Thompson Sampling for CPA model

In a CPA payment model, the expected value of an impression : 

Model 

Notation 

CTR Model 

of the training set is given by: 

 

Choose a simple prior of Wck , 

 

Then we use Laplace approximation which enables us to approximate the posterior with a Gaussian distribution.

: the local minimum of the log-posterior

Also we can derive the following covariance :

      

However, since the sampling time for every variable can be problematic that can leads to a problem of prediction time, we better go more in depth.

Let 

CVR Model 

of the training set is given by: 

 

Choose a simple prior of Wcv , Wd

 

Then we use Laplace approximation which enables us to approximate the posterior with a Gaussian distribution.

, 

: the local minimum of the log-posterior

Also, we can derive the following covariance :

      

However, since the sampling time for every variable can be problematic that can leads to a problem of prediction time, we better go more in depth.

Let 

Thompson Sampling

1. Sample a set of parameters according to the posterior after seeing the training data: 

2. Choose the action that is optimal with respect to the sampled set of parameters: