Gamma-Normal distribution

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The Gamma-Normal distribution is a bivariate compound probability distribution that has gained widespread popularity. It’s a fundamental component of Bayesian statistics, playing an essential role as a conjugate prior of a normal distribution with unknown mean and precision [1]. The distribution has been studied extensively and used in countless applications, from genetics to finance. In this blog post, we’ll dive into the details and explore the Gamma-Normal distribution to better understand its properties and significance in statistical analysis.

Gamma-Normal distribution properties

The Gamma-Normal distribution is a bivariate probability distribution often called the GN Distribution, Gaussian-Normal distribution, or Normal-Gamma distribution.

If X is a normal random variable with probability density function (PDF) φ(x) and CDF Φ(x), then the PDF of the gamma-normal distribution with parameters α, β,µ and σ is defined as [2]

gamma normal distribution pdf

where α > 0, β > 0, σ > 0 and −∞ < µ < ∞.

When α = β = 1, the gamma-normal distribution reduces to the normal distribution with parameters µ and σ.

The cumulative distribution function (CDF) can be written as:

G(x) = γ {α, − log(1 − Φ(x)) / β} / Γ(α).

The shape of the Gamma-Normal distribution changes as more sample data is collected. As a result, the parameters of the model are updated to reflect the new information. This process of updating the parameters is known as updating the posterior. The posterior distribution can then be used to make decisions based on the new information.

In Bayesian statistics, the normal-gamma distribution can be defined a little differently [2]:

Given an n x 1 random vector X and positive random variable Y, then X, Y follows a normal-gamma distribution

X, Y ∼ NG(μ, Λ, a, b),

if:

  1. The distribution of X conditional on Y is a multivariate normal distribution with mean vector μ and covariance matrix (yΛ)−1, and
  2. Y follows a gamma distribution with shape parameter α and rate parameter β:

X|Y ∼ N(μ,(YΛ)−1) ; Y∼Gam(a,b).

Uses of the gamma normal distribution

The gamma normal distribution is a combination of a gamma random variable and a normal/Gaussian random variable; The gamma distribution is often used to model the waiting time until a certain number of events occur. In contrast, the normal distribution is ideal for modeling continuous data, such as height or weight. Thus, this distribution has many applications because it can be used to represent a wide range of physical phenomena, including the sums of independent random variables.

The properties of the Gamma-Normal distribution make it an attractive model for Bayesian inference. A conjugate prior is a distribution that, when used to model the prior beliefs of the parameters in a model, produces a posterior distribution of the same form as the prior distribution. In other words, the shape of the Gamma-Normal distribution is preserved when it is used as a prior distribution in Bayesian analysis. This means that the parameters of the distribution can be updated without having to calculate complex integrals, making Bayesian analysis more straightforward.

One of the essential characteristics of the Gamma-Normal distribution is its flexibility. It can accommodate a wide range of shapes and skewness. This makes it a valuable tool for modeling a wide range of data sets, especially in applications where the distribution of the data is unknown. The Gamma-Normal distribution allows us to use prior knowledge to make more informed decisions about the shape and parameters of the distribution.

In summary, the Gamma-Normal distribution is a critical probability distribution that has been studied extensively in the field of statistics. It is widely used in Bayesian statistics, playing a significant role as a conjugate prior of a normal distribution with unknown mean and precision. Its flexibility in accommodating different shapes and approaches makes it an invaluable tool for modeling a wide range of data sets where the distribution of the data is unknown. As more sample data is collected, the parameters of the model are updated, allowing us to make more informed decisions. The Gamma-Normal distribution has numerous applications and is a powerful tool in statistical analysis.

References

[1] Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley. ISBN 0-471-49464-X

[2] Alzaatreha, A et al. The gamma-normal distribution: Properties and applications. Computational Statistics and Data Analysis 69 (2014) 67–80

[3] Koch KR (2007): “Normal-Gamma Distribution” ; in: Introduction to Bayesian Statistics , ch. 2.5.3, pp. 55-56, eq. 2.212 ; URL: https://www.springer.com/gp/book/9783540727231 ; DOI: 10.1007/978-3-540-72726-2 .

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