Thanks @Chihiro2000GitHub — this is correct and faithfully preserves the original behaviour (the two sample_posteriors calls still draw sequentially from a single stream).

One small consistency point: this lecture already has two sibling functions, sequential_update and check_martingale_mean, that take a seed and build the generator locally (rng = np.random.default_rng(seed)). With this change sample_posteriors instead relies on a module-level rng injected via a rng=rng default argument, so the lecture ends up with two different RNG idioms for three very similar functions.

Could we align sample_posteriors with its siblings by building the generator in the driver (plot_simplex_posteriors) and passing it in? That keeps a single shared stream (so the μ and ν clouds stay independent, as now), removes the module-level global, and lets us drop the np.random.seed(0) line at the top. Something like:

def sample_posteriors(μ_matrix, prior, n_draws=3000, rng=None):
    """..."""
    if rng is None:
        rng = np.random.default_rng()
    N, M = μ_matrix.shape
    states = rng.choice(N, size=n_draws, p=prior)
    signals = np.array([rng.choice(M, p=μ_matrix[s]) for s in states])
    ...

and then in plot_simplex_posteriors:

def plot_simplex_posteriors(μ_matrix, ν_matrix, prior3, n_draws=3000, seed=0):
    rng = np.random.default_rng(seed)
    posts_μ = sample_posteriors(μ_matrix, prior3, n_draws, rng)
    posts_ν = sample_posteriors(ν_matrix, prior3, n_draws, rng)
    ...

With that, the rng = np.random.default_rng(0) line in the imports block can be removed. Non-blocking — happy either way, but it would make the three functions consistent.

Many thanks @mmcky, that makes sense.

I agree that creating the generator locally in plot_simplex_posteriors is
cleaner and keeps the RNG pattern consistent with sequential_update and
check_martingale_mean.

Changing to

posts_μ = sample_posteriors(μ_matrix, prior3, n_draws, rng=rng)
posts_ν = sample_posteriors(ν_matrix, prior3, n_draws, rng=rng)

also makes it explicit that both posterior samples are drawn using the same
local generator.

After making this change, we can drop the global

rng = np.random.default_rng(0)

because it was only being used as the default value for rng in
sample_posteriors.

I think it is a strict improvement and should be adopted everywhere else!

@mmcky @HumphreyYang,

Thank you both! I'll go through your comments later, probably this weekend.