from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as pl
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
setup() # initialize the plotting styles
np.random.seed(42)
x = 0.0
chain = np.empty(1e6)
for step in range(len(chain)):
x += np.random.randn()
chain[step] = x
# Thin the chain for plotting purposes.
chain = chain[::1000]
mn, mx = np.min(chain), np.max(chain)
mid = 0.5 * (mn + mx)
rng = 0.5 * (mx - mn) * 1.1
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(np.arange(len(chain)) * 1e-2, chain, color=COLORS["DATA"])
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.set_xlabel("mcmc step [$\\times 10^5$]")
ax.set_ylabel("$x$")
ax.set_ylim(mid - rng, mid + rng)
savefig(fig, "improper.pdf")
print("Warning: running naive autocorrelation function method. "
"This will be slow!")
mu = np.mean(x)
r = x - mu
C = np.empty(len(r) // 2)
for i in range(1, len(C)):
C[i] = np.mean(r[i:] * r[:-i])
C /= np.mean(r**2)
C[0] = 1.0
return C
def autocorr_function(x):
n = len(x)
f = np.fft.fft(x - np.mean(x), n=2 * n)
acf = np.fft.ifft(f * np.conjugate(f))[:n].real
return acf / acf[0]
if __name__ == "__main__":
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
acf = autocorr_function(chain)
ax.plot(acf[:61])
ax.set_xlim(0, 60)
ax.set_ylabel("$C_x(\Delta)$")
ax.set_xlabel("$\Delta$")
savefig(fig, "estimatetau.pdf")
return -0.5 * np.dot(x, alpha)
def run_mcmc(log_p, x, prop_sigma=1.0, nsteps=2e4, thin=1):
lp = log_p(x)
chain = np.empty((nsteps // thin, len(x)))
acc = 0
for step in range(len(chain)):
for i in range(thin):
x_prime = np.array(x)
x_prime[np.random.randint(
len(x))] += prop_sigma * np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
acc += 1
x[:] = x_prime
lp = lp_prime
if i == thin - 1:
chain[step] = x
return chain, acc / (len(chain) * thin)
if __name__ == "__main__":
chain, _ = run_mcmc(log_p_gauss, np.array([0.0, 0.0]))
fig = corner.corner(chain,
labels=["$x$", "$y$"],
range=[(-4.5, 4.5), (-4.5, 4.5)],
plot_density=False,
plot_contours=False)
savefig(fig, "twod_a.pdf", dpi=300)
import numpy as np
import matplotlib.pyplot as pl
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
from itertau import autocorr_function, autocorr_time_iterative
from tuning import log_p_gauss, run_mcmc, sigs
setup() # initialize the plotting styles
np.random.seed(42)
taus = np.empty((len(sigs), 2))
for i, sig in enumerate(sigs):
chain, acc = run_mcmc(log_p_gauss, np.array([0.0, 0.0]), prop_sigma=sig,
nsteps=2e5)
for j in range(2):
acf = autocorr_function(chain[:, j])
taus[i, j] = autocorr_time_iterative(acf)
print(sig, acc, taus[i])
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(sigs, taus[:, 0], ".-", color=COLORS["MODEL_1"])
ax.plot(sigs, taus[:, 1], ".-", color=COLORS["MODEL_2"])
ax.set_xscale("log")
ax.set_xlabel(r"$\sigma_q$")
ax.set_ylabel(r"$\tau_x$")
ax.set_xlim(0.2, 40)
ax.set_ylim(0, 450)
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
savefig(fig, "tuningtau.pdf")
