用 Python 進行貝葉斯模型建模（3）
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編譯：伯樂在線 - JLee
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《
第 0 節：導論
》
《
第 1 節：估計模型參數
》
《
第 2 節：模型檢驗
》
《第 3 節：分層模型》
第3節：分層模型
貝葉斯模型的一個核心優勢就是簡單靈活，可以實現一個分層模型。這一節將實現和比較整體合併模型和局部融合模型。
import
itertools
import
matplotlib
.
pyplot
as
plt
import
numpy
as
np
import
pandas
as
pd
import
pymc3
as
pm
import
scipy
import
scipy
.
stats
as
stats
import
seaborn
.
apionly
as
sns
from
IPython
.
display
import
Image
from
sklearn
import
preprocessing
%
matplotlib
inline
plt
.
style
.
use
(
"bmh"
)
colors
=
[
"#348ABD"
,
"#A60628"
,
"#7A68A6"
,
"#467821"
,
"#D55E00"
,
"#CC79A7"
,
"#56B4E9"
,
"#009E73"
,
"#F0E442"
,
"#0072B2"
]
messages
=
pd
.
read_csv
(
"data/hangout_chat_data.csv"
)
模型合併
讓我們採取一種不同的方式來對我的 hangout 聊天回復時間進行建模。我的直覺告訴我回復的快慢與聊天的對象有關。我很可能回復女朋友比回復一個疏遠的朋友更快。這樣，我可以對每個對話獨立建模，對每個對話i估計參數 μi 和 αi。μi=Uniform(0,100)
一個必須考慮的問題是，有些對話相比其他的含有的消息很少。這樣，和含有大量消息的對話相比，我們對含有較少消息的對話的回復時間的估計，就具有較大的不確定度。下圖表明了每個對話在樣本容量上的差異。
ax
=
messages
.
groupby
(
"prev_sender"
)[
"conversation_id"
].
size
().
plot
(
kind
=
"bar"
,
figsize
=
(
12
,
3
),
title
=
"Number of messages sent per recipient"
,
color
=
colors
[
0
])
_
=
ax
.
set_xlabel
(
"Previous Sender"
)
_
=
ax
.
set_ylabel
(
"Number of messages"
)
_
=
plt
.
xticks
(
rotation
=
45
)
對於每個對話i中的每條消息j，模型可以表示為：
indiv_traces
=
{}
# Convert categorical variables to integer
le
=
preprocessing
.
LabelEncoder
()
participants_idx
=
le
.
fit_transform
(
messages
[
"prev_sender"
])
participants
=
le
.
classes_
n_participants
=
len
(
participants
)
for
p
in
participants
:
with
pm
.
Model
()
as
model
:
alpha
=
pm
.
Uniform
(
"alpha"
,
lower
=
0
,
upper
=
100
)
mu
=
pm
.
Uniform
(
"mu"
,
lower
=
0
,
upper
=
100
)
data
=
messages
[
messages
[
"prev_sender"
]
==
p
][
"time_delay_seconds"
].
values
y_est
=
pm
.
NegativeBinomial
(
"y_est"
,
mu
=
mu
,
alpha
=
alpha
,
observed
=
data
)
y_pred
=
pm
.
NegativeBinomial
(
"y_pred"
,
mu
=
mu
,
alpha
=
alpha
)
start
=
pm
.
find_MAP
()
step
=
pm
.
Metropolis
()
trace
=
pm
.
sample
(
20000
,
step
,
start
=
start
,
progressbar
=
True
)
indiv_traces
[
p
]
=
trace
Applied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
9.7
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.4
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.0
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.0
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
10.3
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
16.4
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.1
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
17.4
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
19.9
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
15.2
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
16.1
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
10.1
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
11.1
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
11.9
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.8
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
13.0
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
12.4
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
10.9
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
22.6
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
18.7
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
13.1
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
13.9
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
14.0
secApplied
interval
-
transform to alpha
and
added transformed alpha_interval to
model
.
