Modelos discretos conjugados

Modelo Beta Bernoulli

\[ p(\theta) \propto \theta^{\alpha-1}(1-\theta)^{\beta-1} I_{(0,1)}(\theta) \]

es un modelo conjugado para la distribución bernoulli.

\[ p(\theta | \bf{x}) \propto \theta^{\alpha+t-1}(1-\theta)^{\beta+n-t-1} I_{(0,1)}(\theta) \] Los parámetros posteriores \[ \alpha_n = \alpha+t \]

\[ \beta_n = \beta+n-t \]

\[ E(\theta | \bf{x}) = \frac{\alpha_n}{\alpha_n+\beta_n} \]

\[ E(\theta | \bf{x}) = \frac{\alpha_n}{\alpha_n+\beta_n} = \frac{\alpha + t}{\alpha+\beta+n} \]

\[ E(\theta | \bf{x}) = \frac{\alpha_n}{\alpha_n+\beta_n} = \frac{\alpha + t}{\alpha+\beta+n}= \frac{\alpha}{\alpha+\beta+n}+\frac{ t}{\alpha+\beta+n} \] \[ E(\theta | \bf{x}) = \frac{\alpha_n}{\alpha_n+\beta_n} = \frac{\alpha + t}{\alpha+\beta+n}= \frac{\alpha}{\alpha+\beta+n}+\frac{ t}{\alpha+\beta+n}= \frac{\alpha}{\alpha+\beta+n}\frac{\alpha+\beta}{\alpha+\beta}+\frac{ t}{\alpha+\beta+n}\frac{n}{n} \]

\[ E(\theta | \bf{x}) = \frac{\alpha}{\alpha+\beta+n}\frac{\alpha+\beta}{\alpha+\beta}+\frac{ t}{\alpha+\beta+n}\frac{n}{n} =\frac{\alpha+\beta}{\alpha+\beta+n}\frac{\alpha}{\alpha+\beta}+\frac{ n}{\alpha+\beta+n}\frac{t}{n} \]

\[ E(\theta | \bf{x}) = \frac{\alpha}{\alpha+\beta+n}\frac{\alpha+\beta}{\alpha+\beta}+\frac{ t}{\alpha+\beta+n}\frac{n}{n} =\frac{\alpha+\beta}{\alpha+\beta+n}E(\theta)+\frac{ n}{\alpha+\beta+n}\bar{x} \]

\[ E(\theta | \bf{x}) = \frac{\alpha}{\alpha+\beta+n}\frac{\alpha+\beta}{\alpha+\beta}+\frac{ t}{\alpha+\beta+n}\frac{n}{n} =\omega E(\theta)+(1-\omega ) \bar{x} \]

x<-rbinom(30,1,0.3)
x
##  [1] 0 0 0 1 0 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 0

Clasico sobre p

\[H_0: p = p_0\] \[H_a: p \neq p_0\] \[ z = \frac{\hat{p}-p_0}{\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}} \] Se tiene la convergencia en distribución a \(N(0,1)\).

\[ IC(p) := (\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}};\hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}) \]

x<-rbinom(30,1,0.3)
x
##  [1] 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1
t<-sum(x)
n<-length(x)
prop.test(t,n)
## 
##  1-sample proportions test with continuity correction
## 
## data:  t out of n, null probability 0.5
## X-squared = 4.0333, df = 1, p-value = 0.04461
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.1541280 0.4955791
## sample estimates:
##   p 
## 0.3

Intervalo de credibilidad bayesiana

a<-1
b<-1
t<-sum(x)
n<-length(x)
ap <-a+t
bp <-b+n-t
B<-10000
theta_p<-rbeta(B,ap,bp)
plot(density(theta_p))

ic<-c(quantile(theta_p,0.025),quantile(theta_p,0.975))
ic
##      2.5%     97.5% 
## 0.1657047 0.4788451
x<-rbinom(1000,1,0.3)
x
##    [1] 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 0
##   [38] 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1
##   [75] 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0
##  [112] 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 0 1
##  [149] 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1
##  [186] 1 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 0 0 0
##  [223] 0 1 0 0 1 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0
##  [260] 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 1 1
##  [297] 1 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0
##  [334] 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0
##  [371] 0 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0
##  [408] 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1
##  [445] 1 1 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 0
##  [482] 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0
##  [519] 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1
##  [556] 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0
##  [593] 1 0 0 0 1 1 1 0 1 0 0 1 0 0 0 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0
##  [630] 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 0 0
##  [667] 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 1 0 1 0
##  [704] 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0
##  [741] 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0
##  [778] 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1
##  [815] 1 0 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0
##  [852] 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0
##  [889] 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0
##  [926] 1 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0
##  [963] 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0
## [1000] 0
t<-sum(x)
n<-length(x)
prop.test(t,n)
## 
##  1-sample proportions test with continuity correction
## 
## data:  t out of n, null probability 0.5
## X-squared = 162.41, df = 1, p-value < 2.2e-16
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.2699841 0.3275891
## sample estimates:
##     p 
## 0.298

Intervalo de credibilidad bayesiana

a<-1
b<-1
t<-sum(x)
n<-length(x)
ap <-a+t
bp <-b+n-t
B<-10000
theta_p<-rbeta(B,ap,bp)
plot(density(theta_p))

ic<-c(quantile(theta_p,0.025),quantile(theta_p,0.975))
ic
##      2.5%     97.5% 
## 0.2701183 0.3271596
x<-rbinom(100,1,0.3)
x
##   [1] 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 1 0 0 0
##  [38] 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 1
##  [75] 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1
t<-sum(x)
n<-length(x)
prop.test(t,n)
## 
##  1-sample proportions test with continuity correction
## 
## data:  t out of n, null probability 0.5
## X-squared = 12.25, df = 1, p-value = 0.0004653
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.2322385 0.4217920
## sample estimates:
##    p 
## 0.32

Intervalo de credibilidad bayesiana

a<-1
b<-1
t<-sum(x)
n<-length(x)
ap <-a+t
bp <-b+n-t
B<-10000
theta_p<-rbeta(B,ap,bp)
plot(density(theta_p))

ic<-c(quantile(theta_p,0.025),quantile(theta_p,0.975))
ic
##      2.5%     97.5% 
## 0.2366880 0.4170067

Otra forma es de manera teorica

\[ \theta | x \sim beta(a+t,b+n-t) \]

qbeta(0.025,ap,bp)
## [1] 0.2366944
qbeta(0.975,ap,bp)
## [1] 0.4169196