Spring 2010
Pstat160b
Handout
Bayesian Statistics example: Binomial/Beta conjugate prior
Setup: Binomial trials (say coin tosses) where the probability of success
p
is unknown. This is a typical
example in statistics where one would want to estimate
p
, but here we will think of
p
as random (so we
denote it by
P
) and quantify the uncertainty (prior knowledge, or lack of) by assuming that
P
follows a
Beta distribution,
P
∼ B
(
α, β
)
, α, β >
0 and given
P
=
p
,
X
follows a Binomial
B
(
n, p
) distribution.
f
P
(
p
)
=
1
B
(
α,β
)
p
α

1
(1

p
)
β

1
0
≤
p
≤
1
0
otherwise
f
X

P
=
p
(
k
)
=
n
k
p
k
(1

p
)
n

k
k
= 0
,
· · ·
, n
The reason for choosing the Beta distribution is its following advantages: first, its range is [0
,
1], so it can be
interpreted as a probability (for
p
, the probability of success in Bernoulli trials); by varying the parameters
(see graphs in handout 1) we can model from no prior knowledge (uniform distribution,
α
=
β
= 1), to
different beliefs about
p
. To see, this recall the following facts: The mode (most likely value, or location
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 Spring '10
 bonnet
 Statistics, Binomial, Probability, Probability theory, coin tosses

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