For the Bayesian Network we covered in class [slide 18 of http://cross-entropy.net/ML310/Frequentist_Statistics_and_Graphical_Models.pdf],
what is the Probability(Tuberculosis = 1 | SputumSmear = 1, LungInfiltrates = 1, XRay = 1)?

   Use one of the following options to answer this question:
   a. use the disease.csv spreadsheet: http://cross-entropy.net/ML310/BayesNet.csv
   b. use the libpgm software [see the notebook: http://cross-entropy.net/ML310/Graphical_Models.zip]
   c. argue your use of conditional independence [based on the notebook: http://cross-entropy.net/ML310/Graphical_Models.zip]

Probability(Tuberculosis = 1 | SputumSmear = 1, LungInfiltrates = 1, XRay = 1)
= Probability(Tuberculosis = 1, SputumSmear = 1, LungInfiltrates = 1, XRay = 1) / Probability(SputumSmear = 1, LungInfiltrates = 1, XRay = 1)
= P(T = 1, I = 1, X = 1, S = 1) / P(I = 1, X = 1, S = 1)
= (P(P = 0, T = 1, I = 1, X = 1, S = 1) + P(P = 1, T = 1, I = 1, X = 1, S = 1)) / (P(P = 0, T = 1, I = 1, X = 1, S = 1) + P(P = 1, T = 1, I = 1, X = 1, S = 1) + P(P = 0, T = 0, I = 1, X = 1, S = 1) + P(P = 1, T = 0, I = 1, X = 1, S = 1))
= (0.002432 + 0.000512) / (0.002432 + 0.000512 + 0.0044688 + 0.014112)
= 0.002944 / 0.0215248
= 0.1367725

... and thanks to conditional independence, this is the same as ...

Probability(Tuberculosis = 1 | SputumSmear = 1, LungInfiltrates = 1)
= P(T = 1, I = 1, S = 1) / P(I = 1, S = 1)
= (P(P = 0, T = 1, I = 1, X = 0, S = 1) + P(P = 0, T = 1, I = 1, X = 1, S = 1) + P(P = 1, T = 1, I = 1, X = 0, S = 1) + P(P = 1, T = 1, I = 1, X = 1, S = 1)) / (P(P = 0, T = 1, I = 1, X = 0, S = 1) + P(P = 0, T = 1, I = 1, X = 1, S = 1) + P(P = 1, T = 1, I = 1, X = 0, S = 1) + P(P = 1, T = 1, I = 1, X = 1, S = 1) + P(P = 0, T = 0, I = 1, X = 0, S = 1) + P(P = 0, T = 0, I = 1, X = 1, S = 1) + P(P = 1, T = 0, I = 1, X = 0, S = 1) + P(P = 1, T = 0, I = 1, X = 1, S = 1))
= (0.000608 + 0.002432 + 0.000128 + 0.000512) / (0.000608 + 0.002432 + 0.000128 + 0.000512 + 0.0011172 + 0.0044688 + 0.003528 + 0.014112)
= 0.00368 / 0.026906
= 0.1367725