Computer Information Systems

To create the R function for the posterior distribution of θ \theta θ , we need to use Bayes' theorem. Given the prior distribution g ( θ ) g(\theta) and the likelihood P ( X = x ∣ θ ) , we can derive the posterior distribution π ( θ ∣ x ) \pi(\theta \mid x) . Prior Distribution g ( θ ) g(\theta) : g ( θ ) = { 2 θ if 0 < θ < 1 0 elsewhere g(\theta) = \begin{cases} 2\theta & \text{if } 0 < \theta < 1 \\ 0 & \text{elsewhere} \end{cases} Likelihood P ( X = x ∣ θ ) P(X = x \mid \theta) for a binomial distribution: P ( X = x ∣ θ ) = ( n x ) θ x ( 1 − θ ) n − x P(X = x \mid \theta) = \binom{n}{x} \theta^x (1 - \theta)^{n - x} Posterior Distribution π ( θ ∣ x ) \pi(\theta \mid x) : Using Bayes' theorem: π ( θ ∣ x ) ∝ g ( θ ) ⋅ P ( X = x ∣ θ ) \pi(\theta \mid x) \propto g(\theta) \cdot P(X = x \mid \theta) Substitute g ( θ ) g(\theta) g ( θ ) and P ( X = x ∣ θ ) P(X = x \mid \theta) P ( X = x ∣ θ ) : π ( θ ∣ x ) ∝ 2 θ ⋅ θ x ( 1 − θ ) n − x \pi(\theta \mid x) \propto 2\theta \cdot \theta^x (1 - \theta)^{n - x} Simplify: π ( θ ∣ x ) ∝ 2 θ x + 1 ( 1 − θ ) n − x \pi(\theta \mid x) \propto 2 \theta^{x + 1} (1 - \theta)^{n - x} This is the form of a Beta distribution, specifically Beta ( x + 2 , n − x + 1 ) \text{Beta}(x + 2, n - x + 1) . R Function: The R function post(n, theta, x) should return the value of the posterior density function up to a proportionality constant. This can be implemented as follows: post = function(n, theta, x) { 2 * theta^(x + 1) * (1 - theta)^(n - x) } This one-liner R function takes n n , θ \theta , and x x as inputs and returns the unnormalized posterior density of θ \theta .

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