def main():
# The parameters are the bounds of the Uniform.
p = pm.Uniform('p', lower=0, upper=1)
# set constants
p_true = 0.05 # remember, this is unknown.
N = 1500
# sample N Bernoulli random variables from Ber(0.05).
# each random variable has a 0.05 chance of being a 1.
# this is the data-generation step
occurrences = pm.rbernoulli(p_true, N)
print occurrences
print occurrences.sum()
# Occurrences.mean is equal to n/N.
print "What is the observed frequency in Group A? %.4f" % occurrences.mean()
print "Does this equal the true frequency? %s" % (occurrences.mean() == p_true)
# include the observations, which are Bernoulli
obs = pm.Bernoulli("obs", p, value=occurrences, observed=True)
# To be explained in chapter 3
mcmc = pm.MCMC([p, obs])
mcmc.sample(18000, 1000)
plt.title("Posterior distribution of $p_A$, the true effectiveness of site A")
plt.vlines(p_true, 0, 90, linestyle="--", label="true $p_A$ (unknown)")
plt.hist(mcmc.trace("p")[:], bins=25, histtype="stepfilled", normed=True)
plt.legend()
plt.show()
def ages_and_data(N_exam, f_samp, correction_factor_array, age_lims):
"""Called by pred_samps. Simulates ages of survey participants and data given f."""
N_samp = len(f_samp)
N_age_samps = correction_factor_array.shape[1]
# Get samples for the age distribution at the observation points.
age_distribution = []
for i in xrange(N_samp):
l = age_lims[i]
age_distribution.append(S_trace[np.random.randint(S_trace.shape[0]), 0,
l[0]:l[1] + 1])
age_distribution[-1] /= np.sum(age_distribution[-1])
# Draw age for each individual, draw an age-correction profile for each location,
# compute probability of positive for each individual, see how many individuals are
# positive.
A = []
pos = []
for s in xrange(N_samp):
A.append(
np.array(pm.rcategorical(age_distribution[s], size=N_exam[s]),
dtype=int) + age_lims[s][0])
P_samp = pm.invlogit(f_samp[s].ravel(
)) * correction_factor_array[:, np.random.randint(N_age_samps)][A[-1]]
pos.append(pm.rbernoulli(P_samp))
return A, pos, age_distribution
def main():
# The parameters are the bounds of the Uniform.
p = pm.Uniform('p', lower=0, upper=1)
# set constants
p_true = 0.05 # remember, this is unknown.
N = 1500
# sample N Bernoulli random variables from Ber(0.05).
# each random variable has a 0.05 chance of being a 1.
# this is the data-generation step
occurrences = pm.rbernoulli(p_true, N)
print occurrences
print occurrences.sum()
# Occurrences.mean is equal to n/N.
print "What is the observed frequency in Group A? %.4f" % occurrences.mean(
)
print "Does this equal the true frequency? %s" % (occurrences.mean()
== p_true)
# include the observations, which are Bernoulli
obs = pm.Bernoulli("obs", p, value=occurrences, observed=True)
# To be explained in chapter 3
mcmc = pm.MCMC([p, obs])
mcmc.sample(18000, 1000)
plt.title(
"Posterior distribution of $p_A$, the true effectiveness of site A")
plt.vlines(p_true, 0, 90, linestyle="--", label="true $p_A$ (unknown)")
plt.hist(mcmc.trace("p")[:], bins=25, histtype="stepfilled", normed=True)
plt.legend()
plt.show()
def dice(data=None):
if data is None:
x = [pymc.rbernoulli(1.0 / 6.0) for i in range(0, 100)]
else:
x = data
prob = pymc.Uniform('prob', lower=0, upper=1)
d = pymc.Bernoulli('d', p=prob, value=x, observed=True)
return locals()
def dice(data=None):
if data is None:
x = [pymc.rbernoulli(1.0 / 6.0) for i in range(0, 100)]
else:
x = data
prob = pymc.Uniform('prob', lower=0, upper=1)
d = pymc.Bernoulli('d', p=prob, value=x, observed=True)
return locals()
def DoSamplingXPrior(self, w, a, M=None):
if M is not None:
xLen = M
else:
try:
xLast = self.xSeq[-1]
xLen = xLast.size
except:
raise UnboundLocalError('Unable to determine length of x')
assert (w >= 0) and (w <= 1), __file__ + ': DoSamplingXPrior: w is out of bounds'
bIsZero = pymc.rbernoulli(w, (xLen, 1))
FLaplaceDistInv = lambda x: (-a * np.sign(x - 0.5) * np.log(1 - 2 * np.abs(x - 0.5)))
plazeSample = FLaplaceDistInv(0.5 + 0.5 * np.random.uniform(size=(xLen, 1)))
xSample = bIsZero * plazeSample
return xSample
def Main():
p = pm.Uniform('p', lower=0, upper=1)
# Define true parameters for experimental purposes
p_true = 0.05
N = 1_500
# Generate fake Data using parameters
occurrences = pm.rbernoulli(p_true, N)
print(occurrences)
print(len(occurrences), occurrences.sum())
# Define observation random variavle
obs = pm.Bernoulli("obs", p, value=occurrences, observed=True)
# Solve using MCMC
mcmc = pm.MCMC([p, obs])
mcmc.sample(18_000, 1_000)
plot_posteriors(mcmc, p_true)
def trees_to_diagnostics(brt_evaluator, fname, species_name, n_pseudopresences, n_pseudoabsences, config_filename):
"""
Takes the BRT evaluator and sees how well it does at predicting the training dataset.
