Beta-Binomial-Example-in-R/BetaBinomial.R at main · JRigh/Beta-Binomial-Example-in-R · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
#--------------------------------------------
# Poisson (likelihood) - Gamma (prior) models
# in R
#--------------------------------------------

# 1. Histogram of Binomial distributed observations

library(tidyverse)
set.seed(2023)
s1 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 1/100))
s2 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 10/100))
s3 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 50/100))
s4 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 60/100))
s5 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 75/100))
s6 <- data.frame('data' = rbinom(n = 10000, size = 10, prob = 90/100))

p1 <- s1 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.01, n = 10') + theme_classic()
scale_color_gradient(low="firebrick1", high="firebrick4")
p2 <- s2 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.1, n = 10')  + theme_classic()
p3 <- s3 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.5, n = 10') + theme_classic()
p4 <- s4 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.6', n = 10) + theme_classic()
scale_color_gradient(low="firebrick1", high="firebrick4")
p5 <- s5 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.75, n = 10')  + theme_classic()
p6 <- s6 %>% ggplot() +
  geom_bar(aes(x = data, y = stat(count / sum(count))), width = 0.75,
           fill = 'darkred') +
  labs(x = 'y', y = 'proportion', title = p~ '= 0.9, n = 10') + theme_classic()

library(gridExtra)
grid.arrange(p1, p2, p3, p4, p5, p6, nrow = 2)


# 2. Density plots of examples of Beta family

par(mfrow = c(2,3))
p = seq(0, 1, length = 500)
plot(p, dbeta(p, 1/2, 1/2), col = 2, lwd = 2, main = 'Beta(1/2, 1/2)', type = 'l')
plot(p, dbeta(p, 1, 1), col = 2, lwd = 2, main = 'Beta(1, 1)', type = 'l')
plot(p, dbeta(p, 5, 1), col = 2, lwd = 2, main = 'Beta(5, 1)', type = 'l')
plot(p, dbeta(p, 1, 5), col = 2, lwd = 2, main = 'Beta(5, 5)', type = 'l')
plot(p, dbeta(p, 5, 5), col = 2, lwd = 2, main = 'Beta(5, 5)', type = 'l')
plot(p, dbeta(p, 5, 40), col = 2, lwd = 2, main = 'Beta(5, 40)', type = 'l')

# 3. Density plots of posteriors

# data
n = 20; p1 = 0.8 # Binomial likelihood parameters
p <- seq(0,1,by=0.001)

# priors
alpha1 = 8; beta1 = 2
prior_mean = alpha1 / (alpha1 + beta1) # [1] 0.05
prior_variance = (alpha1*beta1) / ((alpha1 + beta1)^2 * (alpha1 + beta1 + 1))
prior1 = p^(alpha1 - 1) * (1-p)^(beta1-1) ; prior1 = prior1 / sum(prior1)

# likelihood
set.seed(2023)
data1 = rbinom(n = n, size = 1, prob = p1)
likelihood1 = dbinom(x = p1*n, size = 20, prob = p)

# posterior
lp1 = likelihood1 * prior1 ; posterior1 = lp1 / sum(lp1)
plot(lp1)

# save a copy of entire dataset, training and testing datasets in .csv
write.csv(data1,
          "C:/Users/julia/OneDrive/Desktop/github/2. Jeffrey prior binomial/Binomial_data.csv",
          row.names = FALSE)

# posterior mean and parameters
meandata1 = mean(data1) # [1] 0.8
alpha_posterior = ((alpha1 + n*mean(data1))) # 24
beta_posterior = (n - n*mean(data1) + beta1) # 6


# ggplot

# data
n = 20; p1 = 0.8 # Binomial likelihood parameters

# priors
alpha1 = 8; beta1 = 2
p <- seq(0,1,by=0.001)
prior1 = p^(alpha1 - 1) * (1-p)^(beta1-1) ; prior1 = prior1 / sum(prior1)