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
if __name__ == "__main__":
s = SQUARE_FIGSIZE
s[0] *= 2
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=s)
ax1.plot(np.arange(len(chain)) / 1e3,
chain,
color=COLORS["DATA"],
rasterized=True)
ax1.set_xlim(0, len(chain) / 1e3)
ax1.set_ylim(2 - 5.5, 2 + 5.5)
ax1.set_ylabel("$x$")
ax1.set_xlabel("thousands of steps")
for i in range(4):
a = i * len(chain) / 4
b = (i + 1) * len(chain) / 4
print(i + 1, np.mean(chain[a:b]), np.var(chain[a:b]))
ax2.hist(chain[a:b], 50, histtype="step")
ax2.set_xlim(2 - 5.5, 2 + 5.5)
ax2.set_yticklabels([])
ax2.set_ylabel("$p(x)$")
ax2.set_xlabel("$x$")
savefig(fig, "convergence.pdf", dpi=300)
s_skew = np.mean((x - s_mu)**3) / s_var**(3. / 2)
s_kurt = np.mean((x - s_mu)**4) / s_var**2
return (
(a_mu, a_var, a_skew, a_kurt),
(s_mu, s_var, s_skew, s_kurt),
)
K = 4**np.arange(1, 11)
a_stats, s_stats = map(np.array, zip(*(compute_stats(k) for k in K)))
shape = np.array([2, 1.5]) * SQUARE_FIGSIZE
fig, axes = pl.subplots(2, 2, sharex=True, figsize=shape)
for i, (ax, title) in enumerate(
zip(axes.flat, ("mean", "variance", "skewness", "kurtosis"))):
mu = a_stats[0, i]
ax.axhline(mu, color=COLORS["DATA"])
ax.plot(np.log2(K), s_stats[:, i], "-o", color=COLORS["MODEL_1"], ms=3)
d = np.max(np.abs(np.array(ax.get_ylim()) - mu))
ax.set_ylim(mu + d * np.array([-1, 1]))
ax.set_ylabel(title)
ax.yaxis.set_label_coords(-0.3, 0.5)
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.xaxis.set_major_locator(pl.MaxNLocator(6))
[ax.set_xlabel(r"$\log_2 K$") for ax in axes[1]]
savefig(fig, "simple_stats.pdf")
raise RuntimeError("chain too short to estimate tau reliably")
if __name__ == "__main__":
from estimatetau import chain
m = len(chain)
N = 2 ** np.arange(2, 19)
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
for s in [slice(0, m//2), slice(m//4, 3*m//4), slice(m//2, m),
slice(None)]:
acf = autocorr_function(chain[s])
taus = np.array([autocorr_time_simple(acf, n) for n in N])
tau = autocorr_time_iterative(acf)
ax.plot(N, taus, "-", color=COLORS["DATA"])
print("tau: {0}".format(tau))
ax.plot(N, taus, ".-", color=COLORS["MODEL_2"])
ax.axhline(tau, color=COLORS["MODEL_2"], ls="dashed")
ax.set_title(r"$\tau_x = {0:.2f}$".format(tau))
ax.set_ylim(0, 25)
ax.set_xlim(2, 5e5)
ax.set_xscale("log")
ax.set_ylabel(r"$\tau_x$")
ax.set_xlabel("window size")
savefig(fig, "itertau.pdf")
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5*np.log(2*np.pi*variance)
neg_log_p = lambda x: -log_p(x)
r = minimize(neg_log_p, [1e3])
x = r.x[0]
lp = log_p(x)
chain = np.empty(6000)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(chain, color=COLORS["DATA"])
ax.xaxis.set_major_locator(pl.MaxNLocator(4))
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("step")
ax.set_ylabel("$x$")
savefig(fig, "optimization.pdf")
from plot_setup import setup, COLORS, SQUARE_FIGSIZE, savefig
setup() # initialize the plotting styles
np.random.seed(42)
x0 = [-4.0, 5.0]
s = SQUARE_FIGSIZE
