Conjugate distribution or conjugate pair means a pair of a sampling distribution and a prior distribution for which the resulting posterior distribution belongs into the same parametric family of distributions than the prior distribution. We also say that the prior distribution is a conjugate prior for this sampling distribution. The prior is said to be conjugate to the likelihood. If the prior and the posterior lie in the same family of distributions. For example, if the prior was normal parameterized by some parameters mu and sigma, we'd expect the posterior to be also normal but with some other mean end variance. Useful distribution theory Conjugate prior is equivalent to (μ− γ) √ n0/σ ∼ Normal(0,1). Also 1/σ2|y ∼ Gamma(α,β) is equivalent to 2β/σ2 ∼ χ2 2α. Now if Z ∼Normal(0,1),X χ2ν/ν,thenZ/ √ X tν. Therefore the marginal prior distribution for μ in the bivariate conjugate prior is such that (μ− γ) n0α/β ∼ t2α 6-6 Informative Prior for SPF Construct an informative prior distribution for : I Take prior median SPF to be 16 I P( > 64) = 0:01 I information in prior is worth 25 observations Solve for hyperparameters that are consistent with these quantiles: m0 = log(16), p0 = 25, v0 = p0 1 P( < log(64)) = 0:99 where m0 p SS0=(v0p0) ˘ tv0) SS0 = 185:7 We have a conjugate prior if the posterior as a function of has the same form as the prior. Exponential/Normal posterior: f( jx) = c 1 e ( prior)2 2˙2 prior x The factor of before the exponential means this is not the pdf of a normal distribution. Therefore it is not a conjugate prior. Exponential/Gamma posterior: Note, we have never learned about Gamma sized parameter. Also suppose the prior distribution for θ is one of a family of parametrized distributions. If the posterior distribution for θ is in this family then we say the the prior is a conjugate prior for the likelihood. 3 Beta distribution. In this section, we will show that the beta distribution is a conjugate prior for binomial, Conjugate prior of a normal distribution with unknown mean. Ask Question Asked 3 years, 2 months ago. Active 3 years, 1 month ago. Viewed 3k times 1 $\begingroup$ I'm following these notes to compute the conjugate prior of a normal distribution with unknown mean and known variance. At some point they claim: The Conjugate Prior for the Normal Distribution 5 3 Both variance (˙2) and mean ( ) are random Now, we want to put a prior on and ˙2 together. We could simply multiply the prior densities we obtained in the previous two sections, implicitly assuming and ˙2 are independent. Unfortunately, if we did that, we would not get a conjugate prior. The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be derived in closed form. Unfortunately, different books use different conventions on how to parameterize the various When you know that your prior is a conjugate prior, you can skip the posterior = likelihood * prior computation. Furthermore, if your prior distribution has a closed-form form expression, you already know what the maximum posterior is going to be. In the example above, the beta distribution is a conjugate prior to the binomial likelihood.
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This video provides a short introduction to the concept of 'conjugate prior distributions'; covering its definition, examples and why we may choose to specif... This video provides a short introduction to the concept of 'conjugate prior distributions'; covering its definition, examples and why we may choose to specif... This video provides some intuition for the properties of the posterior distribution for the case of a normal prior and likelihood. If you are interested in s... A normally distributed prior is the conjugate prior for a Normal likelihood function. This video works through the derivation of the parameters of the resul... Putting a Gamma distribution prior on the inverse variance. Also a pre-cursor to Relevance Vector Machines
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