CAS Exam MAS-II Spring 2019, Question 32
Study guide: Exam MAS-II
A Bayesian logistic regression model that predicts if a prospective customer will purchase an insurance policy is fit to a dataset with two cluster variables. The first cluster variable is the channel through which the insurance policy was sold. The channel variable can only take values from one of the following categories:
- Agency
- Broker
- Internet
The second cluster variable is the type of insurance policy. The type variable can only take values from one of the following categories:
- Personal Automobile
- Homeowners
- Workers Compensation
The model also includes two binary categorical variables that are represented as slopes. The first indicates if the offered insurance policy included a discount to the price. The second indicates if the prospective customer had a prior insurance policy.
In the model, let y_i indicate if the i^{th} prospective customer purchased the insurance policy. Let the channel and type of policy from the i^{th} prospective customer be represented by the channel[i] and type[i] subscripts respectively. Let D_i indicate if the offered insurance policy to the i^{th} customer included a discount to the price, and let P_i indicate if the prospective customer had a prior insurance policy.
The systematic portion of the model is specified below.
\begin{align*}
y_i &\sim \text{Bernoulli}(p_i) \\
\text{logit}(p_i) &= \gamma_i + \delta_i D_i + \Lambda_i P_i \\
\gamma_i &= \alpha + \alpha_{channel[i]} + \alpha_{type[i]} \\
\delta_i &= \beta_D + \beta_{D,channel[i]} + \beta_{D,type[i]} \\
\Lambda_i &= \beta_P + \beta_{P,channel[i]} + \beta_{P,type[i]}
\end{align*}Multivariate Gaussian prior distributions are employed to model the covariance between the intercept and slope parameters for each cluster. In particular, an LKJ prior distribution for the correlation parameters is selected for this model.
Determine the number of correlation parameters that need to be estimated in this model.
- A) Less than or equal to 2
- B) Greater than 2 but less than or equal to 5
- C) Greater than 5 but less than or equal to 8
- D) Greater than 8 but less than or equal to 11
- E) Greater than 11
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