Let us have three tram lines - 6, 22 and 24 - regularly coming to the stop in front of the faculty building. Line 22 operates more frequently than line 24, 24 goes more often than line 6 (the ratio is 5:3:2, it is kept during all the hours of operation). Line 6 uses a single car setting in 9 out of 10 cases during the daytime, in the evening it always has the only car. Line 22 has one car rarely and only in evenings (1 out of 10 tramcars). Line 24 can be short whenever, however, it takes a long setting with 2 cars in 8 out of 10 cases. Albertov is available by line 24, lines 6 and 22 are headed in the direction of IP Pavlova. The line changes appear only when a tram goes to depot (let 24 have its depot in the direction of IP Pavlova, 6 and 22 have their depots in the direction of Albertov). Every tenth tram goes to the depot evenly throughout the operation. The evening regime is from 6pm to 24pm, the daytime regime is from 6am to 6pm. a) Draw a correct, efficient and causal Bayesian network. b) Annotate the network with the conditional probability tables. c) It is evening. A short tram is approaching the stop. What is the probability it will go to Albertov? d) There is a tram 22 standing in the stop. How many cars does it have?

Trace the algorithm of belief propagation in the network below knowing that e = {p2}. Show the individual steps, be as detailed as possible. Explain in which way the unevidenced converging node P3 blocks the path between nodes P1 and P2. Obviously, the posterior probabilities are P r (P1) = P r(P1|p2) = P r(P1), P r (p2) = 1 and P r (P3) = P r(P3|p2) = P P1 P r(P1)P r(P3|p2, P1) (P r (p3) = 0.52). The same values must be reached by belief propagation when message passing stops and the node probabilities are computed as follows: P r (Pi) = i (Pi) (Pi). Belief propagation starts with the follo

a C++ program to convert a given integer to a roman numeral.Go to the editor From Wikipedia: Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers in this system are represented by combinations of letters from the Latin alphabet. Modern usage employs seven symbols, each with a fixed integer value:[1]

For the network from Exercise 4 calculate the conditional probability P r(p1|p2, p3) again. Apply a sampling approximate method. Discuss pros and cons of rejection sampling, likelihood weighting and Gibbs sampling. The table shown below gives an output of a uniform random number generator on the interval (0,1), use the table to generate samples. r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 0.2551 0.5060 0.6991 0.8909 0.9593 0.5472 0.1386 0.1493 0.2575 0.8407 r11 r12 r13 r14 r15 r16 r17 r18 r19 r20 0.0827 0.9060 0.7612 0.1423 0.5888 0.6330 0.5030 0.8003 0.0155 0.6917 Let us start with rejection sampling. The variables must be topologically sorted first (current notation meets the definition of topological ordering, P1 < P2 < P3 < P4