RBesT (version 1.8-1)

preddist: Predictive Distributions for Mixture Distributions

Description

Predictive distribution for mixture of conjugate distributions (beta, normal, gamma).

Usage

preddist(mix, ...)
# S3 method for betaMix
preddist(mix, n = 1, ...)
# S3 method for normMix
preddist(mix, n = 1, sigma, ...)
# S3 method for gammaMix
preddist(mix, n = 1, ...)
# S3 method for mvnormMix
preddist(mix, ...)

Value

The function returns for a normal, beta or gamma mixture the matching predictive distribution for \(y_n\).

Arguments

mix

mixture distribution

...

includes arguments which depend on the specific prior-likelihood pair, see description below.

n

predictive sample size, set by default to 1

sigma

The fixed reference scale of a normal mixture. If left unspecified, the default reference scale of the mixture is assumed.

Methods (by class)

• preddist(betaMix): Obtain the matching predictive distribution for a beta distribution, the BetaBinomial.

• preddist(normMix): Obtain the matching predictive distribution for a Normal with constant standard deviation. Note that the reference scale of the returned Normal mixture is meaningless as the individual components are updated appropriatley.

• preddist(gammaMix): Obtain the matching predictive distribution for a Gamma. Only Poisson likelihoods are supported.

• preddist(mvnormMix): Multivariate normal mixtures predictive densities are not (yet) supported.

Supported Conjugate Prior-Likelihood Pairs

Prior/Posterior	Likelihood	Predictive	Summaries
Beta	Binomial	Beta-Binomial	n, r
Normal	Normal (fixed \(\sigma\))	Normal	n, m, se
Gamma	Poisson	Gamma-Poisson	n, m
Gamma	Exponential	Gamma-Exp (not supported)	n, m

Details

Given a mixture density (either a posterior or a prior)

$$p(\theta,\mathbf{w},\mathbf{a},\mathbf{b})$$

and a data likelihood of

$$y|\theta \sim f(y|\theta),$$

the predictive distribution of a one-dimensional summary \(y_n\) of $n$ future observations is distributed as

$$y_n \sim \int p(\theta,\mathbf{w},\mathbf{a},\mathbf{b}) \, f(y_n|\theta) \, d\theta .$$

This distribution is the marginal distribution of the data under the mixture density. For binary and Poisson data \(y_n = \sum_{i=1}^n y_i\) is the sum over future events. For normal data, it is the mean\(\bar{y}_n = 1/n \sum_{i=1}^n y_i\).

Examples

# Example 1: predictive distribution from uniform prior.
bm <- mixbeta(c(1, 1, 1))
bmPred <- preddist(bm, n = 10)
# predictive proabilities and cumulative predictive probabilities
x <- 0:10
d <- dmix(bmPred, x)
names(d) <- x
barplot(d)
cd <- pmix(bmPred, x)
names(cd) <- x
barplot(cd)
# median
mdn <- qmix(bmPred, 0.5)
mdn

# Example 2: 2-comp Beta mixture

bm <- mixbeta(inf = c(0.8, 15, 50), rob = c(0.2, 1, 1))
plot(bm)
bmPred <- preddist(bm, n = 10)
plot(bmPred)
mdn <- qmix(bmPred, 0.5)
mdn
d <- dmix(bmPred, x = 0:10)
# \donttest{
n.sim <- 100000
r <- rmix(bmPred, n.sim)
d
table(r) / n.sim
# }

# Example 3: 3-comp Normal mixture

m3 <- mixnorm(c(0.50, -0.2, 0.1), c(0.25, 0, 0.2), c(0.25, 0, 0.5), sigma = 10)
print(m3)
summary(m3)
plot(m3)
predm3 <- preddist(m3, n = 2)
plot(predm3)
print(predm3)
summary(predm3)