Applied
interval
-
transform to mu
and
added transformed mu_interval to
model
.
[
-----------------
100
%-----------------
]
20000
of
20000
complete
in
13.6
sec
fig
,
axs
=
plt
.
subplots
(
3
,
2
,
figsize
=
(
12
,
6
))
axs
=
axs
.
ravel
()
y_left_max
=
2
y_right_max
=
2000
x_lim
=
60
ix
=
[
3
,
4
,
6
]
for
i
,
j
,
p
in
zip
([
0
,
1
,
2
],
[
0
,
2
,
4
],
participants
[
ix
])
:
axs
[
j
].
set_title
(
"Observed: %s"
%
p
)
axs
[
j
].
hist
(
messages
[
messages
[
"prev_sender"
]
==
p
][
"time_delay_seconds"
].
values
,
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
)
axs
[
j
].
set_ylim
([
0
,
y_left_max
])
for
i
,
j
,
p
in
zip
([
0
,
1
,
2
],
[
1
,
3
,
5
],
participants
[
ix
])
:
axs
[
j
].
set_title
(
"Posterior predictive distribution: %s"
%
p
)
axs
[
j
].
hist
(
indiv_traces
[
p
].
get_values
(
"y_pred"
),
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
,
color
=
colors
[
1
])
axs
[
j
].
set_ylim
([
0
,
y_right_max
])
axs
[
4
].
set_xlabel
(
"Response time (seconds)"
)
axs
[
5
].
set_xlabel
(
"Response time (seconds)"
)
plt
.
tight_layout
()
上圖顯示了3例對話的觀測數據分布（左）和後驗預測分布（右）。可以看出，不同對話的後驗預測分布大不相同。這可能準確反映出對話的特點，也可能是樣本容量太小造成的誤差。
如果我們把後驗預測分布聯合起來，我們希望得到的分布和觀測數據分布近似，讓我們來進行後驗預測檢驗。
combined_y_pred
=
np
.
concatenate
([
v
.
get_values
(
"y_pred"
)
for
k
,
v
in
indiv_traces
.
items
()])
x_lim
=
60
y_pred
=
trace
.
get_values
(
"y_pred"
)
fig
=
plt
.
figure
(
figsize
=
(
12
,
6
))
fig
.
add_subplot
(
211
)
fig
.
add_subplot
(
211
)
_
=
plt
.
hist
(
combined_y_pred
,
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
,
color
=
colors
[
1
])
_
=
plt
.
xlim
(
1
,
x_lim
)
_
=
plt
.
ylim
(
0
,
20000
)
_
=
plt
.
ylabel
(
"Frequency"
)
_
=
plt
.
title
(
"Posterior predictive distribution"
)
fig
.
add_subplot
(
212
)
_
=
plt
.
hist
(
messages
[
"time_delay_seconds"
].
values
,
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
)
_
=
plt
.
xlim
(
0
,
x_lim
)
_
=
plt
.
xlabel
(
"Response time in seconds"
)
_
=
plt
.
ylim
(
0
,
20
)
_
=
plt
.
ylabel
(
"Frequency"
)
_
=
plt
.
title
(
"Distribution of observed data"
)
plt
.
tight_layout
()
是的，後驗預測分布和觀測數據的分布近似。但是，我關心的是數據較少的對話，對它們的估計也可能具有較大的方差。一個減小這種風險的方法是分享對話信息，但仍對每個對話單獨估計 μi，我們稱之為局部融合。
局部融合
就像整體合併模型，局部融合模型對每個對話i都有獨自的參數估計。但是，這些參數通過融合參數聯繫在一起。這反映出，我的不同對話間的response_time有相似之處，就是我本性上傾向於回復快或慢。
接著上面的例子，估計負二項分布的參數 μi 和 αi。相比於使用均勻分布作為先驗分布，我使用帶兩個參數（μ，σ）的伽馬分布，從而當能夠預測 μ 和 σ 的值時，可以引入更多的先驗信息到模型中。
首先，我們來看看伽馬分布，下面你可以看到，它非常靈活。
局部融合模型可以表示為：
代碼：
Image("graphics/dag neg poisson gamma hyper.png", width=420)
with
pm
.
