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Posterior predictive Normal distribution w. conjugate prior on the mean.


This model assumes that n observations (with sum s) come from a Normal with unknown mean loc (described by the Normal prior) and known variance scale**2. The "known scale predictive" is the distribution of new observations, conditioned on the existing observations and our prior.

Accepts a prior Normal distribution object, having parameters loc0 and scale0, as well as known scale values of the predictive distribution(s) (also assumed Normal), and statistical estimates s (the sum(s) of the observations) and n (the number(s) of observations).

Calculates the Normal distribution(s) p(x | sigma**2):

p(x | sigma**2) = int N(x | mu, sigma**2)N(mu | prior.loc, prior.scale**2) dmu
                = N(x | prior.loc, 1 / (sigma**2 + prior.scale**2))

Returns the predictive posterior distribution object, with parameters (loc', scale'**2), where:

sigma_n**2 = 1/(1/sigma0**2 + n/sigma**2),
mu' = (mu0/sigma0**2 + s/sigma**2) * sigma_n**2.
sigma'**2 = sigma_n**2 + sigma**2,

Distribution parameters from prior, as well as scale, s, and n. will broadcast in the case of multidimensional sets of parameters.


  • prior: Normal object of type dtype: the prior distribution having parameters (loc0, scale0).
  • scale: tensor of type dtype, taking values scale > 0. The known stddev parameter(s).
  • s: Tensor of type dtype. The sum(s) of observations.
  • n: Tensor of type int. The number(s) of observations.


A new Normal predictive distribution object.


  • TypeError: if dtype of s does not match dtype, or prior is not a Normal object.