In [284]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import pymc as pm2
import pymc3 as pm
import time
import math
import numpy.random as rd
import pandas as pd
from pymc3 import summary
In [271]:
from pymc3.backends.base import merge_traces
import theano.tensor as T
メトロポリス法
（１）パラメーターqの初期値を選ぶ
（２）qを増やすか減らすかをランダムに決める
（３）q（新）において尤度が大きくなるならqの値をq（新）に変更する
（４）q（新）で尤度が小さくなる場合であっても、確率rでqの値をq（新）に変更する
In [116]:
def comb(n, r):
if n == 0 or r == 0: return 1
return comb(n, r - 1) * (n - r + 1) / r
def prob(n, y, q):
p = comb(n, y) * q ** y * (1 - q) ** (n - y)
return p
def likelighood(n, y, q):
p = 1.0
for i in y:
p = p*prob(n, i, q)
return p
def metropolis(n, y, q, b, num):
qlist = np.array([q])
for i in range(num):
old_q = q
q = q+np.random.choice([b, -b])
old_l = likelighood(n, y, old_q)
new_l = likelighood(n, y, q)
if new_l > old_l:
old_q = q
else:
r = new_l/old_l
q = np.random.choice([q, old_q], p=[r, 1.0-r])
q = round(q, 5)
qlist = np.append(qlist, q)
return q, qlist
In [122]:
y = [4, 3, 4, 5, 5, 2, 3, 1, 4, 0, 1, 5, 5, 6, 5, 4, 4, 5, 3, 4]
q, qlist = metropolis(8, y, 0.3, 0.01, 10000)
In [129]:
plt.plot(qlist)
Out[129]:
[<matplotlib.lines.Line2D at 0x10da78550>]
In [128]:
plt.hist(qlist)
Out[128]:
(array([ 8., 11., 202., 996., 2589., 3185., 2002., 796.,
198., 14.]),
array([ 0.3 , 0.329, 0.358, 0.387, 0.416, 0.445, 0.474, 0.503,
0.532, 0.561, 0.59 ]),
<a list of 10 Patch objects>)
In [245]:
qlist.mean()
Out[245]:
0.45695930406959306
In [247]:
N = 40
X = np.random.uniform(10, size=N)
Y = X*30 + 4 + np.random.normal(0, 16, size=N)
plt.plot(X, Y, "o")
Out[247]:
[<matplotlib.lines.Line2D at 0x117cd2910>]
In [252]:
multicore = False
saveimage = False
itenum = 1000
t0 = time.clock()
chainnum = 3
with pm.Model() as model:
alpha = pm.Normal('alpha', mu=0, sd =20)
beta = pm.Normal('beta', mu=0, sd=20)
sigma = pm.Uniform('sigma', lower=0)
y = pm.Normal('y', mu=beta*X + alpha, sd=sigma, observed=Y)
start = pm.find_MAP()
step = pm.NUTS(state=start)
In [261]:
with model:
if(multicore):
trace = pm.sample(itenum, step, start=start,
njobs=chainnum, random_seed=range(chainnum),
progress_bar=False)
else:
ts = [pm.sample(itenum, step, chain=i, progressbar=False) for i in range(chainnum)]
trace = merge_traces(ts)
if(saveimage):
pm.tracepot(trace).savefig("simple_linear_trace.png")
print "Rhat="+str(pm.gelman_rubin(trace))
t1=time.clock()
print "elapsed time=" + str(t1-t0)
Rhat={'alpha': 1.0002636216914187, 'beta': 0.99959538672461912, 'sigma': 1.0032735164477551, 'sigma_interval': 1.0323942771051835}
elapsed time=663.205294
In [263]:
if(not multicore):
trace = ts[0]
with model:
pm.traceplot(trace, model.vars)
In [265]:
pm.forestplot(trace)
Out[265]:
<matplotlib.gridspec.GridSpec at 0x12956a150>
In [286]:
summary(trace[4000:])
p_logodds:
Mean SD MC Error 95% HPD interval
-------------------------------------------------------------------
-0.048 0.065 0.001 [-0.173, 0.080]
Posterior quantiles:
2.5 25 50 75 97.5
|--------------|==============|==============|--------------|
-0.176 -0.092 -0.048 -0.003 0.078
mu0:
Mean SD MC Error 95% HPD interval
-------------------------------------------------------------------
-0.943 0.045 0.000 [-1.030, -0.858]
Posterior quantiles:
2.5 25 50 75 97.5
|--------------|==============|==============|--------------|
-1.031 -0.974 -0.943 -0.912 -0.859
mu1:
Mean SD MC Error 95% HPD interval
-------------------------------------------------------------------
