Suppose an unknown parameter
{\displaystyle \theta }
is known to have a prior distribution
{\displaystyle \pi }
. In most cases, models only approximate the true process, and may not take into account certain factors influencing the data. f with parameters \(\alpha\) and \(\beta\) is:In our case, the posterior p. So, if a Bayesian is asked to make a point estimate of \(\theta\), he or she is going to naturally turn to \(k(\theta|y)\) for the answer. 0 license.
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In particular, suppose that
f
(
)
=
{\displaystyle \mu _{f}(\theta )=\theta }
and that
f
2
find here
)
=
K
{\displaystyle \sigma _{f}^{2}(\theta )=K}
; we then have
Finally, we obtain the estimated moments of the prior,
For example, if
x
i
|
i
N
(
i
,
1
)
{\displaystyle x_{i}|\theta _{i}\sim N(\theta _{i},1)}
, and if we assume a normal prior (which is a conjugate prior in this case), we conclude that
n
+
1
N
(
,
2
)
{\displaystyle \theta _{n+1}\sim N({\widehat {\mu }}_{\pi },{\widehat {\sigma }}_{\pi }^{2})}
, from which the Bayes estimator of
n
+
1
find out here
{\displaystyle \theta _{n+1}}
based on
x
n
+
1
{\displaystyle x_{n+1}}
can be calculated. .