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 The conjugate prior of the multivariate Gaussian is comprised of the multi- plication of two distributions, one for each parameter, with a relationship to be implied later. Over the mean, , is another multivariate Gaussian; over the precision, , is the Wishart distribution. Non-conjugate prior distribution assessment for multivariate normal sampling Paul H. Garthwaite The Open University, Milton Keynes, UK and Shafeeqah A. Al-Awadhi University of Kuwait, Kuwait [Received March 1999. Final revision August 2000] Summary. Elicitation methods are proposed for quantifying expert opinion about a multivariate normal The conjugate prior for a Gaussian likelihood estimated on the variance is the inverse-gamma distribution. Multivariate Normal With known covariance matrix The conjugate prior for a multivariate normal likelihood with a known covariance matrix is the Inverse-Wishart distribution The question comes from the fact that log-normal distribution and gamma distribution are conjugate. And now I am working on the log multivariate normal distribution, but I do not know the the conjugate prior of that. It seems something related to gamma or wishart distribution. But I cannot find anything related. $\endgroup$ – piapple Apr 19 I Moving from univariate to multivariate distributions. I The multivariate normal (MVN) distribution. I Conjugate for the MVN distribution. I The inverse Wishart distribution. I Conjugate for the MVN distribution (but on the covariance matrix). I Combining the MVN with inverse Wishart. 2 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 conjugate prior for the mean term of a multivariate normal distribution is a multivariate normal distribution: p( jX) /p( )p(Xj ); (11) where p( ) is a multivariate normal distribution, ˘N( 0; 0). The implication of this prior is that the mean term has a Gaussian distribution across the space that it might lie in: generally large values of Multivariate Normal prior. For a Normal likelihood with known variance, the conjugate prior is another Normal distribution with parameters $\mu_\beta$ and $\Sigma_\beta$. The parameter $\mu_\beta$ describes the initial values for $\beta$ and $\Sigma_\beta$ describes how uncertain we are of these values. Conjugate Bayesian analysis of the Gaussian distribution Kevin P. Murphy∗ [email protected] Last updated October 3, 2007 1 Introduction 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.
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