"""
from diagnostics import simple_assessments, roc, plot_roc_
din = csv2rec(os.path.join('anopheles-caches',fname))
found = din.found
din = dict([(k,din[k]) for k in brt_evaluator.nice_tree_dict.iterkeys()])
probs = pm.flib.invlogit(brt_evaluator(din))
print 'Species %s: fraction %f correctly classified.'%(species_name, ((probs>.5)*found+(probs<.5)*(True-found)).sum()/float(len(probs)))
result_dirname = get_result_dir(config_filename)
resdict = {}
for f in simple_assessments:
resdict[f.__name__] = f(probs>.5, found)
pstack = np.array([pm.rbernoulli(probs) for i in xrange(10000)])
fp, tp, AUC = roc(pstack, found)
resdict['AUC'] = AUC
fout=file(os.path.join(result_dirname,'simple-diagnostics.txt'),'w')
fout.write('presences: %i\n'%(found.sum()-n_pseudopresences))
fout.write('pseudopresences: %i\n'%n_pseudopresences)
fout.write('pseudoabsences: %i\n'%n_pseudoabsences)
for k in resdict.iteritems():
fout.write('%s: %s\n'%k)
import pylab as pl
pl.clf()
plot_roc_(fp,tp,AUC)
pl.savefig(os.path.join(result_dirname,'roc.pdf'))
r = np.rec.fromarrays([fp,tp],names='false,true')
rec2csv(r,os.path.join(result_dirname,'roc.csv'))
def ages_and_data(N_exam, f_samp, correction_factor_array, age_lims):
"""Called by pred_samps. Simulates ages of survey participants and data given f."""
N_samp = len(f_samp)
N_age_samps = correction_factor_array.shape[1]
# Get samples for the age distribution at the observation points.
age_distribution = []
for i in xrange(N_samp):
l = age_lims[i]
age_distribution.append(S_trace[np.random.randint(S_trace.shape[0]),0,l[0]:l[1]+1])
age_distribution[-1] /= np.sum(age_distribution[-1])
# Draw age for each individual, draw an age-correction profile for each location,
# compute probability of positive for each individual, see how many individuals are
# positive.
A = []
pos = []
for s in xrange(N_samp):
A.append(np.array(pm.rcategorical(age_distribution[s], size=N_exam[s]),dtype=int) + age_lims[s][0])
P_samp = pm.invlogit(f_samp[s].ravel())*correction_factor_array[:,np.random.randint(N_age_samps)][A[-1]]
pos.append(pm.rbernoulli(P_samp))
return A, pos, age_distribution
#--------------------------------------------
# main code
if __name__ == "__main__":
VERBOSE = 0
# measure times
start_time = time.time()
#-----------------------------------
# prior probability
p = pm.Uniform("p", lower=0, upper=1)
# initialize constants
p_true = 0.5
N = 1500
# sample N Bernoulli random variables frpm Ber(0.05)
# Each random variable has a 0.05 chance of being a 1.
# This is the data=generation step.
occurrences = pm.rbernoulli(p_true, N)
print("numbers of 1 = {0}".format(occurrences.sum()))
#-----------------------------------
# observed frequency
print("What is the observed frequency in Group A? %.4f" %
occurrences.mean())
print("Does the observed frequency equal the true frequency? {0}".format(
occurrences.mean() == p_true))
#-----------------------------------
# apply Bayesian method
# Include the observations, which are Bernoulli.
obs = pm.Bernoulli("obs", p, value=occurrences, observed=True)
# to be explained in Chapter 3
mcmc = pm.MCMC([p, obs])
mcmc.sample(20000, 1000)
#
# p_A_true and p_B_true are the **true observed values** of A and B.
#
# We are now simulating values from a bernoulli distribution with the values of **p** as A_true and B_true
# In[22]:
# true value of p_A and p_B (unknown)
p_A_true = 0.05 #Probability of click through rates in set up A
p_B_true = 0.04 #Probability of click through rates in set up B
# number of users visiting page A and B
N_A = 1500
N_B = 700
occurrences_A = pm.rbernoulli(p_A_true, N_A)
occurrences_B = pm.rbernoulli(p_B_true, N_B)
print('Observed frequency for A:', sum(occurrences_A))
print('Observed frequency for B:', sum(occurrences_B))
# Now we will define our prior distributions, which is a Uniform distribution. This implies that we don't have any prior imformation. We also capture the difference between the probabilities of A and B
# In[18]:
p_A = pm.Uniform('p_A', lower=0, upper=1)
p_B = pm.Uniform('p_B', lower=0, upper=1)
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
from matplotlib import pyplot
from pylab import savefig
# A/B testing of users purchasing from two different website frontends
# assign prior distribs to unknowns. Let's assume `p` is uniform over [0, 1],
# since we have no strong conviction about it.
p = pm.Uniform('p', lower=0, upper=1)
