# preddist: Predictive Distributions for Mixture Distributions In RBesT: R Bayesian Evidence Synthesis Tools

## Description

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

## Usage

  1 2 3 4 5 6 7 8 9 10 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, ...) 

## 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.

## Details

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.

## Value

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

## Methods (by class)

• 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.

## Supported Conjugate Prior-Likelihood Pairs

 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

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 # 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) 

RBesT documentation built on Nov. 24, 2021, 5:07 p.m.