Replace np.sum(a * b) with a @ b for better performance and accuracy

[Copilot]

This PR replaces instances of np.sum(a * b) with the more efficient a @ b operator throughout the lecture series, following best practices for NumPy operations.

Changes Made

Educational Enhancement

lectures/linear_algebra.md: Added side-by-side comparison showing both methods:

np.sum(x * y)          # Inner product of x and y, method 1
x @ y                  # Inner product of x and y, method 2 (preferred)

Explains that @ operator is preferred for performance and numerical accuracy.

Direct Replacements

• lectures/career.md: Expected value calculations in probability distributions
• lectures/lake_model.md: Expected value computations in economic models
• lectures/mccall_model_with_separation.md: JAX-based dot products
• lectures/mix_model.md: Posterior mean calculations
• lectures/kalman.md: Squared error computations optimized to dot products
• lectures/rand_resp.md: Mean squared error calculations

Benefits

• Performance: Uses optimized BLAS libraries with fused multiply-add operations
• Accuracy: Reduces rounding errors compared to separate multiply and sum steps
• Best Practices: Follows NumPy documentation recommendations and PEP 465
• Educational Value: Demonstrates modern NumPy usage patterns

Examples

Before:

self.F_mean = np.sum(self.θ * self.F_probs)
e1[t-1] = np.sum((x[:, t] - kn.x_hat.flatten())**2)
d_new = jnp.sum(jnp.maximum(v, u(c) + β * d) * q)

After:

self.F_mean = self.θ @ self.F_probs
diff1 = x[:, t] - kn.x_hat.flatten()
e1[t-1] = diff1 @ diff1
d_new = jnp.maximum(v, u(c) + β * d) @ q

All changes preserve mathematical equivalence while improving performance through optimized linear algebra operations.

Fixes #463.

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[github-actions]

🚀 Deployed on https://689fd4a65cfbe334ca14bb37--nostalgic-wright-5fa355.netlify.app

[HumphreyYang]

Hi @mmcky,

I had a look over this PR. I think it introduces a bug to lake_model.

Other than that, other changes are correct to me.

[HumphreyYang]

Hi @mmcky,

I incorprated John's comments. I will merge this once it is built.

Please let me know if you have further comments or changes you want to make!