# set constants
p_true = 0.05 # this is actually unknown, we are just testing
N = 15000 # users
# sample n bernoulli random variables from Ber(0.05)
# each random var has a 0.05 chance of being a 1!
# this is the data-generation step
occurrences = pm.rbernoulli(p_true, N)
print(occurrences)
print(occurrences.sum()) # True == 1; False == 0
print(occurrences.sum()/N) # True == 1; False == 0
print("What is the observed frequency in Group A? %.4f" % occurrences.mean())
print("Does this equal the observed frequency? %s" % (occurrences.mean() == p_true))
# combine observations into the pymc observed variable:
obs = pm.Bernoulli("obs", p, value=occurrences, observed=True)
# then run inference algorithm (monte carlo simulation?)
mcmc = pm.MCMC([p, obs])
mcmc.sample(18000, 1000)
plot_roc_(*results['roc'])
elif k in ['producer_accuracy','consumer_accuracy']:
pl.hist(results[k][:,0])
pl.title('%s, false')
pl.figure()
pl.hist(results[k][:,1])
pl.title('%s, true'%k)
else:
pl.hist(results[k])
pl.title(k)
plot_validation = compose(plot_validation_, validate)
if __name__ == '__main__':
import pymc as pm
import pylab as pl
n = 1000
a = pm.rbernoulli(.7,size=n).astype('bool')
p = pm.rbernoulli(.7,size=n).astype('bool')
for s in simple_assessments:
print s.__name__, s(p, a)
ps = pm.rbernoulli(.7,size=(100,n)).astype('bool')
for i in xrange(100):
if np.random.random()<.05:
ps[i,:]=a
pl.clf()
fp,tp,auc = roc(ps,a)
plot_roc(ps, a)
# simple AB
import matplotlib.pyplot as plt
import pymc as pm
p = pm.Uniform('p', lower = 0, upper = 1)
# setting constants
p_true = 0.05
N = 1500
# sampling N bern(0.05)
data = pm.rbernoulli(p_true, N)
print(data)
print(data.sum())
print("\nmean: ", data.mean())
print("\nObserved prop equal p_true. ", data.mean() == p_true)
obs = pm.Bernoulli("obs", p, value = data , observed = True)
# yay a quick mcmc
mcmc = pm.MCMC([p, obs])
mcmc.sample(20000, 1000)
plt.title("possible values for the true effectiveness of version A")
plt.vlines(p_true, 0, 90, linestyle = '--', label = "true $p_$A (not known)")
plt.hist( mcmc.trace("p")[:] , bins = 35, histtype = "stepfilled", normed = True)
import pymc as pm
import matplotlib.pyplot as plt
true_p_A = 0.05
true_p_B = 0.04
N_A = 1500
N_B = 750
A_data = pm.rbernoulli(true_p_A, N_A)
B_data = pm.rbernoulli(true_p_B, N_B)
print("Mean of A: " , A_data.mean())
print("Mean of B: " , B_data.mean())
# priors
p_A = pm.Uniform("p_A", 0, 1)
p_B = pm.Uniform("p_B", 0, 1)
@pm.deterministic
def delta(p_A = p_A , p_B = p_B):
return p_A - p_B
data_A = pm.Bernoulli("data_A", p_A , value = A_data, observed = True)
data_B = pm.Bernoulli("data_B", p_B , value = B_data, observed = True)
# mcmc
mcmc = pm.MCMC([p_A, p_B, delta, data_A, data_B])
mcmc.sample(25000, 5000)
p_A_samples = mcmc.trace("p_A")[:]
def validate_rate_model(rate_type='neg_binom', data_type='epilepsy', replicate=0):
# set random seed for reproducibility
mc.np.random.seed(1234567 + replicate)
# load data
model = dismod3.data.load('/home/j/Project/dismod/output/dm-32377/')
data = model.get_data('p')
#data = data.ix[:20, :]
# replace data with synthetic data if requested
if data_type == 'epilepsy':
# no replacement needed
pass
elif data_type == 'schiz':
import pandas as pd
data = pd.read_csv('/homes/abie/gbd_dev/gbd/tests/schiz.csv')
elif data_type == 'binom':
N = 1.e6
data['effective_sample_size'] = N
mu = data['value'].mean()
data['value'] = mc.rbinomial(N, mu, size=len(data.index)) / N
elif data_type == 'poisson':
N = 1.e6
data['effective_sample_size'] = N
mu = data['value'].mean()
data['value'] = mc.rpoisson(N*mu, size=len(data.index)) / N
elif data_type == 'normal':
mu = data['value'].mean()
sigma = .125*mu
data['standard_error'] = sigma
data['value'] = mc.rnormal(mu, sigma**-2, size=len(data.index))
elif data_type == 'log_normal':
mu = data['value'].mean()
sigma = .25
data['standard_error'] = sigma*mu
data['value'] = pl.exp(mc.rnormal(pl.log(mu), sigma**-2, size=len(data.index)))
else:
raise TypeError, 'Unknown data type "%s"' % data_type
# sample prevalence data
i_test = mc.rbernoulli(.25, size=len(data.index))
i_nan = pl.isnan(data['effective_sample_size'])
data['lower_ci'] = pl.nan
data['upper_ci'] = pl.nan
data.ix[i_nan, 'effective_sample_size'] = 0.
data['standard_error'] = pl.sqrt(data['value']*(1-data['value'])) / data['effective_sample_size']
data.ix[pl.isnan(data['standard_error']), 'standard_error'] = pl.inf
data['standard_error'][i_test] = pl.inf
data['effective_sample_size'][i_test] = 0.
data['value'] = pl.maximum(data['value'], 1.e-12)
model.input_data = data
# create model
# TODO: set parameters in model.parameters['p'] dict
# then have simple method to create age specific rate model
#model.parameters['p'] = ...
