example

library(imprecise101)

Imprecise Dirichlet Model

4. Bag of Marbles Example

This example is taken from Walley (1996).

4.1. Probabilities of Future Outcomes

For \(s=2\), \(\overline{P}(R|n)=0.375\) and \(\underline{P}(R|n)=0.125\).

op <- idm(nj=1, s=2, N=6, k=4)
c(op$p.lower, op$p.upper)
#> [1] 0.125 0.375

For \(s=1\), \(\overline{P}(R|n)=0.286\) and \(\underline{P}(R|n)=0.143\).

op <- idm(nj=1, s=1, N=6, k=4)
round(c(op$p.lower, op$p.upper),3)
#> [1] 0.143 0.286

For \(s=0\), \(P(R|n)=0.167\) irrespective of \(\Omega\).

op <- idm(nj=1, s=0, N=6, k=4)
round(c(op$p.lower, op$p.upper),3)
#> [1] 0.167 0.167

Table 1. \(P(R|n)\) for different choices of \(\Omega\) and \(s\) (on page 20)

r1 <- c(idm(nj=1, s=1, N=6, k=4)$p, 
        idm(nj=1, s=2, N=6, k=4)$p, 
        idm(nj=1, s=4/2, N=6, k=4)$p, 
        idm(nj=1, s=4, N=6, k=4)$p) # Omega 1

r2 <- c(idm(nj=1, s=1, N=6, k=2)$p, 
        idm(nj=1, s=2, N=6, k=2)$p, 
        idm(nj=1, s=2/2, N=6, k=2)$p, 
        idm(nj=1, s=2, N=6, k=2)$p) # Omega 2 

r3 <- c(idm(nj=1, s=1, N=6, k=3, cA=2)$p, 
        idm(nj=1, s=2, N=6, k=3, cA=2)$p, 
        idm(nj=1, s=3/2, N=6, k=3, cA=2)$p, 
        idm(nj=1, s=3, N=6, k=3, cA=2)$p) # Omega 3

tb1 <- rbind(r1, r2, r3)
rownames(tb1) <- c("Omega1", "Omega2", "Omega3")
colnames(tb1) <- c("s=1", "s=2", "s=k/2", "s=k")
round(tb1,3)
#>          s=1   s=2 s=k/2   s=k
#> Omega1 0.179 0.188 0.188 0.200
#> Omega2 0.214 0.250 0.214 0.250
#> Omega3 0.238 0.292 0.267 0.333

For \(M=6\) and \(s=1\), the CDF are:

mat <- cbind(
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=0)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=1)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=2)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=3)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=4)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=5)),
  unlist(pbetabinom(M=6, x=1, s=1, N=6, y=6))
)
colnames(mat) <- c("y=0", "y=1", "y=2", "y=3", "y=4", "y=5", "y=6")
round(mat, 3)
#>       y=0   y=1   y=2   y=3   y=4   y=5 y=6
#> p.l 0.227 0.500 0.727 0.879 0.960 0.992   1
#> p.u 0.500 0.773 0.909 0.970 0.992 0.999   1

4.2 Means and Standard Deviations for \(\theta_R\)

For \(s=2\), \(\overline{E}(\theta_R|n) = 0.375\), \(\underline{E}(\theta_R|n)=0.125\), \(\overline{\sigma}(\theta_R|n)=0.188\), and \(\underline{\sigma}(\theta_R|n)=0.110\).

For \(s=1\), \(\overline{E}(\theta_R|n) = 0.286\), \(\underline{E}(\theta_R|n)=0.143\), \(\overline{\sigma}(\theta_R|n)=0.164\), and \(\underline{\sigma}(\theta_R|n)=0.124\).

op <- idm(nj=1, s=2, N=6, k=4)
round(c(op$p.upper, op$p.lower, op$s.upper, op$s.lower),3)
#> [1] 0.375 0.125 0.188 0.110

op <- idm(nj=1, s=1, N=6, k=4)
round(c(op$p.upper, op$p.lower, op$s.upper, op$s.lower),3)
#> [1] 0.286 0.143 0.164 0.124

4.3 Credible Intervals for \(\theta_R\)

For \(s=2\), 95%, 90%, and 50% credible intervals are \([0.0031, 0.6587]\), \([0.0066, 0.5962]\), and \([0.0481, 0.3656]\), respectively.

