degross_lpost {degross}

Log-posterior (with gradient and Fisher information) for given spline parameters, small bin frequencies, tabulated sample moments and roughness penalty parameter. This function is maximized during the M-step of the EM algorithm to estimate the B-spline parameters entering the density specification.

Description

Log-posterior (with gradient and Fisher information) for given spline parameters, small bin frequencies, tabulated sample moments and roughness penalty parameter. This function is maximized during the M-step of the EM algorithm to estimate the B-spline parameters entering the density specification.

Usage

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

Arguments

phi

Vector of K B-spline parameters \phi 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.

penalize

Logical indicating whether a roughness penalty of order pen.order is required (with \tau \sim G(aa,bb)). Default: TRUE.

aa

Positive real giving the first parameter in the Gamma prior for tau. Default: 2.

bb

Positive 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.

Value

A list containing :

• lpost, 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.

• moments.penalty : ⁠ ⁠log of the joint (asymptotic) density for the observed sample moments.

• penalty : ⁠ ⁠\log p(\phi|\tau) + \log p(\tau).

• Score, Score.ni : ⁠ ⁠score (w.r.t. \phi) of lpost.ni.

• Score.mj : ⁠ ⁠score (w.r.t. \phi) of lpost.mj.

• Fisher & Fisher.ni: ⁠ ⁠information matrix (w.r.t. \phi) of lpost.ni.

• Fisher.mj : ⁠ ⁠information matrix (w.r.t. \phi) of lpost.mj.

• M.j : ⁠ ⁠theoretical moments of the density (resulting from \phi) 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 \tau.

• phi : ⁠ ⁠reminder of the vector of spline parameters (defining the density).

• n.i : ⁠ ⁠reminder of the small bin frequencies given as input.

Author(s)

Philippe Lambert [e-mail]

References

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

See Also

degross_lpostBasic, degross, degross.object.

Examples

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$m.j)
print(obj.data)
obj.fit = degross(obj.data) ## Estimate the underlying density
## Evaluate the log-posterior at convergence
res = with(obj.fit, degross_lpost(phi, tau, n.i, obj.data, diag.only=diag.only))
print(res$Score) ## Score of the log posterior at convergence