Predicting Click-Through Rates of New Advertisements Based on the Bayesian Network

Algorithm 2

KBN inference.

Input:  𝐺=(𝐾,𝐸), the DAG of the KBN;

    𝐾_𝐵: the evidence nodes in 𝐺 (the new ad’s keywords);

    𝑍: the non-evidence nodes in 𝐺 (all keywords except for the new ad’s keywords);

    𝑒: the set of values of the evidence (set 1 as the value);

    𝑞: the query variable (the keyword will be inferred);

    𝑠: the total times of sampling.

Output: The estimates of 𝑃(𝑞=1∣𝑒)

Variables:

   𝑧: the set of the values of the non-evidence;

   𝑘_(−𝑖)={𝑘₁,𝑘₂,…,𝑘_𝑖−1,𝑘_𝑖+1,…,𝑘_𝑔};

   𝑣=𝑒∪𝑧: the current state of the KBN;

   𝑁(𝑞₁): the number of the samples when the value of 𝑞 is 1

Steps:

    (1) Initialization:

   𝑧←random values of 𝑘_𝑖 in Z

   𝑣⁽⁰⁾←𝑒∪𝑧,𝑐⁽⁰⁾←𝑣⁽⁰⁾

   𝑁(𝑞₁)←0

    (2) For 𝑚←1 to 𝑠 Do

   (i) Compute the probabilities of the selected variables based on the state 𝑐^(𝑚−1),

     Randomly select one non-evidence variable 𝑘_𝑖 from 𝑍:

       𝐵←𝑃(𝑘_𝑖=0∣𝑐_{𝑀𝐵(𝑘𝑖)})+𝑃(𝑘_𝑖=1∣𝑐_{𝑀𝐵(𝑘𝑖)}), where 𝑐_{𝑀𝐵(𝑘𝑖)} is the set of the values in the

     Markov chain of 𝑘_𝑖 in the current state 𝑐

   (ii) Generate a random number 𝑟_𝑘∈[0,𝐵] and we determine the value of 𝑘_𝑖:

      𝑘_𝑖=0𝑟_𝑘≤𝑃𝑘_𝑖=0∣𝑐_{𝑀𝐵(𝑘𝑖)}1𝑃𝑘_𝑖=0∣𝑐_{𝑀𝐵(𝑘𝑖)}<𝑟_𝑘≤𝑃𝑘_𝑖=0∣𝑐_{𝑀𝐵(𝑘𝑖)}+𝑃𝑘_𝑖=1∣𝑐_{𝑀𝐵(𝑘𝑖)} (5)

      𝑣^(𝑘)←𝑣_(−𝑖)^(𝑘−1),𝑘_𝑖,𝑐^(𝑘)←𝑣^(𝑘)

   (iii) Count

      If 𝑐_𝑞^(𝑘)=1 Then 𝑁(𝑞₁)←𝑁(𝑞₁)+1

   End For

    (3) Estimate 𝑃(𝑞=1∣𝑒)

   𝑃(𝑞=1∣𝑒)←𝑁(𝑞₁)/𝑠