The vector-parameter form over a single scalar-valued random variable can be trivially expanded to cover a joint distribution over a vector of random variables. e.
Even when find this is a scalar, and there is only a single parameter, the functions η(θ) and T(x) can still be vectors, as described below.
First, assume that the probability of a single observation follows an exponential family, parameterized using its natural parameter:
Then, for data

X

=
(

x

1

look at here ,

,

x

n

)

{\displaystyle \mathbf {X} =(x_{1},\ldots ,x_{n})}

, the likelihood is computed as follows:
Then, for the above conjugate prior:
We can then compute the posterior as follows:
The last line is the kernel of the posterior distribution, i. 7 This can be used to exclude a parametric family distribution from being an exponential family.

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. This makes the computation of the posterior particularly simple. a. However, if one’s belief about the likely value of the theta parameter of a binomial is represented by (say) a bimodal (two-humped) prior distribution, then this cannot be represented by a beta distribution.
Exponential families arise naturally as the answer to the following question: what is the maximum-entropy distribution consistent with given constraints on expected values?
The information entropy of a probability distribution dF(x) can only be computed with respect to some other probability distribution (or, more generally, a positive measure), and both measures must be mutually absolutely continuous.

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This is important because the dimension of the sufficient statistic does not grow with the data size — it has only as many components as the components of

{\displaystyle {\boldsymbol {\eta }}}

(equivalently, the number of parameters of the distribution of a single data point).
Next, consider the case of a normal distribution with unknown mean and unknown variance.
In the one-dimensional case, we have
This must be normalized, so
Take the derivative of both sides with respect to η:
Therefore,
As an introductory example, consider the gamma distribution, whose distribution is defined by
Referring to Find Out More above table, we can see that the natural parameter is given by
the reverse substitutions are
the sufficient statistics are

(
log

x
,
x
)
,

{\displaystyle (\log x,x),}

and the log-partition function is
We can find the mean of the sufficient statistics as follows. .

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