s[0] *= 2
s[1] *= 0.8
fig, axes = pl.subplots(1, 3, sharex=True, sharey=True, figsize=s)
for n, ax in zip([0.0, -1.0, 2.0], axes):
chain, _ = run_mcmc(log_p_gauss, np.array(x0), nsteps=2e3,
prop_sigma=10**n)
ax.plot(chain[:, 0], chain[:, 1], "o-", color=COLORS["DATA"], ms=2)
ax.plot(x0[0], x0[1], "o", color=COLORS["MODEL_1"])
ax.set_xlim(-6.3, 6.3)
ax.set_ylim(-6.3, 6.3)
ax.set_xlabel("$x$")
ax.annotate(r"$\sigma_q = 10^{{{0:.0f}}}$".format(n),
(1, 0), xycoords="axes fraction",
xytext=(-5, 5), textcoords="offset points",
ha="right", va="bottom")
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.xaxis.set_major_locator(pl.MaxNLocator(5))
axes[0].set_ylabel("$y$")
savefig(fig, "MH_sigma.pdf")
np.random.seed(42)
x0 = [-4.0, 5.0]
s = SQUARE_FIGSIZE
s[0] *= 2
s[1] *= 0.8
fig, axes = pl.subplots(1, 3, sharex=True, sharey=True, figsize=s)
for n, ax in zip([0.0, -1.0, 2.0], axes):
chain, _ = run_mcmc(log_p_gauss,
np.array(x0),
nsteps=2e3,
prop_sigma=10**n)
ax.plot(chain[:, 0], chain[:, 1], "o-", color=COLORS["DATA"], ms=2)
ax.plot(x0[0], x0[1], "o", color=COLORS["MODEL_1"])
ax.set_xlim(-6.3, 6.3)
ax.set_ylim(-6.3, 6.3)
ax.set_xlabel("$x$")
ax.annotate(r"$\sigma_q = 10^{{{0:.0f}}}$".format(n), (1, 0),
xycoords="axes fraction",
xytext=(-5, 5),
textcoords="offset points",
ha="right",
va="bottom")
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.xaxis.set_major_locator(pl.MaxNLocator(5))
axes[0].set_ylabel("$y$")
savefig(fig, "MH_sigma.pdf")
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
if __name__ == "__main__":
s = SQUARE_FIGSIZE
s[0] *= 2
fig, (ax1, ax2) = pl.subplots(1, 2, figsize=s)
ax1.plot(np.arange(len(chain))/1e3, chain, color=COLORS["DATA"],
rasterized=True)
ax1.set_xlim(0, len(chain) / 1e3)
ax1.set_ylim(2-5.5, 2+5.5)
ax1.set_ylabel("$x$")
ax1.set_xlabel("thousands of steps")
for i in range(4):
a = i*len(chain)/4
b = (i+1)*len(chain)/4
print(i+1, np.mean(chain[a:b]), np.var(chain[a:b]))
ax2.hist(chain[a:b], 50, histtype="step")
ax2.set_xlim(2-5.5, 2+5.5)
ax2.set_yticklabels([])
ax2.set_ylabel("$p(x)$")
ax2.set_xlabel("$x$")
savefig(fig, "convergence.pdf", dpi=300)
delta = x[ind] - x_prime[ind]
r += -0.5 * ((delta - 0.5)**2 - (delta + 0.5)**2)
if np.random.rand() <= np.exp(r):
x[:] = x_prime
lp = lp_prime
chain[step] = x
return chain
x0 = [-4.0, 5.0]
s = SQUARE_FIGSIZE
s[0] *= 2
fig, axes = pl.subplots(1, 2, sharex=True, sharey=True, figsize=s)
for broken, ax in zip([True, False], axes):
chain = run_broken_mcmc(log_p_gauss, np.array(x0), broken=broken)
chain = np.concatenate(([x0], chain))
ax.plot(chain[:, 0], chain[:, 1], "o-", color=COLORS["DATA"], ms=2)
ax.plot(x0[0], x0[1], "o", color=COLORS["MODEL_1"])
ax.set_xlim(-6.3, 6.3)
ax.set_ylim(-6.3, 6.3)
ax.set_xlabel("$x$")
ax.annotate("incorrect acceptance" if broken else "corrected acceptance",
(1, 0), xycoords="axes fraction",
xytext=(-5, 5), textcoords="offset points",
ha="right", va="bottom")
axes[0].set_ylabel("$y$")
