compute_pdf_and_cdf {aghq}

Density and Cumulative Distribution Function

Description

Compute the density and cumulative distribution function of the approximate posterior. The density is approximated on a find grid using a polynomial interpolant. The CDF can't be computed exactly (if it could, you wouldn't be using quadrature!), so a fine grid is laid down and the CDF is approximated at each grid point using a simpler integration rule and a polynomial interpolant. This method tends to work well, but won't always.

Usage

compute_pdf_and_cdf(obj, ...)

## Default S3 method:
compute_pdf_and_cdf(
  obj,
  transformation = default_transformation(),
  finegrid = NULL,
  interpolation = "auto",
  ...
)

## S3 method for class 'list'
compute_pdf_and_cdf(obj, transformation = default_transformation(), ...)

## S3 method for class 'aghq'
compute_pdf_and_cdf(obj, transformation = obj$transformation, ...)

Arguments

obj

Either the output of aghq::aghq(), its list of marginal distributions (element marginals), or an individual data.frame containing one of these marginal distributions as output by aghq::marginal_posterior().

...

Used to pass additional arguments.

transformation

Optional. Calculate pdf/cdf for a transformation of the parameter whose posterior was normalized using adaptive quadrature. transformation is either: a) an aghqtrans object returned by aghq::make_transformation, or b) a list that will be passed to that function internally. See ?aghq::make_transformation for details.

finegrid

Optional, a grid of values on which to compute the CDF. The default makes use of the values in margpost but if the results are unsuitable, you may wish to modify this manually.

interpolation

Which method to use for interpolating the marginal posterior, 'polynomial' (default) or 'spline'? If k > 3 then the polynomial may be unstable and you should use the spline, but the spline doesn't work unless k > 3 so it's not the default. See interpolate_marginal_posterior().

Value

A tbl_df/tbl/data.frame with columns theta, pdf and cdf corresponding to the value of the parameter and its estimated PDF and CDF at that value.

See Also

Other summaries: compute_quantiles(), interpolate_marginal_posterior(), marginal_posterior()

Examples

logfteta2d <- function(eta,y) {
  # eta is now (eta1,eta2)
  # y is now (y1,y2)
  n <- length(y)
  n1 <- ceiling(n/2)
  n2 <- floor(n/2)
  y1 <- y[1:n1]
  y2 <- y[(n1+1):(n1+n2)]
  eta1 <- eta[1]
  eta2 <- eta[2]
  sum(y1) * eta1 - (length(y1) + 1) * exp(eta1) - sum(lgamma(y1+1)) + eta1 +
    sum(y2) * eta2 - (length(y2) + 1) * exp(eta2) - sum(lgamma(y2+1)) + eta2
}
set.seed(84343124)
n1 <- 5
n2 <- 5
n <- n1+n2
y1 <- rpois(n1,5)
y2 <- rpois(n2,5)
objfunc2d <- function(x) logfteta2d(x,c(y1,y2))
funlist2d <- list(
  fn = objfunc2d,
  gr = function(x) numDeriv::grad(objfunc2d,x),
  he = function(x) numDeriv::hessian(objfunc2d,x)
)
opt_sparsetrust_2d <- optimize_theta(funlist2d,c(1.5,1.5))
margpost <- marginal_posterior(opt_sparsetrust_2d,3,1) # margpost for theta1
thepdfandcdf <- compute_pdf_and_cdf(margpost)
with(thepdfandcdf,{
  plot(pdf~theta,type='l')
  plot(cdf~theta,type='l')
})