Conjugate Priors for Gamma Distribution with Both Alpha and Beta Unknown

In Bayesian inference, identifying a conjugate prior simplifies posterior calculations by ensuring the posterior distribution belongs to the same probability family as the prior. When sampling data from a Gamma distribution where both the shape parameter α (alpha) and the rate parameter β (beta) are unknown, a simple single-variable conjugate prior does not exist. Instead, the parameters must be treated dependently using a joint conjugate prior distribution. This joint prior framework structures the problem by defining a conditional distribution for β given α alongside a marginal distribution for α.

The joint prior density uses a combination of a Gamma distribution and an exponential-type structure. Given hyperparameters a, b, c, and d, the joint prior density p(α, β) is proportional to: [c^α × β^(α - 1) × e^(-β × b)] / [Γ(α)^d]. When observing a sample dataset of size n with a sum of observations ∑x and a product of observations ∏x, the posterior updates analytically. The updated hyperparameter b becomes (b + ∑x), while the hyperparameter d scales linearly to (d + n). The product multiplier c shifts to (c × ∏x), creating an exact mathematical bridge from prior assumptions to posterior reality without requiring iterative simulation loops.

To implement this setup and visually inspect the joint posterior surface, researchers can use programming environments to map the parameter space. Below are identical implementations in Python and R that calculate the joint log-posterior density grid for a sample dataset, providing the foundational engine needed to evaluate unknown Gamma parameters simultaneously.

Python Implementation

import numpy as np



# Sample data

data = np.array([1.5, 2.3, 3.1, 1.8, 2.7])

n = len(data)

sum_x = np.sum(data)

log_prod_x = np.sum(np.log(data))



# Prior Hyperparameters

a_prior, b_prior, c_prior, d_prior = 2.0, 1.0, 1.0, 1.0



# Define evaluation grids for parameters

alpha_grid = np.linspace(0.5, 5.0, 100)

beta_grid = np.linspace(0.5, 5.0, 100)

log_posterior = np.zeros((100, 100))



# Calculate analytical joint log-posterior grid

for i, a in enumerate(alpha_grid):

    for j, b in enumerate(beta_grid):

        # Updated analytical components

        log_prior = (a * np.log(c_prior) + (a - 1) * np.log(b) - b * b_prior) - (d_prior * np.log(np.math.gamma(a)))

        log_likelihood = n * a * np.log(b) + (a - 1) * log_prod_x - b * sum_x - n * np.log(np.math.gamma(a))

        log_posterior[i, j] = log_prior + log_likelihood



print("Python grid calculation complete. Max log-posterior value:", np.max(log_posterior))

R Implementation

# Sample data

data <- c(1.5, 2.3, 3.1, 1.8, 2.7)

n <- length(data)

sum_x <- sum(data)

log_prod_x <- sum(log(data))



# Prior Hyperparameters

a_prior <- 2.0; b_prior <- 1.0; c_prior <- 1.0; d_prior <- 1.0



# Define evaluation grids for parameters

alpha_grid <- seq(0.5, 5.0, length.out = 100)

beta_grid <- seq(0.5, 5.0, length.out = 100)

log_posterior <- matrix(0, nrow = 100, ncol = 100)



# Calculate analytical joint log-posterior grid

for(i in 1:length(alpha_grid)) {

  for(j in 1:length(beta_grid)) {

    a <- alpha_grid[i]

    b <- beta_grid[j]

    

    log_prior <- (a * log(c_prior) + (a - 1) * log(b) - b * b_prior) - (d_prior * lgamma(a))

    log_likelihood <- n * a * log(b) + (a - 1) * log_prod_x - b * sum_x - n * lgamma(a)

    log_posterior[i, j] <- log_prior + log_likelihood

  }

}



print(paste("R grid calculation complete. Max log-posterior value:", max(log_posterior)))