Model
()
as
model
:
hyper_alpha_sd
=
pm
.
Uniform
(
"hyper_alpha_sd"
,
lower
=
0
,
upper
=
50
)
hyper_alpha_mu
=
pm
.
Uniform
(
"hyper_alpha_mu"
,
lower
=
0
,
upper
=
10
)
hyper_mu_sd
=
pm
.
Uniform
(
"hyper_mu_sd"
,
lower
=
0
,
upper
=
50
)
hyper_mu_mu
=
pm
.
Uniform
(
"hyper_mu_mu"
,
lower
=
0
,
upper
=
60
)
alpha
=
pm
.
Gamma
(
"alpha"
,
mu
=
hyper_alpha_mu
,
sd
=
hyper_alpha_sd
,
shape
=
n_participants
)
mu
=
pm
.
Gamma
(
"mu"
,
mu
=
hyper_mu_mu
,
sd
=
hyper_mu_sd
,
shape
=
n_participants
)
y_est
=
pm
.
NegativeBinomial
(
"y_est"
,
mu
=
mu
[
participants_idx
],
alpha
=
alpha
[
participants_idx
],
observed
=
messages
[
"time_delay_seconds"
].
values
)
y_pred
=
pm
.
NegativeBinomial
(
"y_pred"
,
mu
=
mu
[
participants_idx
],
alpha
=
alpha
[
participants_idx
],
shape
=
messages
[
"prev_sender"
].
shape
)
start
=
pm
.
find_MAP
()
step
=
pm
.
Metropolis
()
hierarchical_trace
=
pm
.
sample
(
200000
,
step
,
progressbar
=
True
)
Applied
interval
-
transform to hyper_alpha_sd
and
added transformed hyper_alpha_sd_interval to
model
.
Applied
interval
-
transform to hyper_alpha_mu
and
added transformed hyper_alpha_mu_interval to
model
.
Applied
interval
-
transform to hyper_mu_sd
and
added transformed hyper_mu_sd_interval to
model
.
Applied
interval
-
transform to hyper_mu_mu
and
added transformed hyper_mu_mu_interval to
model
.
Applied
log
-
transform to alpha
and
added transformed alpha_log to
model
.
Applied
log
-
transform to mu
and
added transformed mu_log to
model
.
[
-----------------
100
%-----------------
]
200000
of
200000
complete
in
593.0
sec
對 μ 和 α 的估計有多條曲線，每個對話i對應一條。整體合併和局部融合模型的不同之處在於，局部融合模型的參數（μ和 α）擁有一個被所有對話共享的融合參數。這帶來兩個好處：
信息在對話間共享，所以對於含有有限樣本容量的對話來說，在估計過程中從別的對話處「借」信息來減小估計的方差。
我們對每個對話單獨做了估計，也對所有對話整體做了估計。
我們快速看一下後驗預測分布
x_lim
=
60
y_pred
=
hierarchical_trace
.
get_values
(
"y_pred"
)[
::
1000
].
ravel
()
fig
=
plt
.
figure
(
figsize
=
(
12
,
6
))
fig
.
add_subplot
(
211
)
fig
.
add_subplot
(
211
)
_
=
plt
.
hist
(
y_pred
,
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
,
color
=
colors
[
1
])
_
=
plt
.
xlim
(
1
,
x_lim
)
_
=
plt
.
ylabel
(
"Frequency"
)
_
=
plt
.
title
(
"Posterior predictive distribution"
)
fig
.
