import numpy as np

import matplotlib.pyplot as plt

from scipy.optimize import minimize

np.random.seed(0)

```

## Experiments and stochastic transformations

caption: Sampled posterior points on the 2-simplex

name: fig-blackwell-simplex-clouds

---

def sample_posteriors(μ_matrix, prior, n_draws=3000):

def sample_posteriors(μ_matrix, prior, n_draws=3000, rng=None):

"""

Simulate n_draws observations from the experiment and compute

the resulting posterior beliefs.

Returns array of shape (n_draws, N).

"""

if rng is None:

rng = np.random.default_rng()

N, M = μ_matrix.shape

states = np.random.choice(N, size=n_draws, p=prior)

signals = np.array([np.random.choice(M, p=μ_matrix[s]) for s in states])

states = rng.choice(N, size=n_draws, p=prior)

signals = np.array([rng.choice(M, p=μ_matrix[s]) for s in states])

posteriors, _ = compute_posteriors(μ_matrix, prior)

return posteriors[signals]

return pts @ corners

def plot_simplex_posteriors(μ_matrix, ν_matrix, prior3, n_draws=3000):

posts_μ = sample_posteriors(μ_matrix, prior3, n_draws)

posts_ν = sample_posteriors(ν_matrix, prior3, n_draws)

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=rng)

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

cart_μ = simplex_to_cart(posts_μ)

cart_ν = simplex_to_cart(posts_ν)