AM_find_gamma_NegBin {AntMAN}

Given that the prior on M is a Negative Binomial, find the \gamma hyperparameter of the weights prior to match E(K)=K*, where K* is user-specified

Description

Once the prior on the number of mixture components M is assumed to be a Negative Binomial with parameter r>0 and 0<p<1, with mean is 1+ r*p/(1-p), this function adopts a bisection method to find the value of gamma such that the induced distribution on the number of clusters is centered around a user specifed value K^{*}, i.e. the function uses a bisection method to solve for \gamma (Argiento and Iorio 2019). The user can provide a lower \gamma_{l} and an upper \gamma_{u} bound for the possible values of \gamma. The default values are \gamma_l= 10^{-3} and \gamma_{u}=10. A defaault value for the tolerance is \epsilon=0.1. Moreover, after a maximum number of iteration (default is 31), the function stops warning that convergence has not bee reached.

Usage

AM_find_gamma_NegBin(
  n,
  r,
  p,
  Kstar = 6,
  gam_min = 0.001,
  gam_max = 10000,
  tolerance = 0.1
)

Arguments

n

The sample size.

r

The dispersion parameter r of the Negative Binomial.

p

The probability of failure parameter p of the Negative Binomial.

Kstar

The mean number of clusters the user wants to specify.

gam_min

The lower bound of the interval in which gamma should lie.

gam_max

The upper bound of the interval in which gamma should lie.

tolerance

Level of tolerance of the method.

Value

A value of gamma such that E(K)=K^{*}

Examples

n <- 82
r <- 1
p <- 0.8571
gam_nb= AM_find_gamma_NegBin(n,r,p,Kstar=6, gam_min=0.001,gam_max=10000, tolerance=0.1)
prior_K_nb= AM_prior_K_NegBin(n,gam_nb, r, p)
prior_K_nb%*%1:n