found by taking the formula for mean and solving it for Let be the sample mean. Then an initial estimate for is given by: Bayesian estimation of the variance of a normal distribution The scaled inverse chi-squared distribution has a second important application, in the Bayesian estimation of the variance of a Normal distribution. According to Bayes' theorem, the posterior probability distribution for quantities of interest is proportional to the product of a prior distribution for the quantities and a likelihood function: where D represents the data and I represents any initial information about σ2 that we may already have. The simplest scenario arises if the mean μ is already known; or, alternatively, if it is the conditional distribution of σ2 that is sought, for a particular assumed value of μ. Then the likelihood term L(σ2|D) = p(D|σ2) has the familiar form Combining this with the rescaling-invariant prior p(σ2|I) = 1/σ2, which can be argued (e.g. following Jeffreys) to be the least informative possible prior for σ2 in this problem, gives a combined posterior probability This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ2 = s2 = (1/n) Σ (xi-μ)2 Gelman et al remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior the "result is not surprising". In particular, the choice of a rescaling-invariant prior for σ2 has the result that the probability for the ratio of σ2 / s2 has the same form (independent of the conditioning variable) when conditioned on s2 as when conditioned on σ2: In the sampling-theory case, conditioned on σ2, the probability distribution for (1/s2) is a scaled inverse chi-squared distribution; and so the probability distribution for σ2 conditioned on s2, given a scale-agnostic prior, is also a scaled inverse chi-squared distribution. Use as an informative prior If more is known about the possible values of σ2, a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ2(n0, s02) can be a convenient form to represent a more informative prior for σ2, as if from the result of n0 previous observations (though n0 need not necessarily be a whole number): Such a prior would lead to the posterior distribution which is itself a scaled inverse chi-squared distribution. The scaled inverse chi-squared distributions are thus a convenient conjugate prior family for σ2