We would now like to investigate what posterior distributions are obtained, as a function of the parameters a and B. a) (5 credits) Briefly comment about how the camera behaves for a = = 1, for a = B = 1/2, and for a = B = 0. For each of these cases, how would you expect this would change how the agent updates it's prior to a posterior on 0, given an observation of X? (No equations required.) You shouldn't need any assumptions about p(0) for this question. b) (10 credits) Compute p(X = x|0) for all x E {0, 1}. "The errors made by the camera are i.i.d, in that past camera outputs do not affect future camera outputs. c) (15 credits) The coin is flipped, and the camera reports seeing a one. (i.e. that X = 1.) Given an arbitrary prior p(0), compute the posterior p(0|X = 1). What does p(0|X = 1) simplify to when a = = 1? When a = = 1/2? When a = 3 = 0? Explain your observations. d) Compute p(0|X = 1) for the uniform prior p(0) = 1. Simplify it under the assumption that B := a. e) (10 credits) Let B = a. Plot p(0|X = 1) as a function of 0, for all a E {0, 4, 4, 7, 1} on the same graph to compare them. Comment on how the shape of the distribution changes with o. Explain your observations. (Assume p(0) = 1.)We have a Bayesian agent running on a computer, trying to learn information about what the pa- rameter 6 could be in the coin ip problem, based on observations through a noisy camera. The noisy camera takes a photo of each coin ip and reports back if the result was a 0 or a 1. Unfortunately, the camera is not perfect, and sometimes reports the wrong value The probability that the camera makes mistakes is controlled by two parameters a and ,8, that control the likelihood of correctly reporting a zero, and a one, respectively. Letting X denote the true outcome of the coin, and 2 denoting what the camera reported back, we can draw the relationship between X and if as shown. So, we have p(}?=0|X=0)=oz pl=0|X=1)=1 p(f=1IX=1)= p()?=1|X=0)=1a We would now like to investigate what posterior distributions are obtained, as a function of the parameters a and )3