1.037 0.046 0.000 [0.949, 1.124]
Posterior quantiles:
2.5 25 50 75 97.5
|--------------|==============|==============|--------------|
0.948 1.006 1.036 1.069 1.124
p:
Mean SD MC Error 95% HPD interval
-------------------------------------------------------------------
0.488 0.016 0.000 [0.457, 0.520]
Posterior quantiles:
2.5 25 50 75 97.5
|--------------|==============|==============|--------------|
0.456 0.477 0.488 0.499 0.519
mu:
Mean SD MC Error 95% HPD interval
-------------------------------------------------------------------
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
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1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
-0.943 0.045 0.000 [-1.030, -0.858]
1.037 0.046 0.000 [0.949, 1.124]
1.037 0.046 0.000 [0.949, 1.124]
Posterior quantiles:
2.5 25 50 75 97.5
|--------------|==============|==============|--------------|
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
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-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
-1.031 -0.974 -0.943 -0.912 -0.859
0.948 1.006 1.036 1.069 1.124
0.948 1.006 1.036 1.069 1.124
In [266]:
data = pd.read_csv("http://hosho.ees.hokudai.ac.jp/~kubo/stat/iwanamibook/fig/hbm/data7a.csv")
plt.bar(range(9), data.groupby('y').sum().id)
data.groupby('y').sum().T
Out[266]:
y
0
1
2
3
4
5
6
7
8
id
1074
691
459
137
315
192
402
830
950
In [ ]:
plt.plot(y, 'o')
In [267]:
Y = np.array(data.y)[:6]
In [279]:
def invlogit(v):
return T.exp(v)/(T.exp(v) + 1)
with pm.Model() as model_hier:
s = pm.Uniform('s', 0, 1.0E+2)
beta = pm.Normal('beta', 0, 1.0E+2)
r = pm.Normal('r', 0, s, shape=len(Y))
q = invlogit(beta+r)
y = pm.Binomial('y', 8, q, observed=Y)
step = pm.Slice([s, beta, r])
trace_hier = pm.sample(1000, step)
with model_hier:
pm.traceplot(trace_hier, model_hier.vars)
[-----------------100%-----------------] 1000 of 1000 complete in 3.9 sec
In [234]:
# Intialize random number generator
np.random.seed(123)
# True parameter values
alpha, sigma = 1, 1
beta = [1, 2.5]
# Size of dataset
size = 100
# Predictor variable
X1 = np.linspace(0, 1, size)
X2 = np.linspace(0,.2, size)
# Simulate outcome variable
Y = alpha + beta[0]*X1 + beta[1]*X2 + np.random.randn(size)*sigma
In [243]:
basic_model = Model()
with basic_model:
# Priors for unknown model parameters
alpha = Normal('alpha', mu=0, sd=10)
beta = Normal('beta', mu=0, sd=10, shape=2)
sigma = HalfNormal('sigma', sd=1)
# Expected value of outcome
mu = alpha + beta[0]*X1 + beta[1]*X2
# Likelihood (sampling distribution) of observations
Y_obs = Normal('Y_obs', mu=mu, sd=sigma, observed=Y)
In [241]:
x_sample = np.random.normal(loc=1.0, scale=1.0, size=1000)
with pm.Model() as model:
mu = pm.Normal('mu', mu=0., sd=0.1)
x = pm.Normal('x', mu=mu, sd=1., observed=x_sample)
with model:
start = pm.find_MAP()
step = pm.NUTS()
trace = pm.sample(10000, step, start)
[-----------------100%-----------------] 10000 of 10000 complete in 3.7 sec
In [285]:
ndims = 2
nobs = 20
n = 1000
y_sample = np.random.binomial(1, 0.5, size=(n,))
x_sample=np.empty(n)
x_sample[y_sample==0] = np.random.normal(-1, 1, size=(n, ))[y_sample==0]
x_sample[y_sample==1] = np.random.normal(1, 1, size=(n, ))[y_sample==1]
with pm.Model() as model:
p = pm.Beta('p', alpha=1.0, beta=1.0)
y = pm.Bernoulli('y', p=p, observed=y_sample)
mu0 = pm.Normal('mu0', mu=0., sd=1.)
