Seonglae ChoConjugate Distribution, Conjugate Relation
In calculating the posterior probability, when the posterior probability belongs to the same distribution family as the prior probability distribution with likelihood, that prior probability distribution is called a Conjugate Prior. By using a conjugate prior distribution, the posterior probability can be calculated by updating the hyperparameters of the prior probability distribution, making the calculation simple.
Unlike cases where prior distributions cannot be used and numerical integration is necessary, analytical integration is possible. As a result, complex formulas can be handled simply.
Assuming the same distribution, the evidence is the probability of observing the data under given conditions, which can be expressed as a constant.
Marginalization over parameters
The product becomes a constant times a distribution (prior + posterior) → Integral becomes easy
Conjugate prior
In Bayesian probability theory, if, given a likelihood function
p
(
x
∣
θ
)
{\displaystyle p(x\mid \theta )}
, the posterior distribution
p
(
θ
∣
x
)
{\displaystyle p(\theta \mid x)}
is in the same probability distribution family as the prior probability distribution
p
(
θ
)
{\displaystyle p(\theta )}
, the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function
p
(
x
∣
θ
)
{\displaystyle p(x\mid \theta )}
.
https://en.wikipedia.org/wiki/Conjugate_prior