#model.vars += dismod3.ism.age_specific_rate(model, 'p')
model.parameters['p']['parameter_age_mesh'] = [0,100]
model.parameters['p']['heterogeneity'] = 'Very'
model.vars['p'] = dismod3.data_model.data_model(
'p', model, 'p',
'all', 'total', 'all',
None, None, None,
rate_type=rate_type,
interpolation_method='zero',
include_covariates=False)
# add upper bound on sigma in log normal model to help convergence
#if rate_type == 'log_normal':
# model.vars['p']['sigma'].parents['upper'] = 1.5
# add upper bound on sigma, zeta in offset log normal
#if rate_type == 'offset_log_normal':
# model.vars['p']['sigma'].parents['upper'] = .1
# model.vars['p']['p_zeta'].value = 5.e-9
# model.vars['p']['p_zeta'].parents['upper'] = 1.e-8
# fit model
dismod3.fit.fit_asr(model, 'p', iter=20000, thin=10, burn=10000)
#dismod3.fit.fit_asr(model, 'p', iter=100, thin=1, burn=0)
# compare estimate to hold-out
data['mu_pred'] = model.vars['p']['p_pred'].stats()['mean']
data['lb_pred'] = model.vars['p']['p_pred'].stats()['95% HPD interval'][:,0]
data['ub_pred'] = model.vars['p']['p_pred'].stats()['95% HPD interval'][:,1]
import data_simulation
model.test = data[i_test]
data = model.test
data['true'] = data['value']
data_simulation.add_quality_metrics(data)
data_simulation.initialize_results(model)
data_simulation.add_to_results(model, 'test')
data_simulation.finalize_results(model)
return model
for i in range(4):
plt.subplot( 4, 1, i )
plot_artificial_sms_dataset()
plt.show()
#
# A/B testing
#
#
# A only
p = pm.Uniform( 'p', lower=0, upper=1 )
p_true = 0.05
N = 1500
occurrences = pm.rbernoulli( p_true, N )
occurrences.sum()
print "observed frequency: %.4f" % occurrences.mean()
# run inference alg
obs = pm.Bernoulli( "obs", p, value=occurrences, observed=True )
mcmc = pm.MCMC( [p, obs] )
mcmc.sample( 18000, 1000 )
plt.title( 'Posterior Dist of $p_A$: the true effectiveness of site A' )
plt.vlines( p_true, 0, 90, linestyles="--", label="true $p_A$ (unknown)" )
plt.hist( mcmc.trace( "p" )[:], bins=25, histtype="stepfilled", normed=True )
plt.legend()
plt.show()
import pymc as pm
from IPython.core.pylabtools import figsize
from matplotlib import pyplot as plt
p = pm.Uniform('p', lower=0, upper=1)
p_ture = 0.05 # unknown in reality
N = 1500
# sample N Bernoulli random variables from Ber(0.05)
# each random variable has a 0.05 chance of being a 1.
# this is the data-generation step
occurrences = pm.rbernoulli(p_ture, N)
print occurrences
print occurrences.sum()
print "What is the observed frequency in Group A? %.4f" % occurrences.mean()
print "Does this equal the true frequency? %s" % (occurrences.mean() == p_ture)
# include the boservations, which are Bernoulli
obs = pm.Bernoulli('obs', p, value=occurrences, observed=True)
mcmc = pm.MCMC([p, obs])
mcmc.sample(18000, 1000)
figsize(12.5, 4)
plt.title("Posterior distribution of $p_A$, the true effectiveness of site A")
plt.vlines(p_ture, 0, 90, linestyle="--", label="true $p_A$ (unknown)")
plt.hist(mcmc.trace('p')[:], bins=25, histtype="stepfilled", normed=True)
plt.legend()
def validate_rate_model(rate_type='neg_binom',
data_type='epilepsy',
replicate=0):
# set random seed for reproducibility
mc.np.random.seed(1234567 + replicate)
# load data
model = dismod3.data.load('/home/j/Project/dismod/output/dm-32377/')
data = model.get_data('p')
#data = data.ix[:20, :]
# replace data with synthetic data if requested
if data_type == 'epilepsy':
# no replacement needed
pass
elif data_type == 'schiz':
import pandas as pd
data = pd.read_csv('/homes/abie/gbd_dev/gbd/tests/schiz.csv')
elif data_type == 'binom':
N = 1.e6
data['effective_sample_size'] = N
mu = data['value'].mean()
data['value'] = mc.rbinomial(N, mu, size=len(data.index)) / N
elif data_type == 'poisson':
N = 1.e6
data['effective_sample_size'] = N
mu = data['value'].mean()
data['value'] = mc.rpoisson(N * mu, size=len(data.index)) / N
elif data_type == 'normal':
mu = data['value'].mean()
sigma = .125 * mu
data['standard_error'] = sigma
data['value'] = mc.rnormal(mu, sigma**-2, size=len(data.index))
elif data_type == 'log_normal':
mu = data['value'].mean()
sigma = .25
data['standard_error'] = sigma * mu
data['value'] = pl.exp(
mc.rnormal(pl.log(mu), sigma**-2, size=len(data.index)))
else:
raise TypeError, 'Unknown data type "%s"' % data_type
# sample prevalence data
i_test = mc.rbernoulli(.25, size=len(data.index))
i_nan = pl.isnan(data['effective_sample_size'])
data['lower_ci'] = pl.nan
data['upper_ci'] = pl.nan
data.ix[i_nan, 'effective_sample_size'] = 0.
data['standard_error'] = pl.sqrt(
data['value'] * (1 - data['value'])) / data['effective_sample_size']
data.ix[pl.isnan(data['standard_error']), 'standard_error'] = pl.inf
data['standard_error'][i_test] = pl.inf
data['effective_sample_size'][i_test] = 0.
data['value'] = pl.maximum(data['value'], 1.e-12)
model.input_data = data
# create model
# TODO: set parameters in model.parameters['p'] dict
# then have simple method to create age specific rate model
#model.parameters['p'] = ...
