import numpy as np

def sample_odds_path(p, s=0.05, N=2000):
    x = []
    xc = np.log(p) - np.log(1-p)

    for k in range(N):
        x.append(xc)
        f = np.exp(xc)
        xc += np.random.normal(loc=0, scale=s) + s**2 * (f-1)/(f+1)

    x = np.array(x)
    L = 1 - 1/(np.exp(x) + 1)

    return x, L

mean_scores = []
log_scores = []

for _ in range(1000):
    x, L = sample_odds_path(0.5, s = 0.15, N = 500)
    q = np.random.choice(L, size=10)

    indicator = np.random.choice([0, 1], p=[1 - L[-1], L[-1]])

    mean_forecast = np.mean(q)
    mean_log_odds = 1 - 1/(np.exp(np.mean(np.log(q/(1-q)))) + 1)

    mean_scores.append(np.log(mean_forecast) if indicator == 1 else np.log(1 - mean_forecast))
    log_scores.append(np.log(mean_log_odds) if indicator == 1 else np.log(1 - mean_log_odds))

print(np.mean(mean_scores), np.mean(log_scores))