When do you call a posterior a conjugate prior?

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.

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.

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 think of conditioning on conjugate priors?

One can think of conditioning on conjugate priors as defining a kind of (discrete time) dynamical system: from a given set of hyperparameters, incoming data updates these hyperparameters, so one can see the change in hyperparameters as a kind of “time evolution” of the system, corresponding to “learning”.

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.

How is the conjugate prior of an exponential determined?

All members of the exponential family have conjugate priors. The form of the conjugate prior can generally be determined by inspection of the probability density or probability mass function of a distribution. For example, consider a random variable which consists of the number of successes

When does the posterior have the same algebraic form as the prior?

For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameter values). Such a choice is a conjugate prior . A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise numerical integration may be necessary.

How are hyperparameters used in conjugate prior distributions?

In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. This can help both in providing an intuition behind the often messy update equations, as well as to help choose reasonable hyperparameters for a prior.

How are conjugate priors useful in machine learning?

Concept of conjugate prior is extremely useful because, conjugate priors reduce the Bayesian updating to only modifying some parameters in the prior distribution. They are also necessary to learn and understand Machine Learning from Bayesian approach. Jupyter notebook used for this post can be found in my GitHub (link below).

How are conjugate priors similar to eigenfunctions?

Analogy with eigenfunctions. Conjugate priors are analogous to eigenfunctions in operator theory, in that they are distributions on which the “conditioning operator” acts in a well-understood way, thinking of the process of changing from the prior to the posterior as an operator.