[kalman] update RV notation#900
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Update the two "Constant value of state x_t" comments to X_t so the code comments match the uppercase random-variable convention used in the surrounding prose. The realized initial value x_0 comment stays lowercase by design. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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Reviewed — the uppercase/lowercase RV convention is applied correctly and consistently, and the new I pushed a small follow-up commit (dc7ca23) updating the two |
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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
LinearStateSpacemodel 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$X_t$ ).
| The measurement equation is $Y_t = \theta + v_t$ where $v_t$ is $N(0,1)$ and IID. | ||
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| The task of this exercise to simulate the model and, using the code from `kalman.py`, plot the first five predictive densities $p_t(x) = N(\hat x_t, \Sigma_t)$. | ||
| The task of this exercise to simulate the model and, using the code from `kalman.py`, plot the first five predictive densities $p_t(x) = N(\hat x_t, \Sigma_t)$ for $X_t$. |
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| Our horse race will be assessed in terms of realized squared error. | ||
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| In particular, your task is to generate a graph plotting observations of both $\| x_t - A x_{t-1} \|^2$ and $\| x_t - \hat x_t \|^2$ against $t$ for $t = 1, \ldots, 49$. | ||
| In particular, your task is to generate a graph plotting simulated realizations of both $\| X_t - A X_{t-1} \|^2$ and $\| X_t - \hat x_t \|^2$ against $t$ for $t = 1, \ldots, 49$. | ||
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| In the code below, `x[:, t]` is the realized value of $X_t$ along the simulated path. |
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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:
Best,
Longye