savefig(fig, "prop_mean.pdf")
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5 * np.log(2 * np.pi * variance)
neg_log_p = lambda x: -log_p(x)
r = minimize(neg_log_p, [1e3])
x = r.x[0]
lp = log_p(x)
chain = np.empty(6000)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(chain, color=COLORS["DATA"])
ax.xaxis.set_major_locator(pl.MaxNLocator(4))
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("step")
ax.set_ylabel("$x$")
savefig(fig, "optimization.pdf")
# -*- coding: utf-8 -*-
from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as pl
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
from twod_a import log_p_gauss, run_mcmc
setup() # initialize the plotting styles
np.random.seed(42)
sigs = 2.0 ** np.arange(-2, 6)
if __name__ == "__main__":
acc_fracs = np.empty_like(sigs)
for i, sig in enumerate(sigs):
_, acc_fracs[i] = run_mcmc(log_p_gauss, np.array([0.0, 0.0]),
prop_sigma=sig)
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.axhline(0.25, color=COLORS["DATA"])
ax.plot(sigs, acc_fracs, ".-", color=COLORS["MODEL_2"])
ax.set_xscale("log")
ax.set_xlabel(r"$\sigma_q$")
ax.set_ylabel("acceptance fraction")
ax.set_xlim(0.2, 40)
savefig(fig, "tuning.pdf")
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5*np.log(2*np.pi*variance)
x = 0.0
lp = log_p(x)
chain = np.empty(2e4)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.hist(chain, 100, histtype="step", color=COLORS["DATA"], normed=True)
x = 2 + np.linspace(-4.5, 4.5, 5000)
ax.plot(x, np.exp(log_p(x)), color=COLORS["MODEL_1"])
ax.set_xlim(x.min(), x.max())
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("$x$")
ax.set_ylabel("$p(x)$")
savefig(fig, "MH.pdf")
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5 * np.log(2 * np.pi * variance)
x = 0.0
lp = log_p(x)
chain = np.empty(2e4)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.hist(chain, 100, histtype="step", color=COLORS["DATA"], normed=True)
x = 2 + np.linspace(-4.5, 4.5, 5000)
ax.plot(x, np.exp(log_p(x)), color=COLORS["MODEL_1"])
ax.set_xlim(x.min(), x.max())
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("$x$")
ax.set_ylabel("$p(x)$")
savefig(fig, "MH.pdf")
#!/usr/bin/env python
# -*- coding: utf-8 -*-
from __future__ import division, print_function
import corner
import numpy as np
from twod_a import run_mcmc
from plot_setup import setup, savefig
setup() # initialize the plotting styles
np.random.seed(42)
def log_p_uniform(x):
if 3 <= x[0] <= 7 and 1 <= x[1] <= 9:
return 0.0
return -np.inf
chain, _ = run_mcmc(log_p_uniform, np.array([5.0, 5.0]), nsteps=1e5)
fig = corner.corner(chain, labels=["$x$", "$y$"],
range=[(2.5, 7.5), (0.5, 9.5)],
plot_density=False, plot_contours=False)
savefig(fig, "twod_b.pdf", dpi=300)
import numpy as np
import matplotlib.pyplot as pl
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
setup() # initialize the plotting styles
np.random.seed(42)
x = 0.0
chain = np.empty(1e6)
for step in range(len(chain)):
x += np.random.randn()
