# Normal likelihood with conjugate prior.

### tf.contrib.distributions.normal_conjugates_known_scale_posterior(prior, scale, s, n)

Posterior Normal distribution with 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 posterior" is the distribution of the unknown loc.

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).

Returns a posterior (also Normal) distribution object, with parameters (loc', scale'^2), where:

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


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

##### Args:
• 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.
##### Returns:

A new Normal posterior distribution object for the unknown observation mean loc.

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

### tf.contrib.distributions.normal_conjugates_known_scale_predictive(prior, scale, s, n)

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.

##### Args:
• 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.
##### Returns:

A new Normal predictive distribution object.

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