(However, a form of this sort is a member of a curved exponential family, which allows multiple factorized terms in the exponent. In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below.
Even when x is a scalar, and there is only a single parameter, the functions η(θ) and T(x) can still be vectors, as described below.
Note that x is often a vector of measurements, in which case T(x) is a function from the space of possible values of x to the real numbers.
In Bayesian statistics, a priori probability distribution is multiplied by a likelihood function and then normalized to get to the a posteriori probability distribution.
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This is why the above cases (e. However, a value of 0 suggests that the mean and variance of all the sufficient statistics are uniformly 0, whereas in fact the mean of the
i
{\displaystyle i}
th sufficient statistic should be
p
view i
{\displaystyle p_{i}\ }
.
A factor consisting of a sum where both types of variables are involved (e. .
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Next, consider the case of a normal distribution with unknown mean and unknown variance.
We choose this particular form to take advantage of certain computational advantages or reasons of generalization. A conjugate prior π for the parameter
{\displaystyle {\boldsymbol {\eta }}}
of an exponential family
is given by
or equivalently
where s is the dimension of
{\displaystyle {\boldsymbol {\eta }}}
and
0
{\displaystyle \nu 0}
and
{\displaystyle {\boldsymbol {\chi }}}
are hyperparameters (parameters controlling parameters).
In the case of an exponential family where
the kernel is
and the partition function is
Since the distribution must be normalized, we have
In other words,
or equivalently
This justifies calling A the log-normalizer or log-partition function.
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.