, Mikelis Bickis
[ctb], Angela McCourt [ctb]| Title: | Introduction to Imprecise Probabilities |
|---|---|
| Description: | An imprecise inference presented in the study of Walley (1996) <doi:10.1111/j.2517-6161.1996.tb02065.x> is one of the statistical reasoning methods when prior information is unavailable. Functions and utils needed for illustrating this inferential paradigm are implemented for classroom teaching and further comprehensive research. Two imprecise models are demonstrated using multinomial data and 2x2 contingency table data. The concepts of prior ignorance and imprecision are discussed in lower and upper probabilities. Representation invariance principle, hypothesis testing, decision-making, and further generalization are also illustrated. |
| Authors: | Chel Hee Lee [aut, cre]
, Mikelis Bickis
[ctb], Angela McCourt [ctb] |
| Maintainer: | Chel Hee Lee <[email protected]> |
| License: | GPL-3 |
| Version: | 0.2.2.4 |
| Built: | 2024-11-29 09:04:20 UTC |
| Source: | CRAN |
This function computes the predictive posterior density of the outcome of interest under the imprecise Dirichlet prior distribution. It follows a beta-binomial distribution.
dbetabinom(i, M, x, s, N, tA) pbetabinom(M, x, s, N, y)dbetabinom(i, M, x, s, N, tA) pbetabinom(M, x, s, N, y)
i |
number of occurrences of event A in the M future trials |
M |
number of future trials |
x |
number of occurrence of event A in the N previous trials |
s |
learning parameter |
N |
total number of previous trials |
tA |
prior probability of event A under the Dirichlet prior |
y |
maximum number of occurrences of event A in the M future trials |
dbetabinom returns a scalar value of density and pdetabinom returns a list of scalars corresponding to the lower and upper probabilities of the distribution.
pbetabinom(M=6, x=1, s=1, N=6, y=0)pbetabinom(M=6, x=1, s=1, N=6, y=0)
Distribution of Difference of Two Proportions
dbetadif(x, a1, b1, a2, b2)dbetadif(x, a1, b1, a2, b2)
x |
difference of two beta distributions |
a1 |
shape 1 parameter of Beta distribution with control |
b1 |
shape 2 parameter of Beta distribution with control |
a2 |
shape 1 parameter of Beta distribution with treatment |
b2 |
shape 2 parameter of Beta distribution with treatment |
betadif gives a scalar value of density.
Chen, Y., & Luo, S. (2011). A few remarks on 'Statistical distribution of the difference of two proportions' by Nadarajah and Kotz, Statistics in Medicine 2007; 26 (18): 3518-3523. Statistics in Medicine, 30(15), 1913-1915.
This function computes lower and upper posterior probabilities under an imprecise Beta model when prior information is not available.
ibm(n = 10, m = 6, s0 = 2, showplot = TRUE, xlab1 = NA, main1 = NA)ibm(n = 10, m = 6, s0 = 2, showplot = TRUE, xlab1 = NA, main1 = NA)
n |
total of trials |
m |
number of observations realized |
s0 |
learning parameter |
showplot |
logical, TRUE by default |
xlab1 |
x axis text |
main1 |
main title text |
ibm returns data.frame containing posterior probabilities on the mean parameter space.
Walley, P. (1996), Inferences from Multinomial Data: Learning About a Bag of Marbles. Journal of the Royal Statistical Society: Series B (Methodological), 58: 3-34. https://doi.org/10.1111/j.2517-6161.1996.tb02065.x
tc <- seq(0,1,0.1) s <- 2 ibm(n=10, m=6)tc <- seq(0,1,0.1) s <- 2 ibm(n=10, m=6)
This function computes lower and upper posterior probabilities under an imprecise Dirichlet model when prior information is not available.
This function searches for the lower and upper bounds of a given level of the highest posterior density interval under the imprecise Dirichlet prior.
idm(nj, s = 1, N, tj = NA_real_, k, cA = 1) hpd( alpha = 3, beta = 5, p = 0.95, tolerance = 1e-04, maxiter = 100, verbose = FALSE )idm(nj, s = 1, N, tj = NA_real_, k, cA = 1) hpd( alpha = 3, beta = 5, p = 0.95, tolerance = 1e-04, maxiter = 100, verbose = FALSE )
nj |
number of observations in the j th category |
s |
learning parameter |
N |
total number of drawings |
tj |
mean probability associated with the j th category |
k |
number of elements in the sample space |
cA |
the number of elements in the event A |
alpha |
shape1 parameter of beta distribution |
beta |
shape2 parameter of beta distribution |
p |
level of credible interval |
tolerance |
level of error allowed |
maxiter |
maximum number of iterations |
verbose |
logical option suppressing messages |
idm returns a list of lower and upper probabilities.
p.lower |
Minimum of imprecise probabilities |
p.upper |
Maximum of imprecise probabilities |
v.lower |
Variance of lower bound |
v.upper |
Variance of upper bound |
s.lower |
Standard deviation of lower bound |
s.upper |
Standard deviation of upper bound |
p |
Precise probabilty |
p.delta |
Degree of imprecision |
hpd gives a list of scalar values corresponding to the lower and upper bounds of highest posterior probability density region.
Walley, P. (1996), Inferences from Multinomial Data: Learning About a Bag of Marbles. Journal of the Royal Statistical Society: Series B (Methodological), 58: 3-34. https://doi.org/10.1111/j.2517-6161.1996.tb02065.x
idm(nj=1, N=6, s=2, k=4) x <- hpd(alpha=3, beta=5, p=0.95) # c(0.0031, 0.6587) when s=2 # round(x,4); x*(1-x)^5idm(nj=1, N=6, s=2, k=4) x <- hpd(alpha=3, beta=5, p=0.95) # c(0.0031, 0.6587) when s=2 # round(x,4); x*(1-x)^5