degross_lpostBasic: Log-posterior for given spline parameters, big bin (and...
In degross: Density Estimation from GROuped Summary Statistics

Description Usage Arguments Value Author(s) References See Also Examples

Log-posterior for given spline parameters, big bin (and optional: small bin) frequencies, tabulated sample moments and roughness penalty parameter. Compared to degross_lpost, no Fisher information matrix is computed and the gradient evaluation is optional, with a resulting computational gain.

degross_lpostBasic(phi, tau, n.i, degross.data,
                          use.moments = rep(TRUE,4), freq.min = 20, diag.only=FALSE,
                          gradient=FALSE,
                          penalize = TRUE, aa = 2, bb = 1e-6, pen.order = 3)

`phi`	Vector of K B-spline parameters φ to specify the log-density.
`tau`	Roughness penalty parameter.
`n.i`	Small bin frequencies.
`degross.data`	A degrossData.object created using the degrossData function.
`use.moments`	Vector with 4 logicals indicating which tabulated sample moments to use as soft constraints. Defaults: `rep(TRUE,4)`.
`freq.min`	Minimal big bin frequency required to use the corresponding observed moments as soft constraints. Default: `20`.
`diag.only`	Logical indicating whether to ignore the off-diagonal elements of the variance-covariance matrix of the sample central moments. Default: FALSE.
`gradient`	Logical indicating if the gradient (Score) of the \log p(φ\|τ,data) should be computed (default: FALSE).
`penalize`	Logical indicating whether a roughness penalty of order `pen.order` is required (with tau \sim G(aa,bb)). Default: `TRUE`.
`aa`	Real giving the first parameter in the Gamma prior for `tau`. Default: `2`.
`bb`	Real giving the second parameter in the Gamma prior for `tau`. Default: `1e-6`.
`pen.order`	Integer giving the order of the roughness penalty. Default: `3`.

A list containing :

lpost.ni : value of the log-posterior based on the given small bin frequencies n.i and the tabulated sample moments.
lpost.mj : value of the log-posterior based on the big bin frequencies degross.data$freq.j and the tabulated sample moments.
llik.ni : multinomial log-likelihood based on the given small bin frequencies n.i.
llik.mj : multinomial log-likelihood based on the big bin frequencies degross.data$freq.j resulting from n.i.
moments.penalty : log of the joint (asymptotic) density for the observed sample moments.
penalty : \log p(φ|τ) + \log p(τ).
M.j : theoretical moments of the density (resulting from φ) within a big bin.
pi.i : small bin probabilities.
ui : small bin midpoints.
delta : width of the small bins.
gamma.j : big bin probabilities.
tau : reminder of the value of the roughness penalty parameter τ.
phi : reminder of the vector of spline parameters (defining the density).
n.i : reminder of the small bin frequencies given as input.
freq.j : reminder of the big bin frequencies in degross.data$freq.j.

Philippe Lambert p.lambert@uliege.be

Lambert, P. (2021) Moment-based density and risk estimation from grouped summary statistics. arXiv:2107.03883.

degross_lpost, degross, degross.object.

sim = simDegrossData(n=3500, plotting=TRUE,choice=2) ## Generate grouped data
obj.data = degrossData(Big.bins=sim$Big.bins, freq.j=sim$freq.j, m.j=sim$m.j)
print(obj.data)
obj.fit = degross(obj.data) ## Estimate the underlying density
phi.hat = obj.fit$phi ; tau.hat = obj.fit$tau
## Evaluate the log-posterior at convergence
res = degross_lpostBasic(phi=phi.hat, tau=tau.hat, degross.data=obj.data,
                         gradient=TRUE)
print(res)