Click Models for Web Search (2)-Parameter estimation

There are two ways to estimate parameters in the Click Model: Maximum Likelihood (MLE) and expected maximum (EM). The method of estimating which parameter to use for each click model depends on the characteristics of the random variable in this model. If the random variables in the model are observed, then the MLE is undoubtedly used, and the EM algorithm should be used if the model contains some hidden variables.

1. The MLE algorithm

The likelihood function is:

The value of the maximum likelihood function for the parameter that needs to be estimated is:

1) MLE for the RCM and CTR MODELS

Rcm:

RCTR:

DCTR:

These are simple, simple statistics-based approaches. The numerator is the number of clicks of each event, and the denominator is the number of occurrences of the respective event.

2) MLE for DCM

In DCM, the attractiveness variable cannot be obsesrved, starting with the document at the last click location. We do not know is the user because the last click of the document is not satisfy and stop examine the next document, or because the next document is not enough attractive. And if we assume that the user is satisfy for the last click of the document, then the attractiveness variable and the satisfaction variable are observed. This is simplified DCM, which has:

3) MLE for SDBN

2. The EM algorithm

Consider the random variable in the Bayesian network and its parent node. The probability is the parameter

Bernoulli distribution. An EM algorithm can be used to estimate the parameters of a variable in the parent node if it cannot be observe.

1) Expectation (E-step)

2) maximization (M-step)

3) EM estimation for UBM

3. Formulas for CLICK MODEL PARAMETERS

Click Models for Web Search (2)-Parameter estimation