chain[step] = x
# Thin the chain for plotting purposes.
chain = chain[::1000]
mn, mx = np.min(chain), np.max(chain)
mid = 0.5 * (mn + mx)
rng = 0.5 * (mx - mn) * 1.1
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(np.arange(len(chain)) * 1e-2, chain, color=COLORS["DATA"])
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.set_xlabel("mcmc step [$\\times 10^5$]")
ax.set_ylabel("$x$")
ax.set_ylim(mid-rng, mid+rng)
savefig(fig, "improper.pdf")
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5 * np.log(2 * np.pi * variance)
x = 1e3
lp = log_p(x)
chain = np.empty(6000)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(chain, color=COLORS["DATA"])
ax.xaxis.set_major_locator(pl.MaxNLocator(4))
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("step")
ax.set_ylabel("$x$")
savefig(fig, "initialization.pdf")
def log_p(x, a=3.0, b=7.0):
if a <= x <= b:
return -np.log(b - a)
return -np.inf
x = 5.0
lp = log_p(x)
chain = np.empty(2e4)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.axhline(0.0, color="k")
ax.hist(chain, 50, histtype="step", color=COLORS["DATA"], normed=True)
x = np.linspace(2.5, 7.5, 5000)
y = np.exp([log_p(_) for _ in x])
ax.plot(x, y, color=COLORS["MODEL_1"])
ax.set_xlim(x.min(), x.max())
ax.set_ylim(-0.01, y.max() + 0.05)
ax.set_xlabel("$x$")
ax.set_ylabel("$p(x)$")
savefig(fig, "MH2.pdf")
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5*np.log(2*np.pi*variance)
x = 1e3
lp = log_p(x)
chain = np.empty(6000)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(chain, color=COLORS["DATA"])
ax.xaxis.set_major_locator(pl.MaxNLocator(4))
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("step")
ax.set_ylabel("$x$")
savefig(fig, "initialization.pdf")
setup() # initialize the plotting styles
np.random.seed(42)
def log_p(x, mean=2.0, variance=2.0):
return -0.5 * (x - mean)**2 / variance - 0.5 * np.log(2 * np.pi * variance)
log_x = 0.0
lp = log_p(log_x)
chain = np.empty(2e6)
for step in range(len(chain)):
log_x_prime = log_x + np.random.randn()
lp_prime = log_p(log_x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
log_x = log_x_prime
lp = lp_prime
chain[step] = np.exp(log_x)
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
x = np.linspace(0, 20, 5000)
ax.plot(x, np.exp(log_p(np.log(x))) / x, color=COLORS["MODEL_1"])
ax.hist(chain, 10000, histtype="step", color=COLORS["DATA"], normed=True)
ax.set_xlim(x.min(), x.max())
ax.yaxis.set_major_locator(pl.MaxNLocator(4))
ax.set_xlabel("$x$")
ax.set_ylabel("$p(x)$")
savefig(fig, "logprior.pdf")
print("Warning: running naive autocorrelation function method. "
"This will be slow!")
mu = np.mean(x)
r = x - mu
C = np.empty(len(r) // 2)
for i in range(1, len(C)):
C[i] = np.mean(r[i:] * r[:-i])
C /= np.mean(r ** 2)
C[0] = 1.0
return C
def autocorr_function(x):
n = len(x)
f = np.fft.fft(x - np.mean(x), n=2*n)
acf = np.fft.ifft(f * np.conjugate(f))[:n].real
return acf / acf[0]
if __name__ == "__main__":
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
acf = autocorr_function(chain)
ax.plot(acf[:61])
ax.set_xlim(0, 60)
ax.set_ylabel("$C_x(\Delta)$")
ax.set_xlabel("$\Delta$")
savefig(fig, "estimatetau.pdf")
def log_p(x, a=3.0, b=7.0):
if a <= x <= b:
