Predictive Distributions for Mixture Distributions
Predictive distribution for mixture of conjugate distributions (beta, normal, gamma).
preddist(mix, ...) ## S3 method for class 'betaMix' preddist(mix, n = 1, ...) ## S3 method for class 'normMix' preddist(mix, n = 1, sigma, ...) ## S3 method for class 'gammaMix' preddist(mix, n = 1, ...)
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. |
Given a mixture density (either a posterior or a prior)
p(θ,w,a,b)
and a data likelihood of
y|θ ~ f(y|θ),
the predictive distribution of a one-dimensional summary y_n of $n$ future observations is distributed as
y_n ~ \int p(u,w,a,b) \, f(y_n|u) du .
This distribution is the marginal distribution of the data under the mixture density. For binary and Poisson data y_n = ∑_{i=1}^n y_i is the sum over future events. For normal data, it is the mean\bar{y}_n = 1/n ∑_{i=1}^n y_i.
The function returns for a normal, beta or gamma mixture the matching predictive distribution for y_n.
betaMix: Obtain the matching predictive distribution
for a beta distribution, the BetaBinomial.
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.
gammaMix: Obtain the matching predictive distribution
for a Gamma. Only Poisson likelihoods are supported.
| Prior/Posterior | Likelihood | Predictive | Summaries |
| Beta | Binomial | Beta-Binomial | n, r |
| Normal | Normal (fixed σ) | Normal | n, m, se |
| Gamma | Poisson | Gamma-Poisson | n, m |
| Gamma | Exponential | Gamma-Exp (not supported) | n, m
|
# 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) 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)
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