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An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for ...
That is, the relative probability assigned to a volume of a probability space using a Jeffreys prior will be the same regardless of the parameterization used to define the Jeffreys prior. This makes it of special interest for use with scale parameters. [2]
The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities.The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or "degrees of belief") equally among all the possible outcomes under consideration.
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 ().
A standard choice of uninformative prior for this problem is the Jeffreys prior, () /, which is equivalent to adopting a rescaling-invariant flat prior for ln(σ 2). One consequence of adopting this prior is that S 2 /σ 2 remains a pivotal quantity , i.e. the probability distribution of S 2 /σ 2 depends only on S 2 /σ 2 , independent of the ...
This indicates that the data are so uninformative about the parameters that the prior probability of arbitrarily large values still matters in the final answer. In some sense, an improper posterior means that the information contained in the data has not "ruled out" arbitrarily large values.
Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. [1]
For example, this choice of hypotheses and prior probabilities implies the statement "if > 0.49 and < 0.51, then the prior probability of being exactly 0.5 is 0.50/0.51 ≈ 98%".