# likelihood
likelihood1 = dbinom(x = p1*n, size = 20, prob = p)

# posterior
lp1 = likelihood1 * prior1 ; posterior1 = lp1 / sum(lp1)

# dataframe
data_frame <- data.frame('p' = p, 'prior' = prior1, 'likelihood' = likelihood1, 'posterior' = posterior1)

ggplot(data=data_frame) +
  geom_line(aes(x = p, y = prior1, color = 'Be(8, 2) prior'), lwd = 1.2) +
  geom_line(aes(x = p, y = likelihood1/sum(likelihood1),
                color = 'Scaled likelihood Bin(20, 0.8)'), lwd = 1.2) +
  geom_line(aes(x = p, y = posterior1, color = 'Be(24, 6) posterior'), lwd = 1.2) +
  xlim(0, 1) + ylim(0, max(c(prior1, posterior1, likelihood1 / sum(likelihood1)))) +
  scale_color_manual(name = "Distributions", values = c("Be(8, 2) prior" = "darkred",
                                                        "Scaled likelihood Bin(20, 0.8)" = "black",
                                                        'Be(24, 6) posterior' = 'darkblue')) +
  labs(title = 'Posterior distribution in blue - Be(24,6)',
       subtitle = 'Working example data',
       y="density", x="p") +
  theme(axis.text=element_text(size=8),
        axis.title=element_text(size=8),
        plot.subtitle=element_text(size=10, face="italic", color="darkred"),
        panel.background = element_rect(fill = "white", colour = "grey50"),
        panel.grid.major = element_line(colour = "grey90"))

# Posterior mean, posterior variance and 95% Credible Interval including the sample median
xbar = p1
alpha_prior = 2; beta_prior = 8
alpha_posterior = ((alpha1 + n*mean(data1))) # 24
beta_posterior = (n - n*mean(data1) + beta1) # 6

pmean = alpha_posterior / (alpha_posterior + beta_posterior)
pmean
# [1] 0.8

pvariance = (alpha_posterior *beta_posterior) / ((alpha_posterior + beta_posterior)^2 + (alpha_posterior + beta_posterior + 1) )
pvariance
# [1] 0.1546724

# 95% Cedible Interval obtained by direct sampling (simulation)
set.seed(2023)
round(quantile(rbeta(n = 10^8, alpha_posterior, beta_posterior), probs = c(0.025, 0.5, 0.975)),4)
#   2.5%    50%  97.5%
# 0.6423 0.8067 0.9201

# Posterior mean obtained from direct sampling
set.seed(2023)
mean(rbeta(n = 10^8, alpha_posterior, beta_posterior))
# [1] 0.7999991


# Jeffrey's prior

seq=seq(from = 0, to = 1, by = 0.01)
q=dbeta(seq, 1/2, 1/2)

df=data.frame(seq,q)
ggplot(df, aes(seq)) +
  geom_line(aes(y=q), colour="red3", size = 1.5) +
  xlab("theta ")+ylab("density")+
  geom_hline(yintercept=1, size = 1.5) +
  xlim(0, 1)+ ylim(0, 5) +
  geom_text(x=0.7, y=4, label="Beta(1/2, 1/2)", size = 6)+
  labs(title = 'Jeffreys Beta prior',
       subtitle = 'in red Beta(1/2, 1/2) prior for a Binomial likelihood',
       y="density", x="p") +
  theme(axis.text=element_text(size=8),
        axis.title=element_text(size=8),
        plot.subtitle=element_text(size=10, face="italic", color="darkred"),
        panel.background = element_rect(fill = "white", colour = "grey50"),
        panel.grid.major = element_line(colour = "grey90"))

# posterior mean
mean(rbeta(n=100000000, shape1 = 16.5, shape2 = 4.5))
# 0.7857248

# credible interval
quantile(rbeta(n=100000000, shape1 = 16.5, shape2 = 4.5), probs = c(0.025, 0.975))
#      2.5%     97.5%
# 0.5917689 0.9284637

#----
# end
#----