第１３回数学カフェ「素数！！」二次会 LT資料「乱数！！」 2016/5/7 乱数生成の話から確率分布、そして確率モデルの例としてトピックモデルについてご紹介しました。 表紙はネタですw

1. 第353回 確率統計カフェ
「乱数！！」
2016/5/7
Ken ichi Matsui (@kenmatsu4)

3. http://www.slideshare.net/matsukenbook

5. https://twitter.com/_inundata/status/616658949761302528

6. https://twitter.com/_inundata/status/616658949761302528

7. x1, x2, · · · , xn xn+1

12. M
xj+1 := axj + c mod M
a = 1103515245, c = 12345, M = 231
231

13. xj+1 := axj + c mod M
231
Ma = 1103515245, c = 12345, M = 231

14. 219937
1
xk+n := xk+m (xk
u
| xk+1
l
)A k = 0, 1, . . .
219937
1

15. xk+n := xk+m (xk
u
| xk+1
l
)A k = 0, 1, . . .
A =
✓
0 Iw 1
aw 1 (aw 2, . . . , a0)
◆
w
=
0
B
B
B
B
B
B
B
B
B
@
0 1 · · · 0
...
0
...
...
...
...
0 0 · · · 1
aw 1 aw 2 · · · a0
1
C
C
C
C
C
C
C
C
C
A
1  m < n
0  r  w 1
xA =
(
shiftright(x)
shiftright(x) + a
xi = (xi(w 1), xi(w 2), · · · , xi(0)) xi(j) 2 {0, 1}

16. y := x ((x >> u)&d)
y := y ((y << s)&b)
y := y ((y << t)&c)
z := y (y >> l)
x

17. y := x ((x >> u)&d)
y := y ((y << s)&b)
y := y ((y << t)&c)
z := y (y >> l)
(w, n, m, r) = (32, 624, 397, 31)
a = 9908B0DF16
(u, d) = (11, FFFFFFFF16)
(s, b) = (7, 9D2C568016)
(t, c) = (15, EFC6000016)
l = 18
xk+n := xk+m (xk
u
| xk+1
l
)A k = 0, 1, . . .
A =
✓
0 Iw 1
aw 1 (aw 2, . . . , a0)
◆
w = 32, n = 624, m = 397, r = 3
nw r = 19937
2nw r
1
219937
1
w = 32, n = 624, m = 397, r = 31

18. y := x ((x >> u)&d)
y := y ((y << s)&b)
y := y ((y << t)&c)
z := y (y >> l)
(w, n, m, r) = (32, 624, 397, 31)
a = 9908B0DF16
(u, d) = (11, FFFFFFFF16)
(s, b) = (7, 9D2C568016)
(t, c) = (15, EFC6000016)
l = 18
xk+n := xk+m (xk
u
| xk+1
l
)A k = 0, 1, . . .
A =
✓
0 Iw 1
aw 1 (aw 2, . . . , a0)
◆
w = 32, n = 624, m = 397, r = 3
nw r = 19937
2nw r
1
219937
1
w = 32, n = 624, m = 397, r = 31

22. x = 1 x = 0
f(x; p) =
8
<
:
p if x = 1,
1 p if x = 0.
f(x; p) = px
(1 p)1 x
, x = {0, 1}
x
p
p
1 p

26. http://www.math.wm.edu/~leemis/2008amstat.pdf
確率分布
曼荼羅
76個
有り〼

29. P(X = x) = px
(1 p)1 x
(x = 0, 1)

31. p
=
0.7
trial_size
=
10000
set.seed(71)
data
<-­‐
rbern(trial_size,
p)
dens
<-­‐
data.frame(y=c((1-­‐p),p)*trial_size,
x=c(0,
1))
ggplot()
+
layer(data=data.frame(x=data),
mapping=aes(x=x),
geom="bar",
stat="bin",
bandwidth=0.1
)
+
layer(data=dens,
mapping=aes(x=x,
y=y),
geom="bar",
stat="identity",
width=0.05,
fill="#777799",
alpha=0.7)

