legacy/doit2-1d.md

In sieste/doit: Bayesian Computation using Design of Experiments-Based Interpolation Technique

base_name = paste(knitr::current_input(), '_figs/', sep='')
knitr::opts_chunk$set(
  cache.path=paste('_knitr_cache/', base_name, sep='/'),
  fig.path=paste('figure/', base_name, sep='/'),
  dpi=300
)

Reproduce Figure 2 of Joseph 2012, using the improved doit method that avoids negative densities:

The example used is a Bernoulli - Logit-Normal model:

• $y|r \sim Ber(1/(1+exp(-r)))$
• $r \sim N(\mu, \tau^2)$
• using $y = 1, \mu = 1, \tau = 4$

# log h(r), where h(r) = p(y|r)p(r)
h = function(r, y=1) {
  p = 1/(1+exp(-r))
  llik = sum(dbinom(y, 1, p, log=TRUE))
  lprior = dnorm(r, 1, 4, log=TRUE)
  return(exp(llik + lprior))
}

# design points
m = 10
design = data.frame(xx=seq(-10, 20, len=m))
design$hh = sapply(design$xx, h)

the doit2 approximator for 1-dimensional functions

The improved non-negative DoIt method approximated the square root of the unnormalised posterior by kernel density interpolation, and then infers the square. The modifications to the original method are very minor:

doit = function(r, design) {

  m = nrow(design)

  GGfun = function(xx, yy, sigma2) {
    drop(exp(-0.5/sigma2 * outer(xx, yy, '-')^2))
  }

  # leave-one-out MSE as function of the kernel width
  wmscv = function(sigma2) {
    GG_ = GGfun(design$xx, design$xx, sigma2)
    GGinv_ = solve(GG_)
    ee_ = drop(1/diag(GGinv_) * GGinv_ %*% sqrt(design$hh))
    wmscv = 1/m * drop(ee_ %*% diag(diag(GGinv_)) %*% ee_)
    return(wmscv)
  }

  # minimise wmscv wrt sigma2
  opt = optimize(wmscv, c(1e-6, diff(range(design$xx))))
  sigma2_opt = opt$minimum

  # apply approximation
  GG_ = GGfun(design$xx, design$xx, sigma2_opt)
  bb = drop(solve(GG_, sqrt(design$hh)))

  # approximate target function at input points
  bGG2b_ = drop(bb %*% GGfun(design$xx, design$xx, 2*sigma2_opt) %*% bb)
  post_r = sapply(r, function(rr) {
    gg = GGfun(rr, design$xx, sigma2_opt)
    (sum(bb * gg))^2 / (sqrt(pi*sigma2_opt) * bGG2b_)
  })
  return(post_r)
}

rr = seq(-10, 20, .1)
post_r_approx = doit(rr, design)
plot(rr, post_r_approx, type='l')

m2 = 20
design2 = data.frame(xx=seq(-10, 20, len=m2))
design2$hh = sapply(design2$xx, h)
post_r_approx2 = doit(rr, design2)
plot(rr, post_r_approx2, type='l')

sieste/doit documentation built on May 9, 2019, 4:10 p.m.