mu1 = pm.Normal('mu1', mu=0., sd=1.)
mu = pm.Deterministic('mu', mu0 * (1-y_sample) + mu1 * y_sample)
x = pm.Normal('x', mu=mu, sd=1., observed=x_sample)
with model:
start = pm.find_MAP()
step = pm.NUTS()
trace = pm.sample(10000, step, start)
pm.traceplot(trace)
plt.savefig("result2.jpg")
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
<ipython-input-285-72532e73f473> in <module>()
20 with model:
21 start = pm.find_MAP()
---> 22 step = pm.NUTS()
23 trace = pm.sample(10000, step, start)
24
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/step_methods/nuts.pyc in __init__(self, vars, scaling, step_scale, is_cov, state, Emax, target_accept, gamma, k, t0, model, profile, **kwargs)
66
67 if isinstance(scaling, dict):
---> 68 scaling = guess_scaling(Point(scaling, model=model), model=model, vars = vars)
69
70
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/tuning/scaling.pyc in guess_scaling(point, vars, model)
79 def guess_scaling(point, vars=None, model=None):
80 model = modelcontext(model)
---> 81 h = find_hessian_diag(point, vars, model=model)
82 return adjust_scaling(h)
83
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/tuning/scaling.pyc in find_hessian_diag(point, vars, model)
74 """
75 model = modelcontext(model)
---> 76 H = model.fastfn(hessian_diag(model.logpt, vars))
77 return H(Point(point, model=model))
78
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/memoize.pyc in memoizer(*args, **kwargs)
12
13 if key not in cache:
---> 14 cache[key] = obj(*args, **kwargs)
15
16 return cache[key]
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/theanof.pyc in hessian_diag(f, vars)
101
102 if vars:
--> 103 return -t.concatenate([hessian_diag1(f, v) for v in vars], axis=0)
104 else:
105 return empty_gradient
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/theanof.pyc in hessian_diag1(f, v)
86 def hessian_diag1(f, v):
87
---> 88 g = gradient1(f, v)
89 idx = t.arange(g.shape[0])
90
/Users/nigg/anaconda/lib/python2.7/site-packages/pymc3/theanof.pyc in gradient1(f, v)
42 def gradient1(f, v):
43 """flat gradient of f wrt v"""
---> 44 return t.flatten(t.grad(f, v, disconnected_inputs='warn'))
45
46
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in grad(cost, wrt, consider_constant, disconnected_inputs, add_names, known_grads, return_disconnected, null_gradients)
559
560 rval = _populate_grad_dict(var_to_app_to_idx,
--> 561 grad_dict, wrt, cost_name)
562
563 for i in xrange(len(rval)):
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in _populate_grad_dict(var_to_app_to_idx, grad_dict, wrt, cost_name)
1322 return grad_dict[var]
1323
-> 1324 rval = [access_grad_cache(elem) for elem in wrt]
1325
1326 return rval
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_grad_cache(var)
1277 for idx in node_to_idx[node]:
1278
-> 1279 term = access_term_cache(node)[idx]
1280
1281 if not isinstance(term, gof.Variable):
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_term_cache(node)
971 inputs = node.inputs
972
--> 973 output_grads = [access_grad_cache(var) for var in node.outputs]
974
975 # list of bools indicating if each output is connected to the cost
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_grad_cache(var)
1277 for idx in node_to_idx[node]:
1278
-> 1279 term = access_term_cache(node)[idx]
1280
1281 if not isinstance(term, gof.Variable):
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_term_cache(node)
971 inputs = node.inputs
972
--> 973 output_grads = [access_grad_cache(var) for var in node.outputs]
974
975 # list of bools indicating if each output is connected to the cost
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_grad_cache(var)
1277 for idx in node_to_idx[node]:
1278
-> 1279 term = access_term_cache(node)[idx]
1280
1281 if not isinstance(term, gof.Variable):
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gradient.pyc in access_term_cache(node)
1097 new_output_grads)):
1098 orig_output, new_output_grad = packed
-> 1099 if not hasattr(orig_output, 'shape'):
