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In genomics, the gamma distribution was applied in peak calling step (i.e., in recognition of signal) in ChIP-chip [41] and ChIP-seq [42] data analysis. In Bayesian statistics, the gamma distribution is widely used as a conjugate prior. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution.
A Bayes estimator derived through the empirical Bayes method is called an empirical Bayes estimator. Empirical Bayes methods enable the use of auxiliary empirical data, from observations of related parameters, in the development of a Bayes estimator. This is done under the assumption that the estimated parameters are obtained from a common prior.
But since the posterior is a gamma distribution, the MLE of the marginal turns out to be just the mean of the posterior, which is the point estimate we need. Recalling that the mean μ {\displaystyle \mu } of a gamma distribution G ( α ′ , β ′ ) {\displaystyle G(\alpha ',\beta ')} is simply α ′ β ′ {\displaystyle \alpha '\beta ...
In Bayesian probability theory, if, given a likelihood function (), the posterior distribution is in the same probability distribution family as the prior probability distribution (), 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 ().
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often ...
The Gamma distribution, which describes the time until n consecutive rare random events occur in a process with no memory. The Erlang distribution, which is a special case of the gamma distribution with integral shape parameter, developed to predict waiting times in queuing systems; The inverse-gamma distribution; The generalized gamma distribution
The risk is constant, but the ML estimator is actually not a Bayes estimator, so the Corollary of Theorem 1 does not apply. However, the ML estimator is the limit of the Bayes estimators with respect to the prior sequence π n ∼ N ( 0 , n σ 2 ) {\displaystyle \pi _{n}\sim N(0,n\sigma ^{2})\,\!} , and, hence, indeed minimax according to ...
Consider a data set (,), …, (,), where the are Euclidean vectors and the are scalars.The multiple regression model is formulated as = +. where the are random errors. Zellner's g-prior for is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for , similar to a Jeffreys prior.