Quizz 06: Distributions, Estimation

Name: ______________________________

1. Consider a distribution over two discrete variables x, y displayed in the following figure: .
   Provide formulas to compute:

   
   Joint probability:
   
   Marginal probability for x:
   
   Conditional probability given y:
   
   

2. What is the criterion that defines that two random variables X and Y are independent using conditional probability:

   
   
   
   
   

3. Consider a distribution object as a software engineering pure interface. List 5 methods that can be computed over a distribution with their signature:

   
   
   
   
   
   

4. Given a distribution p with parameters w, and an observed dataset D, the Bayes formula states that:

   p(w | D) = p(D | w) p(w) / p(D)
   

   Indicate the definition of the following terms:

   
   Posterior:
   
   Prior:
   
   Likelihood:
   
   

   
   
5. Indicate which formula is optimized for each of the following two estimation methods:

   
   
   Maximum Likelihood Estimator (MLE): w* =
   
   Maximum a posteriori estimator (MAP): w* =
   
   

6. Bayesian estimation differs from MLE and MAP because it does not provide a pointwise estimator of the parameters of a distribution given a dataset. What does it do instead? How can Bayesian estimation be used to perform prediction?