1100 continue
1101 if isinstance(new_output_grad.type, DisconnectedType):
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/tensor/var.pyc in <lambda>(self)
279 return theano.tensor.basic.transpose(self, axes)
280
--> 281 shape = property(lambda self: theano.tensor.basic.shape(self))
282
283 size = property(lambda self: self.shape[0] if self.ndim == 1 else
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/op.pyc in __call__(self, *inputs, **kwargs)
645 # compute output value once with test inputs to validate graph
646 thunk = node.op.make_thunk(node, storage_map, compute_map,
--> 647 no_recycling=[])
648 thunk.inputs = [storage_map[v] for v in node.inputs]
649 thunk.outputs = [storage_map[v] for v in node.outputs]
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/op.pyc in make_thunk(self, node, storage_map, compute_map, no_recycling)
916 try:
917 return self.make_c_thunk(node, storage_map, compute_map,
--> 918 no_recycling)
919 except (NotImplementedError, utils.MethodNotDefined):
920 logger.debug('Falling back on perform')
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/op.pyc in make_c_thunk(self, node, storage_map, compute_map, no_recycling)
834 logger.debug('Trying CLinker.make_thunk')
835 outputs = cl.make_thunk(input_storage=node_input_storage,
--> 836 output_storage=node_output_storage)
837 fill_storage, node_input_filters, node_output_filters = outputs
838
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/cc.pyc in make_thunk(self, input_storage, output_storage, storage_map, keep_lock)
1207 cthunk, in_storage, out_storage, error_storage = self.__compile__(
1208 input_storage, output_storage, storage_map,
-> 1209 keep_lock=keep_lock)
1210
1211 res = _CThunk(cthunk, init_tasks, tasks, error_storage)
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/cc.pyc in __compile__(self, input_storage, output_storage, storage_map, keep_lock)
1145 output_storage,
1146 storage_map,
-> 1147 keep_lock=keep_lock)
1148 return (thunk,
1149 [link.Container(input, storage) for input, storage in
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/cc.pyc in cthunk_factory(self, error_storage, in_storage, out_storage, storage_map, keep_lock)
1594 """
1595 try:
-> 1596 key = self.cmodule_key()
1597 except KeyError:
1598 key = None
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/cc.pyc in cmodule_key(self)
1289 """
1290 return self.cmodule_key_(self.fgraph, self.no_recycling,
-> 1291 compile_args=self.compile_args(),
1292 libraries=self.libraries(),
1293 header_dirs=self.header_dirs(),
/Users/nigg/anaconda/lib/python2.7/site-packages/theano/gof/cc.pyc in compile_args(self)
948 try:
949 try:
--> 950 ret += x.c_compile_args(c_compiler)
951 except TypeError:
952 ret += x.c_compile_args()
KeyboardInterrupt:
In [278]:
x_sample = np.random.normal(loc=1.0, scale=1.0, size=1000)
with pm.Model() as model:
mu = pm.Normal('mu', mu=0., sd=0.1)
x = pm.Normal('x', mu=mu, sd=1., observed=x_sample)
with model:
start = pm.find_MAP()
step = pm.NUTS()
trace = pm.sample(10000, step, start)
pm.traceplot(trace)
plt.savefig("result1.jpg")
[-----------------100%-----------------] 10000 of 10000 complete in 3.1 sec
In [282]:
ndims = 2
nobs = 20
n = 1000
y_sample = np.random.binomial(1, 0.5, size=(n,))
x_sample=np.empty(n)
x_sample[y_sample==0] = np.random.normal(-1, 1, size=(n, ))[y_sample==0]
x_sample[y_sample==1] = np.random.normal(1, 1, size=(n, ))[y_sample==1]
with pm.Model() as model:
p = pm.Beta('p', alpha=1.0, beta=1.0)
y = pm.Bernoulli('y', p=p, observed=y_sample)
mu0 = pm.Normal('mu0', mu=0., sd=1.)
mu1 = pm.Normal('mu1', mu=0., sd=1.)
mu = pm.Deterministic('mu', mu0 * (1-y_sample) + mu1 * y_sample)
x = pm.Normal('x', mu=mu, sd=1., observed=x_sample)
with model:
start = pm.find_MAP()
step = pm.NUTS()
trace = pm.sample(10000, step, start)
pm.traceplot(trace)
plt.savefig("result2.jpg")
[-----------------100%-----------------] 10000 of 10000 complete in 9.8 sec
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