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 $$\alpha,\beta$$.

1. Given the sequence  $$X_1,…,X_n$$ what is the posterior pdf of $$p$$ ?
2. 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

1. $$\lambda \sim \mbox{Geometric}(p)$$
2. $$Y \mid \lambda \sim \mbox{Poisson}(\lambda)$$
Compute $$\mathbf{E}(Y)$$.

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 $$X_1,…,X_n$$.

1. Using Bayes rule provide the formula for
$\mathbf{P}(p = .1 \mid x_1,…,x_n),\quad \mathbf{P}(p = .5 \mid x_1,…,x_n), \quad\text{and}\quad \mathbf{P}(p = .9 \mid x_1,…,x_n)$.
2. 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 ?