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Conjugate priors
Because of computational problems,
modelling priors has been traditionally
a compromise between a realistic assessment of beliefs
and choosing a mathematical function that simplifies
the analytic calculations.
A wellknown strategy is to choose a prior with a suitable form
so the posterior belongs to the same functional family as the prior.
The choice of the family depends on the likelihood. A prior
and posterior chosen in this way are said to be conjugate.
For instance, given a Gaussian likelihood and choosing a Gaussian prior,
the posterior is still Gaussian, as we have seen in
Eqs. (25), (28) and
(29). This is because expressions of the form
can always be written in the form
with suitable values for , and . The Gaussian distribution
is autoconjugate. The mathematics is simplified but, unfortunately,
only one shape is possible.
An interesting case, both for flexibility and practical interest is
offered by the binomial likelihood (see Sect. 5.3).
Apart from the binomial coefficient,
has the
shape
, which has the same structure as the
Beta distribution, well known to statisticians:

(94) 
where stands for the Beta function, defined as

(95) 
which can be expressed in terms of Euler's Gamma function as
.
Expectation
and variance of the Beta distribution are:
If and , then the mode is unique, and it is at
.
Depending on the value of the parameters the Beta pdf
can take a large variety of shapes.
For example, for large values of and ,
the function is very similar to a Gaussian distribution,
while a constant function is obtained
for . Using the Beta pdf as prior function in
inferential problems with a binomial likelihood, we have
The posterior distribution is still a Beta with
and , and expectation and standard
deviation can be calculated easily from Eqs. (96)
and (97).
These formulae demonstrate how the posterior estimates become
progressively independent of the prior information in the limit of
large numbers;
this happens when both and . In this limit, we get
the same result as for a uniform prior ().
Table 2:
Some useful conjugate priors.
and stand for the observed value (continuous or discrete, respectively)
and is the generic
symbol for the
parameter to infer, corresponding to of a Gaussian, of a binomial
and of a Poisson distribution.
likelihood 
conjugate prior 
posterior 



Normal

Normal

Normal
[Eqs. (30)(32)] 
Binomial 
Beta 
Beta 
Poisson 
Gamma 
Gamma 
Multinomial

Dirichlet

Dirichlet

Table 2 lists some of the more useful conjugate priors.
For a more complete collection of conjugate priors,
see e.g. (Bernardo and Smith 1994, Gelman et al 1995).
Next: General principle based priors
Up: Choice of priors 
Previous: Purely subjective assessment of
Giulio D'Agostini
20030513