add_subplot
(
212
)
_
=
plt
.
hist
(
messages
[
"time_delay_seconds"
].
values
,
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
)
_
=
plt
.
xlabel
(
"Response time in seconds"
)
_
=
plt
.
ylabel
(
"Frequency"
)
_
=
plt
.
title
(
"Distribution of observed data"
)
plt
.
tight_layout
()
收縮效果：合併模型 vs 分層模型
如討論的那樣，局部融合模型中 μ 和 α 享有一個融合參數，通過對話間信息共享，它使得參數的估計收縮得更緊密，尤其是對含有少量數據的對話。
下圖顯示了這種收縮效果，可以看出通過融合參數，參數μ 和 α 是怎樣聚集在一起。
hier_mu
=
hierarchical_trace
[
"mu"
][
500
:
].
mean
(
axis
=
0
)
hier_alpha
=
hierarchical_trace
[
"alpha"
][
500
:
].
mean
(
axis
=
0
)
indv_mu
=
[
indiv_traces
[
p
][
"mu"
][
500
:
].
mean
()
for
p
in
participants
]
indv_alpha
=
[
indiv_traces
[
p
][
"alpha"
][
500
:
].
mean
()
for
p
in
participants
]
fig
=
plt
.
figure
(
figsize
=
(
8
,
6
))
ax
=
fig
.
add_subplot
(
111
,
xlabel
=
"mu"
,
ylabel
=
"alpha"
,
title
=
"Pooled vs. Partially Pooled Negative Binomial Model"
,
xlim
=
(
5
,
45
),
ylim
=
(
0
,
10
))
ax
.
scatter
(
indv_mu
,
indv_alpha
,
c
=
colors
[
5
],
s
=
50
,
label
=
"Pooled"
,
zorder
=
3
)
ax
.
scatter
(
hier_mu
,
hier_alpha
,
c
=
colors
[
6
],
s
=
50
,
label
=
"Partially Pooled"
,
zorder
=
4
)
for
i
in
range
(
len
(
indv_mu
))
:
ax
.
arrow
(
indv_mu
[
i
],
indv_alpha
[
i
],
hier_mu
[
i
]
-
indv_mu
[
i
],
hier_alpha
[
i
]
-
indv_alpha
[
i
],
fc
=
"grey"
,
ec
=
"grey"
,
length_includes_head
=
True
,
alpha
=
.
5
,
head_width
=
0
)
_
=
ax
.
legend
()
對後驗分布提問
我們開始利用貝葉斯統計最好的方面之一——後驗分布。不像頻率論方法，我們得到完整的後驗分布而不是點估計。本質上，我們是得到許多可信的參數值，這使得我們可以按照比較自然直觀的方式提問。
What are the chances I』ll respond to my friend in less than 10 seconds?
為了估計這個概率，我們可以看看 Timothy 和 Andrew 的回復時間的後驗預測分布，檢查有多少樣本是小於 10 秒的。當我第一次聽到這個方法時，我以為我理解錯了，因為它太簡單了。
print("Here are some samples from Timothy"s posterior predictive distribution: n %s" % participant_y_pred("Timothy"))
Here are some samples
from
Timothy
"s
posterior predictive
distribution
:
[
24
24
24
...,
19
19
19
]
def
person_plotA
(
person_name
)
:
ix_check
=
participant_y_pred
(
person_name
)
>
10
_
=
plt
.
hist
(
participant_y_pred
(
person_name
)[
~
ix_check
],
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
,
label
=
"<10 seconds"
)
_
=
plt
.
hist
(
participant_y_pred
(
person_name
)[
ix_check
],
range
=
[
0
,
x_lim
],
bins
=
x_lim
,
histtype
=
"stepfilled"
,
label
=
">10 seconds"
)
_
=
plt
.
title
(
"Posterior predictive ndistribution for %s"
%
person_name
)
_
=
plt
.