return -np.log(b-a)
return -np.inf
x = 5.0
lp = log_p(x)
chain = np.empty(2e4)
for step in range(len(chain)):
x_prime = x + np.random.randn()
lp_prime = log_p(x_prime)
if np.random.rand() <= np.exp(lp_prime - lp):
x = x_prime
lp = lp_prime
chain[step] = x
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.axhline(0.0, color="k")
ax.hist(chain, 50, histtype="step", color=COLORS["DATA"], normed=True)
x = np.linspace(2.5, 7.5, 5000)
y = np.exp([log_p(_) for _ in x])
ax.plot(x, y, color=COLORS["MODEL_1"])
ax.set_xlim(x.min(), x.max())
ax.set_ylim(-0.01, y.max() + 0.05)
ax.set_xlabel("$x$")
ax.set_ylabel("$p(x)$")
savefig(fig, "MH2.pdf")
x[:] = x_prime
lp = lp_prime
chain[step] = x
return chain
x0 = [-4.0, 5.0]
s = SQUARE_FIGSIZE
s[0] *= 2
fig, axes = pl.subplots(1, 2, sharex=True, sharey=True, figsize=s)
for broken, ax in zip([True, False], axes):
chain = run_broken_mcmc(log_p_gauss, np.array(x0), broken=broken)
chain = np.concatenate(([x0], chain))
ax.plot(chain[:, 0], chain[:, 1], "o-", color=COLORS["DATA"], ms=2)
ax.plot(x0[0], x0[1], "o", color=COLORS["MODEL_1"])
ax.set_xlim(-6.3, 6.3)
ax.set_ylim(-6.3, 6.3)
ax.set_xlabel("$x$")
ax.annotate("incorrect acceptance" if broken else "corrected acceptance",
(1, 0),
xycoords="axes fraction",
xytext=(-5, 5),
textcoords="offset points",
ha="right",
va="bottom")
axes[0].set_ylabel("$y$")
savefig(fig, "prop_mean.pdf")
s_skew = np.mean((x - s_mu)**3) / s_var**(3./2)
s_kurt = np.mean((x - s_mu)**4) / s_var**2
return (
(a_mu, a_var, a_skew, a_kurt),
(s_mu, s_var, s_skew, s_kurt),
)
K = 4**np.arange(1, 11)
a_stats, s_stats = map(np.array, zip(*(compute_stats(k) for k in K)))
shape = np.array([2, 1.5])*SQUARE_FIGSIZE
fig, axes = pl.subplots(2, 2, sharex=True, figsize=shape)
for i, (ax, title) in enumerate(zip(
axes.flat, ("mean", "variance", "skewness", "kurtosis"))):
mu = a_stats[0, i]
ax.axhline(mu, color=COLORS["DATA"])
ax.plot(np.log2(K), s_stats[:, i], "-o", color=COLORS["MODEL_1"], ms=3)
d = np.max(np.abs(np.array(ax.get_ylim()) - mu))
ax.set_ylim(mu + d * np.array([-1, 1]))
ax.set_ylabel(title)
ax.yaxis.set_label_coords(-0.3, 0.5)
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
ax.xaxis.set_major_locator(pl.MaxNLocator(6))
[ax.set_xlabel(r"$\log_2 K$") for ax in axes[1]]
savefig(fig, "simple_stats.pdf")
import matplotlib.pyplot as pl
from plot_setup import setup, SQUARE_FIGSIZE, COLORS, savefig
from itertau import autocorr_function, autocorr_time_iterative
from tuning import log_p_gauss, run_mcmc, sigs
setup() # initialize the plotting styles
np.random.seed(42)
taus = np.empty((len(sigs), 2))
for i, sig in enumerate(sigs):
chain, acc = run_mcmc(log_p_gauss,
np.array([0.0, 0.0]),
prop_sigma=sig,
nsteps=2e5)
for j in range(2):
acf = autocorr_function(chain[:, j])
taus[i, j] = autocorr_time_iterative(acf)
print(sig, acc, taus[i])
fig, ax = pl.subplots(1, 1, figsize=SQUARE_FIGSIZE)
ax.plot(sigs, taus[:, 0], ".-", color=COLORS["MODEL_1"])
ax.plot(sigs, taus[:, 1], ".-", color=COLORS["MODEL_2"])
ax.set_xscale("log")
ax.set_xlabel(r"$\sigma_q$")
ax.set_ylabel(r"$\tau_x$")
ax.set_xlim(0.2, 40)
ax.set_ylim(0, 450)
ax.yaxis.set_major_locator(pl.MaxNLocator(5))
savefig(fig, "tuningtau.pdf")
