In wiesenfa/ESS: Quantification of prior informativeness in terms of prior effective and effective current sample size

library(RBesT)
library(ESS)
knitr::opts_chunk$set(
  eval=T, 
  collapse = TRUE,
  comment = "#>"
)

This vignette reproduces Section 3 in Wiesenfarth and Calderazzo (2019) and visualizes how plotECSS() can be used to quantify the impact of a prior (including mixture priors and empirical Bayes power priors/commensurate priors) in terms of effective current sample sizes on a grid of true values of the data generating process.

The effective current sample size (ECSS) is defined as follows:

Let $\pi$ be the prior of interest (specified by priorlist) with mean $\theta_\pi$, $\pi_b$ a baseline prior (an objective or reference prior) (specified by prior.base) and $f_n(y_{1:n} | \theta_0)$ be the data distribution.

The ECSS at target sample size $k$ (specified by n.target) is defined as the sample size $m$ which minimizes $$ECSS=argmin_{m} | D^{\theta_0}{MSE}({\pi}(\theta|y{1:(k-m)})) - D^{\theta_0}{MSE}(\pi_b(\theta|y{1:k})) |$$

where $D^{\theta_0}{MSE}$ is the mean squared error measure induced by the posterior mean estimate $\mbox{E}\pi(\theta|y_{1:k})$, $$D^{\theta_0}{MSE}({\pi}(\theta|y{1:k}))=E_{y|\theta_0 } [ E_\pi(\theta|y_{1:k})-\theta_0 ]^2$$ or a different target measure specified by D. The true parameter value $\theta_0$ (specified by grid) is assumed to be known.

Normal Outcome

  # standard deviation
    sigma=1
  # baseline
    rob=c(0,10)
    vague <-mixnorm(vague=c(1, rob), sigma=sigma)
  # prior with nominal prior ESS=50
    inf=c(0,1/sqrt(50))
    info <-mixnorm(informative=c(1, inf), sigma=sigma)
  # robust mixture
    mix50 <-mixnorm(informative=c(.5, inf),vague=c(.5, rob), sigma=sigma)
  # emprirical Bayes power prior / emprirical Bayes commensurate prior
    pp <-as.powerprior(info)

  plotECSS(priorlist=list(informative=info,mixture=mix50,powerprior=pp), 
            grid=((0:50)/50), 
            n.target=200, min.ecss=-150,sigma=sigma,
            prior.base=vague, progress="none", D=MSE)

Binary outcome

# uniform baseline prior
  rob=c(1,1)
  vague <- mixbeta(rob=c(1.0, rob))
# prior with nominal prior ESS=20
  inf=c(4, 16)
  info=mixbeta(inf=c(1, inf))
  prior.mean=inf[1]/sum(inf)

# robust mixture
  mix50 <-mixbeta(informative=c(.5, inf), vague=c(.5, rob))
# emprirical Bayes power prior / emprirical Bayes commensurate prior
    pp <-as.powerprior(info)
## emprirical Bayes commensurate prior (takes long)
#  cp <-as.commensurateEB(info)

plotECSS(priorlist=list(informative=info,mixture=mix50,powerprior=pp), 
          grid=((0:10)/10), #use ((0:40)/40) for better resolution
          n.target=40, min.ecss=-50,sigma=sigma,
          prior.base=vague, progress="none", D=MSE)

References

Wiesenfarth, M., Calderazzo, S. (2019). Quantification of Prior Impact in Terms of Effective Current Sample Size. Submitted.

sessionInfo

sessionInfo()

wiesenfa/ESS documentation built on June 19, 2019, 4:19 p.m.