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Category Archives: Prior/Posterior Distribution
Beta-binomial
You have a sequence of coins \(X_1,…,X_n\) drawn iid from a Bernouli distribution with unknown parameter \(p\) and known fixed \(n\). Assume a priori that the coins parameter \(p\) follows a Beta distribution with parameters \(\alpha,\beta\).
- Given the sequence \(X_1,…,X_n\) what is the posterior pdf of \(p\) ?
- For what value of \(p\) is the maximum of the posterior pdf attained.
Hint: If \(X\) is distributed Bernoulli(p) then for \(x=1,0\) one has \(P(X=x)=p^x(1-p)^{(1-x)}\). Furthermore, if \(X_1,X_2\) are i.i.d. Bernoulli(p) then
\[P(X_1=x_1, X_2=x_2 )=P(X_1=x_1)P(X_2=x_2 )=p^{x_1}(1-p)^{(1-x_1)}p^{x_2}(1-p)^{(1-x_2)}\]
Expectation of hierachical model
Consider the following hierarchical random variable
- \(\lambda \sim \mbox{Geometric}(p)\)
- \(Y \mid \lambda \sim \mbox{Poisson}(\lambda)\)
Biased coins
Given a random variable \(x = \{ 0,1\} \) where \(0\) corresponds to heads and \(1\) corresponds to tails.
For a single coin flip: \( \mathbf{P}(x \mid p) = p^x(1-p)^{1-x}\).
For a sequence of \(n\) coin flips: \( \mathbf{P}(x_1,…,x_n \mid p) = \prod_{i=1}^n p^{x_i}(1-p)^{1-x_{i}}\).
I have a bag with three types of coins with the following probabilities of drawing each type:
\( \mathbf{P}(p=.5) = .7 \), \( \mathbf{P}(p=.1) = .2 \), \( \mathbf{P}(p=.9) = .1 \).
I draw a coin from the bag. I flip it \(n\) times resulting in a sequence \(X_1,…,X_n\).
- Using Bayes rule provide the formula for
\[ \mathbf{P}(p = .1 \mid x_1,…,x_n),\quad
\mathbf{P}(p = .5 \mid x_1,…,x_n), \quad\text{and}\quad
\mathbf{P}(p = .9 \mid x_1,…,x_n) \]. - If \(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)=(1,1,0,1,1,0,0,1)\) what is the most likely value probability \(p \) of the coin the was used ?
