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Is the likelihood function a conjugate prior?
If the likelihood function belongs to the exponential family, then a conjugate prior exists, often also in the exponential family; see Exponential family: Conjugate distributions . This section needs additional citations for verification.
When is a posterior distribution called a conjugate distribution?
In Bayesian probability theory, if the posterior distributions p ( θ | x) are in the same probability distribution family as the prior probability distribution p (θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.
When do you call a posterior a conjugate prior?
Conjugate prior. In Bayesian probability theory, if the posterior distributions p ( θ | x) are in the same probability distribution family as the prior probability distribution p (θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.
How to calculate the marginal likelihood of a conjugate distribution?
The parameter space is discrete and finite: Ω = (θ1,…,θp) Ω = ( θ 1, …, θ p); in this case the marginal likelihood can be computed as a finite sum: f Y (y) = p ∑ i=1f Y|Θ(yi|θi)f Θ(θi). f Y ( y) = ∑ i = 1 p f Y | Θ ( y i | θ i) f Θ ( θ i). The prior distribution is a conjugate prior for the sampling distribution.
Why are posterior and prior distributions called conjugate distributions?
In Bayesian prob a bility theory, if the posterior distribution is in the same family of the prior distribution, then the prior and posterior are called conjugate distributions, and the prior is called the conjugate prior to the likelihood function. Initially, you may think okay that’s cool but how exactly that helps us? Why Conjugate Prior?
How to define conjugate priors for an exponential family?
For exponential families the likelihood is a simple standarized function of the parameter and we can define conjugate priors by mimicking the form of the likelihood. Multiplication of a likelihood and a prior that have the same exponential form yields a posterior that retains that form.