33. P(X = x) = nCrpx
(1 p)n x
(x = 1, 2, · · · , n)

35. p
=
0.7
trial_size
=
10000
sample_size
=
30
set.seed(71)
gen_binom_var
<-­‐
function()
{
return(sum(rbern(sample_size,
p)))
}
result
<-­‐
rdply(trial_size,
gen_binom_var())
dens
<-­‐
data.frame(y=dbinom(seq(sample_size),
sample_size,
0.7))*trial_size
ggplot()
+
layer(data=resuylt,
mapping=aes(x=V1),
geom="bar",
stat
=
"bin",
binwidth=1,
fill="#6666ee",
color="gray"
)
+
layer(data=dens,
mapping=aes(x=seq(sample_size)+.5,
y=y),
geom="line",
stat="identity",
position="identity",colour="red"
)
+
ggtitle("Bernoulli
to
Binomial.")

37. f(x) =
⇢
1 (0  x  1)
0 (otherwise)

38. Z = x1(1/2)1
+ x2(1/2)2
+ · · · + xq(1/2)q

39. width
<-­‐
0.02
p
<-­‐
0.5;
sample_size
<-­‐
1000
trial_size
<-­‐
100000
gen_unif_rand
<-­‐
function()
{
return
(sum(rbern(sample_size,
p)
*
(rep(1/2,
sample_size)
**
seq(sample_size))))
}
gen_rand
<-­‐
function(){
return(
rdply(trial_size,
gen_unif_rand())
)
}
system.time(res
<-­‐
gen_rand())
ggplot()
+
layer(data=res,
mapping=aes(x=V1),
geom="bar",
stat
=
"bin",
binwidth=width,
fill="#6666ee",
color="gray"
)
+
ggtitle("Bernoulli
to
Standard
Uniform")

41. (0 < x < 1)
Xi ⇠ U(0, 1)iid
(i = 1, 2, · · · , ↵ + 1)
f(x, ↵, ) =
1
B(↵, )
x↵ 1
(1 x) 1

43. width
<-­‐
0.03;
p
<-­‐
0.5
digits_length
<-­‐
30;
set_size
<-­‐
3
trial_size
<-­‐
30000
gen_unif_rand
<-­‐
function()
{
return
(sum(rbern(digits_length,
p)
*
(rep(1/2,
digits_length)
**
seq(digits_length))))
}
gen_rand
<-­‐
function(){
return(
rdply(set_size,
gen_unif_rand())$V1
)
}
unif_dataset
<-­‐
rlply(trial_size,
gen_rand,
.progress='text')
p
<-­‐
ceiling(set_size
*
0.5);
q
<-­‐
set_size
-­‐
p
+
1
get_nth_data
<-­‐
function(a){
return(a[order(a)][p])
}
disp_data
<-­‐
data.frame(lapply(unif_dataset,
get_nth_data))
names(disp_data)
<-­‐
seq(length(disp_data));
disp_data
<-­‐
data.frame(t(disp_data))
names(disp_data)
<-­‐
"V1"
x_range
<-­‐
seq(0,
1,
0.001)
dens
<-­‐
data.frame(y=dbeta(x_range,
p,
q)*trial_size*width)
ggplot()
+
layer(data=disp_data,
mapping=aes(x=V1),
geom="bar",
stat
=
"bin",
binwidth=width,
fill="#6666ee",
color="gray"
)
+
layer(data=dens,
mapping=aes(x=x_range,
y=y),
geom="line",
stat="identity",
position="identity",
colour="red"
)
+
ggtitle("Bernoulli
to
Beta")

45. P(X = x) = px
(1 p)1 x
p, 1 p
p ⇠ Beta(↵, )
p, 1 p1 p

46. P

47. i
f(x = i|p) = pi
p = (p1, ..., pk)
kX
i=1
pi = 1

48. trial_num
=
10000
x
=
rd.multinomial(1,
[1/6]*6,
trial_num)
result
=
np.sum(x,
axis=0)
data
=
np.array([result,
np.array([1/6]*6)*trial_num]).T
#
Draw
graph
df
=
pd.DataFrame(data,
columns=["trial","theory"],index=range(1,7))
ax
=
df.plot.bar()
ax.set_ylim(0,2000)
ax.legend(loc='best')

49. P
P

50. ip = (p1, ..., pk)
n
結果の例
([[4, 1, 1, 5, 5, 2],
[3, 3, 2, 4, 3, 3],
[1, 4, 3, 4, 3, 3],
...,
[3, 3, 4, 2, 3, 3],
[3, 3, 2, 3, 4, 3],
[1, 3, 4, 3, 4, 3]])
f(x; p) =
(
n!
x1!···xk! px1
1 · · · pxk
k when
Pk
i=1 xi = n
0 otherwise.

