- The document discusses random number generation and probability distributions. It presents methods for generating random numbers from Bernoulli, binomial, beta, and multinomial distributions using random bits generated from linear congruential generators. - Graphical examples are shown comparing histograms of generated random samples to theoretical probability density functions. Code examples in R demonstrate how to simulate random number generation from various discrete distributions. - The goal is to introduce different methods for random number generation from basic discrete distributions that are important for modeling random phenomena and Monte Carlo simulations.

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
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doc word freq
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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. ✓