词条 | Normal-gamma distribution |
释义 |
name =normal-gamma| type =density| pdf_image =| cdf_image =| parameters = location (real) }} In probability theory and statistics, the normal-gamma distribution (or Gaussian-gamma distribution) is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.[2] DefinitionFor a pair of random variables, (X,T), suppose that the conditional distribution of X given T is given by meaning that the conditional distribution is a normal distribution with mean and precision — equivalently, with variance Suppose also that the marginal distribution of T is given by where this means that T has a gamma distribution. Here λ, α and β are parameters of the joint distribution. Then (X,T) has a normal-gamma distribution, and this is denoted by PropertiesProbability density functionThe joint probability density function of (X,T) is{{citation needed|date=April 2013}} Marginal distributionsBy construction, the marginal distribution over is a gamma distribution, and the conditional distribution over given is a Gaussian distribution. The marginal distribution over is a three-parameter non-standardized Student's t-distribution with parameters .{{citation needed|date=April 2013}} Exponential familyThe normal-gamma distribution is a four-parameter exponential family with natural parameters and natural statistics .{{citation needed|date=April 2013}} Moments of the natural statisticsThe following moments can be easily computed using the moment generating function of the sufficient statistic:{{citation needed|date=April 2013}} where is the digamma function, ScalingIf then for any b > 0, (bX,bT) is distributed as{{citation needed|date=April 2013}} {{dubious|date=April 2013}} Posterior distribution of the parametersAssume that x is distributed according to a normal distribution with unknown mean and precision . and that the prior distribution on and , , has a normal-gamma distribution for which the density {{pi}} satisfies Suppose i.e. the components of are conditionally independent given and the conditional distribution of each of them given is normal with expected value and variance The posterior distribution of and given this dataset can be analytically determined by Bayes' theorem.[3] Explicitly, where is the likelihood of the data given the parameters. Since the data are i.i.d, the likelihood of the entire dataset is equal to the product of the likelihoods of the individual data samples: This expression can be simplified as follows: where , the mean of the data samples, and , the sample variance. The posterior distribution of the parameters is proportional to the prior times the likelihood. The final exponential term is simplified by completing the square. On inserting this back into the expression above, This final expression is in exactly the same form as a Normal-Gamma distribution, i.e., Interpretation of parametersThe interpretation of parameters in terms of pseudo-observations is as follows:
As a consequence, if one has a prior mean of from samples and a prior precision of from samples, the prior distribution over and is and after observing samples with mean and variance , the posterior probability is Note that in some programming languages, such as Matlab, the gamma distribution is implemented with the inverse definition of , so the fourth argument of the Normal-Gamma distribution is . Generating normal-gamma random variatesGeneration of random variates is straightforward:
Related distributions
Notes1. ^1 Bernardo & Smith (1993, p. 434) 2. ^Bernardo & Smith (1993, pages 136, 268, 434) 3. ^{{cite web |url=http://www.trinity.edu/cbrown/bayesweb/ |title=Archived copy |accessdate=2014-08-05 |deadurl=no |archiveurl=https://web.archive.org/web/20140807091855/http://www.trinity.edu/cbrown/bayesweb/ |archivedate=2014-08-07 |df= }} References
3 : Multivariate continuous distributions|Conjugate prior distributions|Normal distribution |
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