for i in range(1, s):
# a) Generovani kandidata theta_ast
theta_ast = np.random.normal() * d + mu
# b) Spocitani akceptacni pravdepodobnosti
alpha = min(
math.exp(-1 / 2 * abs(theta_ast) + 1 / 2 * abs(theta[i - 1]) - 1 /
(2 * d**2) * (theta[i - 1] - mu)**2 + 1 / (2 * d**2) *
(theta_ast - mu)**2), 1)
# c) rozhodnuti o prijeti nebo odmitnuti kandidata
if alpha > np.random.rand():
theta[i] = theta_ast
count += 1
else:
theta[i] = theta[i - 1]
progress_bar(i, s)
# ===== 3. Prezentace vysledku =====
theta = theta[s_0:] # Vyhozeni prvnich s_0 vzorku
# Graficke zobrazeni konvergence
fig, axs = plt.subplots(1, 2, tight_layout=True)
axs[0].plot(theta[::500])
axs[0].set_title("Kazdy 500. odhad theta")
axs[1].hist(theta, bins=50, facecolor="blue", alpha=0.5, edgecolor="black")
axs[1].set_title("Rozdeleni theta")
# Pocitani momentu
e_theta = np.mean(theta)
d_theta = np.var(theta)
## 1 - Deklarace promennych
S = 1_000_000 # pocet simulaci
r = 0 # pocitadlo stavu n/a_win
b_win = 0 # stredni hodnota vyhry Boba pri stavu n/a_win
step = round(S / 100)
## 2 - Simulace hry
for i in range(S):
p = uniform() # pocatecni rozdeleni stolu p~U(0,1)
y = uniform(size=n) # vektor rozmeru 8x1 s nezavislymi tahy z U(0,1)
if sum(y < p) == a_win:
r += 1 # pocitadlo stavu n/a_win inkrementujeme o 1
b_win += (1 - p)**3 # scitani pravdepodobnosti vyhry Boba
# (z predchozich behu pri stavu n/a_win)
progress_bar(i, S)
## 3 - Vypocet a prezentace vysledku
b_win_expected = b_win / r # Stredni (ocekavana) hodnota pravd. vyhry Boba
# pri stavu n/a_win
a_win_expected = 1 - b_win_expected # Stredni (ocekavana) hodnota pravd. vyhry Alice
# pri stavu n/a_win
print(
f"\nPrumerna pravd. vyhry jednotlivych hracu pri stavu {a_win}:{n-a_win}:")
print(
tabulate([["Jmeno", "Pravd."], ["Alice", a_win_expected],
["Bob", b_win_expected]],
headers="firstrow"))
print(f"\nPocet stavu {a_win}:{n-a_win}: {r}")
# Cast pro vypocet citatele SD pomeru hustot
# d) beta_depart = 0 (druhy prvek vektoru beta)
# p(b|h,y) ~ N(beta_1, v_1)
v_1 = inv(inv(v_0) + h[i] * (t(x) @ x)) # (4.4) dle Koop (2003)
beta_1 = v_1 @ (inv(v_0) @ beta_0 + h[i] * (t(x) @ y))
new_sdd_nom = norm.pdf(0, beta_1[1][0], sqrt(v_1[1][1]))
sdd_nom.append(new_sdd_nom)
# e) beta_reds = beta_trains
# p(R*b|h,y) ~ N(R * beta_1, R * v_1 * R')
r = np.array([[0, 0, 1, -1]])
new_sde_nom = mvn.pdf(0, r @ beta_1, r @ v_1 @ t(r))
sde_nom.append(new_sde_nom)
progress_bar(i, s)
# ===== 3. Posteriorni analyza =====
# vyhozeni prvnich s_0 vzorku
beta = np.delete(beta, range(s_0), axis=1)
h = h[s_0:]
sdd_nom = sdd_nom[s_0:]
sde_nom = sde_nom[s_0:]
# Graficke zobrazeni konvergence
k = 100 # delka kroku (neplest si s krokem ve funkci geweke)
fig_1 = plt.figure()
axs = fig_1.subplots(nrows=3, ncols=2)
for i in range(beta.shape[0]):
theta = np.zeros((2, s + 1))
theta[:, 0] = np.array([-1000, 1000])
# ===== 2. Gibbsuv vzorkovac =====
for i in range(1, s + 1):
# generovani theta_1/theta_2 ~ N(mu_12,sigma_12)
mu_12 = mu[0][0] + rho * (theta[1, i - 1] - mu[1][0])
sigma_12 = 1 - rho**2
theta[0, i] = randn() * sqrt(sigma_12) + mu_12
# generovani theta_2/theta_1 ~ N(mu_21,sigma_21)
mu_21 = mu[1][0] + rho * (theta[0, i] - mu[0][0])
sigma_21 = 1 - rho**2
theta[1, i] = randn() * sqrt(sigma_21) + mu_21
progress_bar(i, s + 1)
# ===== 3. Grafy =====
fig, axs = plt.subplots(2, 2)
k = 10
axs[0, 0].plot(theta[0, 0:k], theta[1, 0:k], linewidth=0.5, marker="x")
axs[0, 0].set_title(f"Konvergence prvnich {k} kroku Gibbsova vzorkovace")
axs[0, 0].set_xlabel("theta_1")
axs[0, 0].set_ylabel("theta_2")
theta = theta[:, s_1:]
axs[0, 1].set_title(f"Sdruzena hustota na zaklade {s_1} vzorku")
axs[0, 1].scatter(theta[0], theta[1], marker=".", s=0.75)
axs[0, 1].set_xlabel("theta_1")
log_bf = res_gm_rest["log_ml"] - res_gm["log_ml"]
bf = exp(log_bf) # Bayesuv faktor (odlogaritmujeme predchozi vyraz)
print("Bayesuv faktor porovnavajici omezeny a neomezeny model:")
print(f"BF = {round(bf, 4)}", "\n")
# ===== 3. Hypoteza, ze beta > 1 =====
mc = 100_000 # pocet simulaci (zvyste napr. na 10_000)
beta_sim = np.array([[], []])
print("Vypocet pravd., ze beta > 1 pomoci simulace:")
for i in range(mc):
h_sim = float(gamm_rnd_koop(res_gm["h_1"], res_gm["nu_1"], (1, 1)))
new_column = norm_rnd(1 / h_sim * res_gm["v_1"]) + res_gm["beta_1"]
beta_sim = np.append(beta_sim, new_column, axis=1)
progress_bar(i, mc)
# Vypocet pravd. beta > 1
pr_beta = sum(t(beta_sim > 1))[1] / mc
print(f"Pravdepodobnost, ze beta > 1:")
print(f"Pr. = {round(pr_beta, 4)}")
# Analyticky vypocet pravdepodobnost
# a) standardizace skalovaneho t-rozdeleni (p(beta|y)) pro druhy prvek vektoru parametru beta
zscore = float((1 - res_gm["beta_1"][1]) / res_gm["b1_std"][1])
# b) vypocet odpovidajiciho kvantilu ze standardizovaneho centrovaneho t-rozdeleni
pr_beta_analyticky = 1 - student.cdf(zscore, res_gm["nu_1"])
print(f"Pr. = {round(pr_beta_analyticky, 4)} (analyticky)")
