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pymc_chap1.py
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import numpy as np
from matplotlib import pyplot as plt
import scipy.stats as stats
import pymc as pm
"""SECTION 1
dist = stats.beta
n_trials = [0, 1, 2, 3, 4, 5, 8, 15, 50, 500]
data = stats.bernoulli.rvs(0.5, size=n_trials[-1])
x = np.linspace(0, 1, 100)
# For the already prepared, I'm using Binomial's conj. prior.
for k, N in enumerate(n_trials):
sx = plt.subplot(len(n_trials) / 2, 2, k + 1)
plt.xlabel("$p$, probability of heads") \
if k in [0, len(n_trials) - 1] else None
plt.setp(sx.get_yticklabels(), visible=False)
heads = data[:N].sum()
y = dist.pdf(x, 1 + heads, 1 + N - heads)
plt.plot(x, y, label="observe %d tosses,\n %d heads" % (N, heads))
plt.fill_between(x, 0, y, color="#348ABD", alpha=0.4)
plt.vlines(0.5, 0, 4, color="k", linestyles="--", lw=1)
leg = plt.legend()
leg.get_frame().set_alpha(0.4)
plt.autoscale(tight=True)
plt.suptitle("Bayesian updating of posterior probabilities",
y=1.02,
fontsize=14)
plt.tight_layout()
plt.show()
"""
"""SECTION 2
p = np.linspace(0, 1, 50)
plt.plot(p, 2 * p / (1 + p), color="#348ABD", lw=3)
# plt.fill_between(p, 2*p/(1+p), alpha=.5, facecolor=["#A60628"])
plt.scatter(0.2, 2 * (0.2) / 1.2, s=140, c="#348ABD")
plt.xlim(0, 1)
plt.ylim(0, 1)
plt.xlabel("Prior, $P(A) = p$")
plt.ylabel("Posterior, $P(A|X)$, with $P(A) = p$")
plt.title("Is my code bug-free?")
plt.show()
"""
"""SECTION 3
colours = ["#348ABD", "#A60628"]
prior = [0.20, 0.80]
posterior = [1. / 3, 2. / 3]
plt.bar([0, .7], prior, alpha=0.70, width=0.25,
color=colours[0], label="prior distribution",
lw="3", edgecolor=colours[0])
plt.bar([0 + 0.25, .7 + 0.25], posterior, alpha=0.7,
width=0.25, color=colours[1],
label="posterior distribution",
lw="3", edgecolor=colours[1])
plt.ylim(0,1)
plt.xticks([0.20, .95], ["Bugs Absent", "Bugs Present"])
plt.title("Prior and Posterior probability of bugs present")
plt.ylabel("Probability")
plt.legend(loc="upper left");
plt.show()
"""
"""SECTION 4
a = np.arange(16)
poi = stats.poisson
lambda_ = [1.5, 4.25]
colours = ["#348ABD", "#A60628"]
plt.bar(a, poi.pmf(a, lambda_[0]), color=colours[0],
label="$\lambda = %.1f$" % lambda_[0], alpha=0.60,
edgecolor=colours[0], lw="3")
plt.bar(a, poi.pmf(a, lambda_[1]), color=colours[1],
label="$\lambda = %.1f$" % lambda_[1], alpha=0.60,
edgecolor=colours[1], lw="3")
plt.xticks(a + 0.4, a)
plt.legend()
plt.ylabel("probability of $k$")
plt.xlabel("$k$")
plt.title("Probability mass function of a Poisson random variable; differing $\lambda$ values")
plt.show()
"""
"""SECTION 5
colours = ["#348ABD", "#A60628"]
a = np.linspace(0, 4, 100)
expo = stats.expon
lambda_ = [0.5, 1]
for l, c in zip(lambda_, colours):
plt.plot(a, expo.pdf(a, scale=1. / l), lw=3,
color=c, label="$\lambda = %.1f$" % l)
plt.fill_between(a, expo.pdf(a, scale=1. / l), color=c, alpha=.33)
plt.legend()
plt.ylabel("PDF at $z$")
plt.xlabel("$z$")
plt.ylim(0, 1.2)
plt.title("Probability density function of an Exponential random variable; differing $\lambda$")
plt.show()
"""
count_data = np.loadtxt("txtdata.csv")
n_count_data = len(count_data)
"""SECTION 5
plt.bar(np.arange(n_count_data), count_data, color="#348ABD")
plt.xlabel("Time (days)")
plt.ylabel("count of text-msgs received")
plt.title("Did the user's texting habits change over time?")