#model.vars += dismod3.ism.age_specific_rate(model, 'p')
model.parameters['p']['parameter_age_mesh'] = [0, 100]
model.parameters['p']['heterogeneity'] = 'Very'
model.vars['p'] = dismod3.data_model.data_model(
'p',
model,
'p',
'all',
'total',
'all',
None,
None,
None,
rate_type=rate_type,
interpolation_method='zero',
include_covariates=False)
# add upper bound on sigma in log normal model to help convergence
#if rate_type == 'log_normal':
# model.vars['p']['sigma'].parents['upper'] = 1.5
# add upper bound on sigma, zeta in offset log normal
#if rate_type == 'offset_log_normal':
# model.vars['p']['sigma'].parents['upper'] = .1
# model.vars['p']['p_zeta'].value = 5.e-9
# model.vars['p']['p_zeta'].parents['upper'] = 1.e-8
# fit model
dismod3.fit.fit_asr(model, 'p', iter=20000, thin=10, burn=10000)
#dismod3.fit.fit_asr(model, 'p', iter=100, thin=1, burn=0)
# compare estimate to hold-out
data['mu_pred'] = model.vars['p']['p_pred'].stats()['mean']
data['lb_pred'] = model.vars['p']['p_pred'].stats()['95% HPD interval'][:,
0]
data['ub_pred'] = model.vars['p']['p_pred'].stats()['95% HPD interval'][:,
1]
import data_simulation
model.test = data[i_test]
data = model.test
data['true'] = data['value']
data_simulation.add_quality_metrics(data)
data_simulation.initialize_results(model)
data_simulation.add_to_results(model, 'test')
data_simulation.finalize_results(model)
return model
figsize(12, 4)
#--------------------------------------------
# main code
if __name__ == "__main__":
VERBOSE = 0
# measure times
start_time = time.time()
#-----------------------------------
# initialize constants
true_p_A = 0.05
true_p_B = 0.04
N_A = 1500
N_B = 750
# Generate some observations.
occurrences_A = pm.rbernoulli(true_p_A, N_A)
occurrences_B = pm.rbernoulli(true_p_B, N_B)
print("numbers of Site A = {0}".format(occurrences_A.sum()))
print("numbers of Site B = {0}".format(occurrences_B.sum()))
#-----------------------------------
# observed frequency
print("What is the observed frequency in Group A? %.4f" %
occurrences_A.mean())
print("What is the observed frequency in Group B? %.4f" %
occurrences_B.mean())
# Set up the PyMC model, Again assume Uniform priors for p_A and p_B.
p_A = pm.Uniform("p_A", 0, 1)
p_B = pm.Uniform("p_B", 0, 1)
#Define the deterministic delta function. This is our unknown of interest.
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
import pymc as pm
from IPython.core.pylabtools import figsize
from matplotlib import pyplot as plt
p = pm.Uniform('p', lower=0, upper=1)
p_ture = 0.05 # unknown in reality
N = 1500
# sample N Bernoulli random variables from Ber(0.05)
# each random variable has a 0.05 chance of being a 1.
# this is the data-generation step
occurrences = pm.rbernoulli(p_ture, N)
print occurrences
print occurrences.sum()
print "What is the observed frequency in Group A? %.4f" % occurrences.mean()
print "Does this equal the true frequency? %s" % (occurrences.mean() == p_ture)
# include the boservations, which are Bernoulli
obs = pm.Bernoulli('obs', p, value=occurrences, observed=True)
mcmc = pm.MCMC([p,obs])
mcmc.sample(18000, 1000)
figsize(12.5, 4)
plt.title("Posterior distribution of $p_A$, the true effectiveness of site A")
plt.vlines(p_ture, 0, 90, linestyle="--", label="true $p_A$ (unknown)")
plt.hist(mcmc.trace('p')[:], bins=25, histtype="stepfilled", normed=True)
plt.legend()
# ----------------------------------------------------------------------------------------------------------------------
# Define random variables for probs of A and B
p_A = pm.Uniform('p_A', lower=0, upper=1)
p_B = pm.Uniform('p_B', lower=0, upper=1)
# Define true parameters for experimental purposes
p_true_A = 0.05
p_true_B = 0.04
# Note: Unequal sample sizes are valid in bayesian analysis
N_A = 1_500
N_B = 750
# Generate fake Data using parameters
observations_A = pm.rbernoulli(p_true_A, N_A)
observations_B = pm.rbernoulli(p_true_B, N_B)
# Define observation random variavle
obs_A = pm.Bernoulli("obs_A", p_A, value=observations_A, observed=True)
obs_B = pm.Bernoulli("obs_B", p_B, value=observations_B, observed=True)
# ----------------------------------------------------------------------------------------------------------------------
# Define Functions
# ----------------------------------------------------------------------------------------------------------------------
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
return p_A - p_B
