$ for the random variables; used main's $\{W_t\}$ for the shock sequence and $\mu_t$ for the filter prediction
Exercise 3 task	Kept your "simulated realizations", uppercase $X_t$ / $X_{t-1}$, and your new "x[:, t] is the realized value of $X_t$" note; used main's $\mu_t$ in $| X_t - \mu_t |^2$

The one point worth your eye: #908 set the state shock to uppercase $W_t$ (in kl_xdynam and the Convergence section), so in Exercise 3 I used $\{W_t\}$ to match. Your state-space block still keeps lowercase $w_t$ / $v_t$ for the standard-normal shocks of the general LinearStateSpace form, which I left as you wrote it. If you'd prefer a single convention for shocks throughout, let me know and I can align it.

Could you confirm the resolution reads the way you intended? Thanks!

[jstac]

@longye-tian My bad, I missed this PR and then made some related edits. Sorry about that.

Could you please provide a brief review of the current status of the lecture and this PR?

[longye-tian]

Hi @longye-tian — I've resolved the merge conflict on this branch (merged latest main in via 23a0563), but I'd like you to double-check the resolution since it involved harmonising with another notation change.

The conflict arose because #908 ("[kalman] Fix errors, modernize code, and unify notation") landed on main after this branch was created, and it did a global notation overhaul of the same file: \hat x → \mu (e.g. \hat x_t → \mu_t, code variable x_hat → μ) and the state shock w_t → W_t. That collided with your RV-uppercase changes in three places.

I resolved each by keeping your changes and adopting main's now-established notation where they overlapped, so the file stays internally consistent. Specifically:

Location How I resolved it
Exercise 1 task Kept your grammar fix ("exercise is to") and "**for X t **"; used main's N(\mu_t, \Sigma_t) instead of N(\hat x_t, \Sigma_t)
Exercise 3 intro Kept your uppercase $X_t$ / $X_{t-1}$ for the random variables; used main's $\{W_t\}$ for the shock sequence and $\mu_t$ for the filter prediction
Exercise 3 task Kept your "simulated realizations", uppercase $X_t$ / $X_{t-1}$, and your new "x[:, t] is the realized value of
𝑋
𝑡
" note; used main's $\mu_t$ in $| X_t - \mu_t |^2$
The one point worth your eye: #908 set the state shock to uppercase $W_t$ (in kl_xdynam and the Convergence section), so in Exercise 3 I used $\{W_t\}$ to match. Your state-space block still keeps lowercase $w_t$ / $v_t$ for the standard-normal shocks of the general LinearStateSpace form, which I left as you wrote it. If you'd prefer a single convention for shocks throughout, let me know and I can align it.

Could you confirm the resolution reads the way you intended? Thanks!

Many thanks @mmcky They all looks good to me!

[longye-tian]

@longye-tian My bad, I missed this PR and then made some related edits. Sorry about that.

Could you please provide a brief review of the current status of the lecture and this PR?

No worries @jstac.

I think @mmcky has resolved all the merge conflicts with your PR #908 .

This PR and yours now share the same notations and this PR complements yours by focuses on the Implementation section and Exercises section.

Best,
Longye