For \(s=1\), 95%, 90%, and 50% credible intervals are \([0.0076, 0.5834]\), \([0.0150, 0.5141]\), \([0.0761, 0.2958]\), respectively.

round(hpd(alpha=3, beta=5, p=0.95),4) # s=2
#>      a      b 
#> 0.0031 0.6588
round(hpd(alpha=3, beta=5, p=0.90),4) # s=2
#>      a      b 
#> 0.0066 0.5962
round(hpd(alpha=3, beta=5, p=0.50),4) # s=2 (required for message of failure)
#>      a      b 
#> 0.3641 0.9048
round(hpd(alpha=2, beta=5, p=0.95),4) # s=1
#>      a      b 
#> 0.0076 0.5834
round(hpd(alpha=2, beta=5, p=0.90),4) # s=1
#>     a     b 
#> 0.015 0.514
round(hpd(alpha=2, beta=5, p=0.50),4) # s=1
#>      a      b 
#> 0.0760 0.2957

HPD interval, uniform prior \((s=2) [0.0133, 0.5273]\).

x <- pscl::betaHPD(alpha=2, beta=6, p=0.95, plot=FALSE)
round(x,4)
#> [1] 0.0133 0.5273

4.4 Testing Hypotheses about \(\theta_R\)

Test \(H_0: \theta_R \ge 1/2\) against \(H_a: \theta_R < 1/2\). The posterior lower and upper probabilities of \(H_0\) are \(\underline{P}(H_0|n)=0.00781\) and \(\overline{P}(H_0|n)=0.227\).

fn <- function(x) choose(7,1)*(1-x)^6
integrate(f=fn, lower=1/2, upper=1)$value
#> [1] 0.0078125

fn <- function(x) dbeta(x, 3, 5)
integrate(f=fn, lower=1/2, upper=1)$value 
#> [1] 0.2265625

Imprecise Beta Model

5. CT vs ECMO Example

This example is taken from Walley (1996).

5.5 Deciding when to terminate randomized trials

x <- seq(-0.99, 0.99, 0.02)
ymax <- ymin <- numeric(length(x))
for(i in 1:length(x)) ymin[i] <- dbetadif(x=x[i], a1=9,b1=2,a2=8,b2=4)
for(i in 1:length(x)) ymax[i] <- dbetadif(x=x[i], a1=11,b1=0.01,a2=6,b2=6)

plot(x=x, y=cumsum(ymin)/sum(ymin), type="l", ylab="F(z)", xlab="z",
     main=expression(paste("Fig 1. Posterior upper and lower CDFs for ", psi, 
                           "=", theta[e]-theta[c])))
points(x=x, y=cumsum(ymax)/sum(ymax), type="l")

Further inferences concerning \(\theta_c\), \(\theta_e\), and \(\psi\)

This example is taken from Walley (1996).

Since the imprecision is the area between lower and upper probabilities, \(A = \overline{E} - \underline{E}\) =0.3475766

TODO

  • Add the examples of decision problem from Walley et al. (1996).

References

Bernard, Jean-Marc. 2005. “An Introduction to the Imprecise Dirichlet Model for Multinomial Data.” International Journal of Approximate Reasoning 39 (2): 123–50. https://doi.org/https://doi.org/10.1016/j.ijar.2004.10.002.
Walley, P. 1991. Statistical Reasoning with Imprecise Probabilities. Chapman & Hall/CRC Monographs on Statistics & Applied Probability. Taylor & Francis.
Walley, Peter. 1996. “Inferences from Multinomial Data: Learning about a Bag of Marbles.” Journal of the Royal Statistical Society. Series B (Methodological) 58 (1): 3–57. http://www.jstor.org/stable/2346164.
Walley, Peter, Lyle Gurrin, and Paul Burton. 1996. “Analysis of Clinical Data Using Imprecise Prior Probabilities.” Journal of the Royal Statistical Society. Series D (The Statistician) 45 (4): 457–85. http://www.jstor.org/stable/2988546.

Appendix