def validate_covariate_model_dispersion(N=1000, delta_true=.15, pi_true=.01, zeta_true=[.5, -.5, 0.]):
## generate simulated data
a = pl.arange(0, 100, 1)
pi_age_true = pi_true * pl.ones_like(a)
model = data.ModelData()
model.parameters['p']['parameter_age_mesh'] = [0, 100]
model.input_data = pandas.DataFrame(index=range(N))
initialize_input_data(model.input_data)
Z = mc.rbernoulli(.5, size=(N, len(zeta_true))) * 1.0
delta = delta_true * pl.exp(pl.dot(Z, zeta_true))
for i in range(len(zeta_true)):
model.input_data['z_%d'%i] = Z[:,i]
model.input_data['true'] = pi_true
model.input_data['effective_sample_size'] = mc.runiform(100, 10000, N)
n = model.input_data['effective_sample_size']
p = model.input_data['true']
model.input_data['value'] = mc.rnegative_binomial(n*p, delta*n*p) / n
## Then fit the model and compare the estimates to the truth
model.vars = {}
model.vars['p'] = data_model.data_model('p', model, 'p', 'all', 'total', 'all', None, None, None)
model.map, model.mcmc = fit_model.fit_data_model(model.vars['p'], iter=10000, burn=5000, thin=5, tune_interval=100)
graphics.plot_one_ppc(model.vars['p'], 'p')
graphics.plot_convergence_diag(model.vars)
pl.show()
model.input_data['mu_pred'] = model.vars['p']['p_pred'].stats()['mean']
model.input_data['sigma_pred'] = model.vars['p']['p_pred'].stats()['standard deviation']
add_quality_metrics(model.input_data)
model.zeta = pandas.DataFrame(index=model.vars['p']['Z'].columns)
model.zeta['true'] = zeta_true
model.zeta['mu_pred'] = model.vars['p']['zeta'].stats()['mean']
model.zeta['sigma_pred'] = model.vars['p']['zeta'].stats()['standard deviation']
add_quality_metrics(model.zeta)
print '\nzeta'
print model.zeta
model.delta = pandas.DataFrame(dict(true=[delta_true]))
model.delta['mu_pred'] = pl.exp(model.vars['p']['eta'].trace()).mean()
model.delta['sigma_pred'] = pl.exp(model.vars['p']['eta'].trace()).std()
add_quality_metrics(model.delta)
print 'delta'
print model.delta
print '\ndata prediction bias: %.5f, MARE: %.3f, coverage: %.2f' % (model.input_data['abs_err'].mean(),
pl.median(pl.absolute(model.input_data['rel_err'].dropna())),
model.input_data['covered?'].mean())
print 'effect prediction MAE: %.3f, coverage: %.2f' % (pl.median(pl.absolute(model.zeta['abs_err'].dropna())),
model.zeta.dropna()['covered?'].mean())
model.results = dict(param=[], bias=[], mare=[], mae=[], pc=[])
add_to_results(model, 'delta')
add_to_results(model, 'input_data')
add_to_results(model, 'zeta')
model.results = pandas.DataFrame(model.results, columns='param bias mae mare pc'.split())
return model
from __future__ import print_function
import matplotlib.pyplot as plt
import numpy as np
import pymc as pm
p_a = 0.05
p_b = 0.04
N_a = 1500
N_b = 7500
data_a = pm.rbernoulli(p_a, N_a)
data_b = pm.rbernoulli(p_b, N_b)
p_a = pm.Uniform("p_a", 0, 1)
obs_a = pm.Bernoulli("obs_a", p_a, value=data_a, observed=True)
p_b = pm.Uniform("p_b", 0, 1)
obs_b = pm.Bernoulli("obs_b", p_b, value=data_b, observed=True)
@pm.deterministic
def delta(p_a=p_a, p_b=p_b):
return p_a - p_b
mcmc = pm.MCMC([p_a, obs_a, p_b, obs_b, delta])
mcmc.sample(50000, 1000)
'''
print ( "mean:" + str(data.mean()) \
+ " true:" + str(p_t) \
+ " form data:" + str( mcmc.trace("p")[:].mean() )
def generate(self, N_A, N_B):
obsA = pymc.rbernoulli(self.true_pA, N_A)
obsB = pymc.rbernoulli(self.true_PB, N_B)
This is an example of using Bayesian A/B testing
"""
import pymc as pm
#these two quantities are unknown to us.
true_p_A = 0.05
true_p_B = 0.04
#notice the unequal sample sizes -- no problem in Bayesian analysis.
N_A = 1500
N_B = 1000
#generate data
observations_A = pm.rbernoulli(true_p_A, N_A)
observations_B = pm.rbernoulli(true_p_B, N_B)
#set up the pymc model. Again assume Uniform priors for p_A and p_B
p_A = pm.Uniform("p_A", 0, 1)
p_B = pm.Uniform("p_B", 0, 1)
#define the deterministic delta function. This is our unknown of interest.
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
return p_A - p_B
def main():
# these two quantities are unknown to us.
true_p_A = 0.05
true_p_B = 0.04
# notice the unequal sample sizes -- no problem in Bayesian analysis.
N_A = 1500
N_B = 750
# generate some observations
observations_A = pm.rbernoulli(true_p_A, N_A)
observations_B = pm.rbernoulli(true_p_B, N_B)
print observations_A.mean()
print observations_B.mean()
# Set up the pymc model. Again assume Uniform priors for p_A and p_B.
p_A = pm.Uniform("p_A", 0, 1)
p_B = pm.Uniform("p_B", 0, 1)
