Gradient of the Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization)

Computes the gradient vector (vector of first partial derivatives) of the negative log-likelihood function for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by gamma ($\gamma$) and delta ($\delta$), corresponding to the standard Beta distribution with shape parameters shape1 = gamma and shape2 = delta + 1. The gradient is useful for optimization algorithms.

Usage

grbeta(par, data)

Arguments

par: A numeric vector of length 2 containing the distribution parameters in the order: gamma ($\gamma > 0$), delta ($\delta \ge 0$).
data: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive).

Value

Returns a numeric vector of length 2 containing the partial derivatives of the negative log-likelihood function $-\ell(\theta | \mathbf{x})$ with respect to each parameter: $(-\partial \ell/\partial \gamma, -\partial \ell/\partial \delta)$. Returns a vector of NaN if any parameter values are invalid according to their constraints, or if any value in data is not in the interval (0, 1).

Details

This function calculates the gradient of the negative log-likelihood for a Beta distribution with parameters shape1 = gamma ($\gamma$) and shape2 = delta + 1 ($\delta+1$). The components of the gradient vector ($-\nabla \ell(\theta | \mathbf{x})$) are:

$$ -\frac{\partial \ell}{\partial \gamma} = n[\psi(\gamma) - \psi(\gamma+\delta+1)] - \sum_{i=1}^{n}\ln(x_i) $$ $$ -\frac{\partial \ell}{\partial \delta} = n[\psi(\delta+1) - \psi(\gamma+\delta+1)] - \sum_{i=1}^{n}\ln(1-x_i) $$

where $\psi(\cdot)$ is the digamma function (digamma). These formulas represent the derivatives of $-\ell(\theta)$, consistent with minimizing the negative log-likelihood. They correspond to the relevant components of the general GKw gradient (grgkw) evaluated at $\alpha=1, \beta=1, \lambda=1$. Note the parameterization: the standard Beta shape parameters are $\gamma$ and $\delta+1$.

References

Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume 2 (2nd ed.). Wiley.

Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. Journal of Statistical Computation and Simulation, 81(7), 883-898. doi:10.1080/00949650903530745

(Note: Specific gradient formulas might be derived or sourced from additional references).

Author

Lopes, J. E.

Examples

set.seed(123)
x <- rbeta_(200, gamma = 2, delta = 3)
par <- c(gamma = 2, delta = 3)

## Gradient of the negative log-likelihood llbeta(), not of the log-likelihood
g <- grbeta(par, x)
g
#> [1] -2.704143 -2.096572

## A small step against the gradient lowers llbeta()
llbeta(par - 1e-4 * g, x) < llbeta(par, x)
#> [1] TRUE

## Agrees with a numerical derivative of llbeta()
if (requireNamespace("numDeriv", quietly = TRUE))
  all.equal(g, numDeriv::grad(llbeta, par, data = x), tolerance = 1e-6)
#> [1] TRUE