51. P
P

52. ↵ = (↵1, ↵2, · · · , ↵K)
f(p; ↵) =
1
B(↵)
KY
i=1
p↵i 1
i
pi 0,
X
pi = 1

56. z wθα
β φ
N D
:"Observed"variables
:"Unknown"parameters
:"Hyper"parameters
wordtopictopic"generate
distribu;on
word"generate
distribu;on

57. doc word freq
0 128 2
0 129 2
0 130 2
0 131 1
0 5 1
0 134 2
0 7 2
0 137 1
0 139 1
0 140 1
0 141 1
0 14 1
0 16 2
0 18 2
0 19 3
0 20 1
0 23 1
0 26 6
0 28 3
0 31 2
0 32 7
0 36 1
0 37 1
0 38 1
0 42 5
0 44 1
0 45 4
0 46 2
0 47 3
0 49 1
0 52 5
0 53 1
0 9 1
0 57 1
0 6 1
0 59 2
0 60 1
0 61 1
0 66 3
0 67 1
0 68 1
0 69 1
0 70 6
0 72 2
0 75 1
0 76 1
0 78 1
0 79 5
0 81 2
0 82 1
0 83 2
0 84 2
0 85 2
0 55 1
0 89 2
0 90 1
0 91 1
0 92 1
0 93 1
0 94 4
0 95 2
0 96 3
0 98 14
0 99 1
0 100 2
0 101 5
0 103 7
0 104 4
0 105 3
0 106 1
0 107 1
doc word freq
98 142 1
99 129 1
99 131 4
99 5 2
99 134 3
99 1 2
99 136 1
99 137 1
99 10 1
99 139 1
99 13 1
99 16 1
99 3 1
99 20 1
99 22 1
99 25 1
99 27 1
99 28 2
99 29 1
99 30 2
99 133 1
99 36 3
99 37 1
99 42 3
99 45 6
99 46 1
99 47 2
99 8 1
99 115 2
99 52 1
99 53 1
99 138 1
99 55 4
99 57 2
99 61 1
99 63 1
99 67 1
99 69 1
99 70 1
99 72 2
99 73 1
99 74 2
99 75 3
99 76 4
99 77 1
99 79 3
99 84 3
99 85 1
99 89 1
99 91 3
99 94 5
99 144 1
99 98 2
99 99 3
99 101 1
99 102 2
99 103 4
99 105 1
99 107 1
99 108 2
99 109 1
99 111 1
99 114 2
99 19 2
99 116 2
99 118 3
99 119 1
99 121 1
99 9 1
99 123 1
99 127 1

58. data
{
int<lower=2>
K;
#
num
topics
int<lower=2>
V;
#
num
words
int<lower=1>
M;
#
num
docs
int<lower=1>
N;
#
total
word
instances
int<lower=1,upper=V>
W[N];
#
word
n
int<lower=1>
Freq[N];
#
frequency
of
word
n
int<lower=1,upper=N>
Offset[M,2];
#
range
of
word
index
per
doc
vector<lower=0>[K]
Alpha;
#
topic
prior
vector<lower=0>[V]
Beta;
#
word
prior
}
parameters
{
simplex[K]
theta[M];
#
topic
dist
for
doc
m
simplex[V]
phi[K];
#
word
dist
for
topic
k
}
model
{
#
prior
for
(m
in
1:M)
theta[m]
~
dirichlet(Alpha);
for
(k
in
1:K)
phi[k]
~
dirichlet(Beta);
#
likelihood
for
(m
in
1:M)
{
for
(n
in
Offset[m,1]:Offset[m,2])
{
real
gamma[K];
for
(k
in
1:K)
gamma[k]
<-­‐
log(theta[m,k])
+
log(phi[k,W[n]]);
increment_log_prob(Freq[n]
*
log_sum_exp(gamma));
}
}
}

63. ✓

64. ✓