章节大纲


    • Lecture : slides part II - 9-14 (complete data)
    • Assignment
      1. Let's use the data and the network provided in the file example-parameter-learning-complete.xlsx (cf. below)
        • The CPDs (conditional probability distributions) corresponding to this network are P(A), P(B|A), P(D), P(C | A, D).
        • The contingency tables associated to these CPDs are N(A), N(B,A), N(D), N(C, A, D). Fill these tables by counting the number of corresponding events in the dataset.
        • Maximum of likelihood : how can we estime P(A) from N(A) ? P(D) from N(D) ? P(B|A) from N(B,A) ? P(C | A,D) from N(C, A,D) ? What are the results ?
        • Let's consider the following Dirichlet coefficient αA=[1 1], αD=[3 7], αBA=[1 1; 1 1], αCAD=[1 1 1 1; 1 1 1 1]. What are the CPDs estimation with Expectation a posteriori approach ?

    • Lecture: slides part II - 15-17 (incomplete data)
    • Assignment
      1. Let's use the data and the network provided in the file example-parameter-learning-incomplete.xlsx (cf. below)
        • How many of the 50 lines of data will we consider by applying Complete Case Analysis approach ?
        • If we now consider Available Case Analysis approach, how many lines will we consider for the estimation of P(A) ? P(D) ? P(B | A) ?
        • EM algorithm :
          • initialize the parameters by applying (by hand) Available Case Analysis approach
          • use Genie to implement this network
          • First E step :
            • the first missing data is the first sample, with A=0, C=0, D=0 and B missing.
              • use your Genie implementation to ask your actual network (initialized with ACA) what is P(B | A=0, C=0, D=0).
              • what are the contingency tables impacted by this missing value ?
              • if B surely observed and equal to 0, N(B=0) would be increased by 1. if B surely observed and equal to 1, N(B=1) would be increased by 1. Here B is missing, but we have an estimation of its probability, so N(B=0) and N(B=1) will be increased by the respective probability of P(B=0| A=0, C=0, D=0) and P(B=1| A=0, C=0, D=0). Same reasoning for N(A,B) where two values are impacted: N(A=0, B=0) and N(A,0, B=1).
          • First M step : apply Maximum of Likelihood estimation to estimate one new version for each of your CPDs, and update your BN
          • And officially, you can repeat this until convergence :-)