plt.xlim(0, n_count_data)
plt.show()
"""
alpha = 1.0 / count_data.mean() # Recall count_data is the
# variable that holds our txt counts
lambda_1 = pm.Exponential("lambda_1", alpha)
lambda_2 = pm.Exponential("lambda_2", alpha)
tau = pm.DiscreteUniform("tau", lower=0, upper=n_count_data)
print("Random output:", tau.random(), tau.random(), tau.random())
@pm.deterministic
def lambda_(tau=tau, lambda_1=lambda_1, lambda_2=lambda_2):
out = np.zeros(n_count_data)
out[:tau] = lambda_1 # lambda before tau is lambda1
out[tau:] = lambda_2 # lambda after (and including) tau is lambda2
return out
observation = pm.Poisson("obs", lambda_, value=count_data, observed=True)
model = pm.Model([observation, lambda_1, lambda_2, tau])
# Mysterious code to be explained in Chapter 3.
mcmc = pm.MCMC(model)
mcmc.sample(40000, 10000, 1)
lambda_1_samples = mcmc.trace('lambda_1')[:]
lambda_2_samples = mcmc.trace('lambda_2')[:]
tau_samples = mcmc.trace('tau')[:]
ax = plt.subplot(311)
ax.set_autoscaley_on(False)
plt.hist(lambda_1_samples, histtype='stepfilled', bins=30, alpha=0.85,
label="posterior of $\lambda_1$", color="#A60628", normed=True)
plt.legend(loc="upper left")
plt.title(r"""Posterior distributions of the variables
$\lambda_1,\;\lambda_2,\;\tau$""")
plt.xlim([15, 30])
plt.xlabel("$\lambda_1$ value")
ax = plt.subplot(312)
ax.set_autoscaley_on(False)
plt.hist(lambda_2_samples, histtype='stepfilled', bins=30, alpha=0.85,
label="posterior of $\lambda_2$", color="#7A68A6", normed=True)
plt.legend(loc="upper left")
plt.xlim([15, 30])
plt.xlabel("$\lambda_2$ value")
plt.subplot(313)
w = 1.0 / tau_samples.shape[0] * np.ones_like(tau_samples)
plt.hist(tau_samples, bins=n_count_data, alpha=1,
label=r"posterior of $\tau$",
color="#467821", weights=w, rwidth=2.)
plt.xticks(np.arange(n_count_data))
plt.legend(loc="upper left")
plt.ylim([0, .75])
plt.xlim([35, len(count_data) - 20])
plt.xlabel(r"$\tau$ (in days)")
plt.ylabel("probability");
plt.show()
plt.figure()
# tau_samples, lambda_1_samples, lambda_2_samples contain
# N samples from the corresponding posterior distribution
N = tau_samples.shape[0]
expected_texts_per_day = np.zeros(n_count_data)
for day in range(0, n_count_data):
# ix is a bool index of all tau samples corresponding to
# the switchpoint occurring prior to value of 'day'
ix = day < tau_samples
# Each posterior sample corresponds to a value for tau.
# for each day, that value of tau indicates whether we're "before"
# (in the lambda1 "regime") or
# "after" (in the lambda2 "regime") the switchpoint.
# by taking the posterior sample of lambda1/2 accordingly, we can average
# over all samples to get an expected value for lambda on that day.
# As explained, the "message count" random variable is Poisson distributed,
# and therefore lambda (the poisson parameter) is the expected value of
# "message count".
expected_texts_per_day[day] = (lambda_1_samples[ix].sum()
+ lambda_2_samples[~ix].sum()) / N
plt.plot(range(n_count_data), expected_texts_per_day, lw=4, color="#E24A33",
label="expected number of text-messages received")
plt.xlim(0, n_count_data)
plt.xlabel("Day")
plt.ylabel("Expected # text-messages")
plt.title("Expected number of text-messages received")
plt.ylim(0, 60)
plt.bar(np.arange(len(count_data)), count_data, color="#348ABD", alpha=0.65,
label="observed texts per day")
plt.legend(loc="upper left");
plt.show()