# Define the deterministic delta function. This is our unknown of interest.
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
return p_A - p_B
# Set of observations, in this case we have two observation datasets.
obs_A = pm.Bernoulli("obs_A", p_A, value=observations_A, observed=True)
obs_B = pm.Bernoulli("obs_B", p_B, value=observations_B, observed=True)
# To be explained in chapter 3.
mcmc = pm.MCMC([p_A, p_B, delta, obs_A, obs_B])
mcmc.sample(20000, 1000)
p_A_samples = mcmc.trace("p_A")[:]
p_B_samples = mcmc.trace("p_B")[:]
delta_samples = mcmc.trace("delta")[:]
# histogram of posteriors
ax = plt.subplot(311)
plt.xlim(0, .1)
plt.hist(p_A_samples, histtype='stepfilled', bins=25, alpha=0.85,
label="posterior of $p_A$", color="#A60628", normed=True)
plt.vlines(true_p_A, 0, 80, linestyle="--", label="true $p_A$ (unknown)")
plt.legend(loc="upper right")
plt.title("Posterior distributions of $p_A$, $p_B$, and delta unknowns")
ax = plt.subplot(312)
plt.xlim(0, .1)
plt.hist(p_B_samples, histtype='stepfilled', bins=25, alpha=0.85,
label="posterior of $p_B$", color="#467821", normed=True)
plt.vlines(true_p_B, 0, 80, linestyle="--", label="true $p_B$ (unknown)")
plt.legend(loc="upper right")
ax = plt.subplot(313)
plt.hist(delta_samples, histtype='stepfilled', bins=30, alpha=0.85,
label="posterior of delta", color="#7A68A6", normed=True)
plt.vlines(true_p_A - true_p_B, 0, 60, linestyle="--",
label="true delta (unknown)")
plt.vlines(0, 0, 60, color="black", alpha=0.2)
plt.legend(loc="upper right")
plt.show()
# Count the number of samples less than 0, i.e. the area under the curve
# before 0, represent the probability that site A is worse than site B.
print "Probability site A is WORSE than site B: %.3f" % \
(delta_samples < 0).mean()
print "Probability site A is BETTER than site B: %.3f" % \
(delta_samples > 0).mean()
def validate_covariate_model_dispersion(N=1000,
delta_true=.15,
pi_true=.01,
zeta_true=[.5, -.5, 0.]):
## generate simulated data
a = pl.arange(0, 100, 1)
pi_age_true = pi_true * pl.ones_like(a)
model = data.ModelData()
model.parameters['p']['parameter_age_mesh'] = [0, 100]
model.input_data = pandas.DataFrame(index=range(N))
initialize_input_data(model.input_data)
Z = mc.rbernoulli(.5, size=(N, len(zeta_true))) * 1.0
delta = delta_true * pl.exp(pl.dot(Z, zeta_true))
for i in range(len(zeta_true)):
model.input_data['z_%d' % i] = Z[:, i]
model.input_data['true'] = pi_true
model.input_data['effective_sample_size'] = mc.runiform(100, 10000, N)
n = model.input_data['effective_sample_size']
p = model.input_data['true']
model.input_data['value'] = mc.rnegative_binomial(n * p, delta * n * p) / n
## Then fit the model and compare the estimates to the truth
model.vars = {}
model.vars['p'] = data_model.data_model('p', model, 'p', 'all', 'total',
'all', None, None, None)
model.map, model.mcmc = fit_model.fit_data_model(model.vars['p'],
iter=10000,
burn=5000,
thin=5,
tune_interval=100)
graphics.plot_one_ppc(model.vars['p'], 'p')
graphics.plot_convergence_diag(model.vars)
pl.show()
model.input_data['mu_pred'] = model.vars['p']['p_pred'].stats()['mean']
model.input_data['sigma_pred'] = model.vars['p']['p_pred'].stats(
)['standard deviation']
add_quality_metrics(model.input_data)
model.zeta = pandas.DataFrame(index=model.vars['p']['Z'].columns)
model.zeta['true'] = zeta_true
model.zeta['mu_pred'] = model.vars['p']['zeta'].stats()['mean']
model.zeta['sigma_pred'] = model.vars['p']['zeta'].stats(
)['standard deviation']
add_quality_metrics(model.zeta)
print '\nzeta'
print model.zeta
model.delta = pandas.DataFrame(dict(true=[delta_true]))
model.delta['mu_pred'] = pl.exp(model.vars['p']['eta'].trace()).mean()
model.delta['sigma_pred'] = pl.exp(model.vars['p']['eta'].trace()).std()
add_quality_metrics(model.delta)
print 'delta'
print model.delta
print '\ndata prediction bias: %.5f, MARE: %.3f, coverage: %.2f' % (
model.input_data['abs_err'].mean(),
pl.median(pl.absolute(model.input_data['rel_err'].dropna())),
model.input_data['covered?'].mean())
print 'effect prediction MAE: %.3f, coverage: %.2f' % (
pl.median(pl.absolute(model.zeta['abs_err'].dropna())),
model.zeta.dropna()['covered?'].mean())
model.results = dict(param=[], bias=[], mare=[], mae=[], pc=[])
add_to_results(model, 'delta')
add_to_results(model, 'input_data')
add_to_results(model, 'zeta')
model.results = pandas.DataFrame(model.results,
columns='param bias mae mare pc'.split())
return model
def generate(self, N_A, N_B):
obsA = pymc.rbernoulli(self.true_pA, N_A)
obsB = pymc.rbernoulli(self.true_PB, N_B)
foos.value
samples = [data_generator.random() for _ in xrange(10000)]
plt.hist(samples, bins = 70, normed = True, histtype = "stepfilled")
plt.title("FooBar")
plt.xlim(0, 8)
plt.show()
# AB testing example ---------------
# Preliminary -- just A
p = pm.Uniform("p", lower = 0, upper = 1)
p_true = 0.05
N = 1500
occurrences = pm.rbernoulli(p_true, N)
obs = pm.Bernoulli("obs", p, value = occurrences, observed = True)
model = pm.Model([p, obs])
mcmc = pm.MCMC(model)
mcmc.sample(18000, 1000)
samples = mcmc.trace("p")[:]
# AB
true_p_A = 0.05
true_p_B = 0.04
n_A = 1500
n_B = 750
"""
This is an example of using Bayesian A/B testing
"""
