Environmental Engineering Reference
In-Depth Information
Table 13.1
Analytical results for Bayesian updating with random sampling conjugate pair distributions
Unknown
parameter
Prior and
posterior
Data
Likelihood function
Conjugate density
Posterior parameter
Lp
()
PX
(
xnp
,
|
)
(,)
()()
x
nx
x
Occurrences in
n trials from
Bernoulli
Sequence
p
(frequency)
=
=
Γαβ
ΓαΓβ
αα
ββ
″
=+
=+−
′
Beta
α
,
β
f
p
()
p
p
α
−
1
(
1
p
)
β
−
1
=
×
−
(
)
″
′
n
x
=
x
n
x
pp
(
1
−
)
−
x
Occurrences
within interval t
from Poisson
Process
Gamma
α
,
β
λ
(rate)
1
x
e
t
()
t
x
αα
ββ
″
=+
=+
′
−λ
λ
f
()
α
×
−−
α
1
e
λβ
L
()
P Xxt
(
,
|
λ=
)
λ
=
Γα
βλ
λ= =
λ
()
x
!
1
″
′
L
()
P xn
(,
|
)
f
µ
()
(
,
)
n
Observations
with s
a
mple
mean
x
from
normal variate
Normal
μ
μ
,
σ
μ
µ
=
µ
µ
=
φ
µ σ
µσ σ
σ
′
(
2
/
nx
n
)
+
+
(
′2
)
μ
(mean)
µ µµ
µ
µ
µ
″
=
µ
(
2
/
)
(
)
1
σ
′2
n
1
∏
µ
=
=
2
πσ
(
2
2
/
/
n
n
+
(
)
(
)
σ
×
σ
′2
2
2
πσ
µ
µ
″
2
σ
=
i
1
=
µ
(
)
σ
′2
)
σ
2
‡
‰
‰
Ž
‰
‰
1
2
µµ
σ
−
2
−
‡
Ž
×
−−
1
2
x
µ
‰
‰
‰
‰
µ
exp
×
i
exp
σ
µ
K
φ
(
x
,
σ
/
n
)
=×
µ
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