The following Bayesian Network represents two causes and two effects related to Lung Cancer. Each variable takes the value true (t) or false (f). We will abbreviate the five variable names using their leading letters: P, S, C, X, and D. The probabilities shown are all for the "is true" outcome, e.g. read P(P=t) =0.90 as the probability that the variable Pollution takes the value true is 0.90. The probability that it is false is not shown, but is easily derived. 3 Requirements Note regarding the calculation, you should show your working process of the calculation to demonstrate that you know how to calculate them.

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1. Using inference by enumeration to calculate the probability P(P = t|X = t) (i) describe what are the evidence, hidden and query variables in this inference, (ii) describe how would you use variable elimination in this inference, i.e. to perform the join operation and the elimination operation on which variables and in what order, and (iii) report the probability,?

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2. Given the Bayesian Network , find the variables that are independent to each other or conditionally independent given another variable. Find at least three pairs or groups of such variables.?

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3. If given the variable order as , draw a new Bayesian Network structure (nodes and connections only) to describe the same problem/domain as shown in the above given Bayesian Network. [hint: considering the above (conditionally) independent variables, the network should keep the original dependence between variables, which are that (conditionally) independent variables should remain being independent to each other, and dependent variables remain being dependent]. For each connection, explain why it is needed. 2 Relevant Data Files and Program Files?

(a)|This question asks you to consider a Bayesian approach to inference about .1, the mortality rate in an exponential model for survival time. In order to take a Bayesian approach, we specify a prior distribution for A which is gamma distribution. I Show that the gamma distribution is a conjugate prior distribution for the exponential model; i.e. if we specify that 1"\"Gamma (a, ,3} a priori, then the posterior distribution is also Gamma, with parameters that depend on a, [3,115. 0 Provide expressions for the parameters of this Gamma posterior distribution, and for the mean and mode of the posterior distribution. 1. Briefly describe the steps involved in conducting phylogenetic inference using parsimony 2. Why is using parsimony with morphological data problematic? (Hint: imagine character - 'wings' of bats, humans, mice and butterflies. Based on parsimony, bats and butterfly will be put together. How is that a problem?) 3. The eyes of an octopus and human look very similar but they do not share a common ancestor. This is an example of a(n) a. Homoplasy b. Homology C. Synapomorphy d. Plesiomorphy e. Apomorphy 4. What does it mean for a trait to have an appropriate amount of variation for a use in a phylogenetic analysis? Use examples 5. Which statement(s) is/are true about Bayesian method? a. Bayesian searches for trees with highest posterior probability b. It is based on Bayes theorem C. Does not use models for tree inference d. Utilizes prior information/probability C. Same principle as maximum parsimonyQUESTION 21 Consider the Probability Density Function (PDF) of a continuous random variable x in the schematic below. Which statement is TRUE? (a) If integrated, it gives the probability of x in the range [(1, b]. (b) It describes a uniformly distributed random number (c) It gives the probability that x = a or x = b (d) This is not a PDF of a continuous random variable (e) None of the above pdfo)