import pymc as pm
# these two quantities are unknown to us.
true_p_A = 0.05
true_p_B = 0.04
# notice the unequal sample sizes -- no problem in Bayesian analysis.
N_A = 1500
N_B = 1000
# generate data
observations_A = pm.rbernoulli(true_p_A, N_A)
observations_B = pm.rbernoulli(true_p_B, N_B)
# set up the pymc model. Again assume Uniform priors for p_A and p_B
p_A = pm.Uniform("p_A", 0, 1)
p_B = pm.Uniform("p_B", 0, 1)
# define the deterministic delta function. This is our unknown of interest.
@pm.deterministic
def delta(p_A=p_A, p_B=p_B):
return p_A - p_B
def DoSamplingSpecificXConditionedAll(self, w, a, ind, fitErrExcludingInd, varLast, bLogDebug=False):
assert (a > 0) and (w >= 0) and (varLast >= 0), __file__ + ': DoSamplingSpecificXConditionedAll: invalid inputs'
etaIndSquared = varLast / self._hNormSquared[ind]
assert (etaIndSquared > 0), __file__ + ': etaIndSquared is not strictly positive'
dotProduct = np.sum(fitErrExcludingInd * self._h[:, ind])
muIndComponents = (dotProduct / self._hNormSquared[ind], -etaIndSquared / a)
muInd = sum(muIndComponents)
"""
When y is big, calculating uInd is a challenge since there are numerical issues. We're
trying to multiply a very small number (which equals 0 due to finite floating point
representation) and a very large number, which is prone to returning 0.
"""
y = -muInd / PlazeGibbsSamplerReconstructor.CONST_SQRT_2 / math.sqrt(etaIndSquared)
uInd = (w / a) * \
mpmath.sqrt(etaIndSquared) * PlazeGibbsSamplerReconstructor.CONST_SQRT_HALF_PI * mpmath.erfc(y) * \
mpmath.exp(y * y)
uIndFloat = float(uInd) # Convert to an ordinary Python float
assert (uIndFloat >= 0), __file__ + ': uIndFloat is negative: ' + str(uIndFloat) + \
' w=' + str(w) + \
' a=' + str(a) + \
' etaIndSquared=' + str(etaIndSquared) + \
' y=' + str(y) + \
' muIndComp=(' + str(muIndComponents[0]) + ',' + str(muIndComponents[1]) + ')' + \
' dotProduct=' + str(dotProduct) + \
' hNormSquared=' + str(self._hNormSquared[ind])
if uIndFloat == float('inf'):
wInd = 1
else:
wInd = uIndFloat / (uIndFloat + (1 - w))
if ((wInd < 0) or (wInd > 1)):
raise ValueError('uInd is {0} and wInd is {1}'.format(uInd, wInd))
if pymc.rbernoulli(wInd):
# With probability wInd, generate a sample from a truncated Gaussian r.v. (support (0,Inf))
# xSample = pymc.rtruncated_normal(muInd, 1/etaIndSquared, a=0)[0]
try:
xSample = NumericalHelper.RandomNonnegativeNormal(muInd, etaIndSquared)
except:
fmtString = "Caught exception at {0}: NNN({1}, {2}). Intm. calc.: {3}, {4}, {5}. Exception: {6}"
msg = fmtString.format(ind,
muInd, etaIndSquared,
muIndComponents[0],
muIndComponents[1],
varLast,
sys.exc_info()[0])
logging.error(msg)
xSample = 0; # XXX: Due to a numerical problem
else:
# Check the value of xSample
if (xSample < 0):
fmtString = "Invalid xSample at {0}: NNN({1}, {2}) ~> sample {3}. Intm. calc.: {4}, {5}, {6}"
logging.error(fmtString.format(ind,
muInd, etaIndSquared, xSample,
muIndComponents[0],
muIndComponents[1],
varLast))
# Don't throw an exception
# raise ValueError('xSample cannot be negative')
xSample = 0; # XXX: Also due to a numerical problem, but no exception raised
else:
# With probability (1-wInd), generate 0
xSample = 0
if bLogDebug:
fmtString = ' {0}/{1}: {2:.5e}, {3:.5f}={4:.5f}-{5:.5f}, {6:.5e}, {7:.5e}: {8:.5e}'
logging.debug(fmtString.format(self._samplerIter,
ind,
etaIndSquared,
muInd,
muIndComponents[0],
-muIndComponents[1],
uIndFloat,
wInd,
xSample))
return xSample
import pymc as pm
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
# A-B test
p = pm.Uniform('p', lower=0, upper=1)
p_true = 0.05 # Unknown
N = 1500
# Ber(0.05) simmulation
occur = pm.rbernoulli(p_true, N)
print(occur)
print(occur.sum())
print(occur.mean())
obs = pm.Bernoulli("obs", p, value=occur, observed=True)
mcmc = pm.MCMC([p, obs])
mcmc.sample(20000, 1000)
plt.figure(figsize=(12.5, 4))
plt.vlines(p_true, 0, 90, linestyle='--', label="real $p_A$ unknown value")
plt.hist(mcmc.trace("p")[:], bins=25, histtype="stepfilled", normed=True)
plt.legend()
plt.show()
true_p_a = 0.05
true_p_b = 0.04
n_a = 1500
n_b = 750