xlabel
(
"Response time"
)
_
=
plt
.
ylabel
(
"Frequency"
)
_
=
plt
.
legend
()
def
person_plotB
(
person_name
)
:
x
=
np
.
linspace
(
1
,
60
,
num
=
60
)
num_samples
=
float
(
len
(
participant_y_pred
(
person_name
)))
prob_lt_x
=
[
100
*
sum
(
participant_y_pred
(
person_name
)
<
i
)
/
num_samples
for
i
in
x
]
_
=
plt
.
plot
(
x
,
prob_lt_x
,
color
=
colors
[
4
])
_
=
plt
.
fill_between
(
x
,
prob_lt_x
,
color
=
colors
[
4
],
alpha
=
0.3
)
_
=
plt
.
scatter
(
10
,
float
(
100
*
sum
(
participant_y_pred
(
person_name
)
<
10
))
/
num_samples
,
s
=
180
,
c
=
colors
[
4
])
_
=
plt
.
title
(
"Probability of responding nto %s before time (t)"
%
person_name
)
_
=
plt
.
xlabel
(
"Response time (t)"
)
_
=
plt
.
ylabel
(
"Cumulative probability t"
)
_
=
plt
.
ylim
(
ymin
=
0
,
ymax
=
100
)
_
=
plt
.
xlim
(
xmin
=
0
,
xmax
=
60
)
fig
=
plt
.
figure
(
figsize
=
(
11
,
6
))
_
=
fig
.
add_subplot
(
221
)
person_plotA
(
"Timothy"
)
_
=
fig
.
add_subplot
(
222
)
person_plotB
(
"Timothy"
)
_
=
fig
.
add_subplot
(
223
)
person_plotA
(
"Andrew"
)
_
=
fig
.
add_subplot
(
224
)
person_plotB
(
"Andrew"
)
plt
.
tight_layout
()
participant_y_pred("Andrew")
array([13, 13, 13, ..., 28, 28, 28])
我發現這個方法非常直觀而且靈活。上圖左邊根據大於 10 秒或小於 10 秒把後驗預測的樣本分為兩部分，通過計算小於 10 的樣本比例來計算概率。右邊的圖對回復時間小於每個 0 到 60 間的值的概率進行了計算。所以，看起來，Timothy 和 Andrew 分別有 38% 和 8% 的可能性在 10 秒內得到回復。
我的朋友們兩兩之間比較如何呢？
def
prob_persona_faster
(
persona
,
personb
)
:
return
sum
(
participant_y_pred
(
persona
)
<
participant_y_pred
(
personb
))
/
len
(
participant_y_pred
(
persona
))
(
"Probability that Tom is responded to faster than Andrew: {:.2%}"
.
format
(
prob_persona_faster
(
"Tom"
,
"Andrew"
)))
Probability that Tom is responded to faster than Andrew: 33.05%
# Create an empty dataframe
ab_dist_df
=
pd
.
DataFrame
(
index
=
participants
,
columns
=
participants
,
dtype
=
np
.
float
)
# populate each cell in dataframe with persona_less_personb()
for
a
,
b
in
itertools
.
permutations
(
participants
,
2
)
:
ab_dist_df
.
ix
[
a
,
b
]
=
prob_persona_faster
(
a
,
b
)
# populate the diagonal
for
a
in
participants
:
ab_dist_df
.
ix
[
a
,
a
]
=
0.5
# Plot heatmap
f
,
ax
=
plt
.
subplots
(
figsize
=
(
12
,
9
))
cmap
=
plt
.
get_cmap
(
"Spectral"
)
_
=
sns
.
heatmap
(
ab_dist_df
,
square
=
True
,
cmap
=
cmap
)
_
=
plt
.
title
(
"Probability that Person A will be responded to faster than Person B"
)
_
=
plt
.
ylabel
(
"Person A"
)
_
=
plt
.
xlabel
(
"Person B"
)
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