Reference no: EM133030241

IST 557 Data Mining - Techniques and Applications - Pennsylvania State University

Homework

Consider a study of M families, the m-th family containing Dm individuals. For each individual, in each family, we measure their height (in centimeters). Your friend, who is working with you to analyze this data suggests the following Bayesian model for these data:
y_i ~ N (α_z , σ²) (3)
α_m ~ N (µ, τ²) (4)

where i refers to the i-th individual in the study and z_i refers to the family (z_i ∈ {1, . . . , M}) to which individual i belongs, and α = {α₁, . . . , α_M }.

Problem 1

1. Briefly, describe an interpretation of the parameter σ² (1-2 sentences). If you were to set this parameter yourself, what value would you choose and why (1-2 sentences)? Hint: remember y is measured in centimeters.

2. Briefly, describe an interpretation of the parameter τ².

3. Briefly, describe an interpretation of the parameter µ². If you were to set this parameter yourself, what value would you choose and why?

4. Briefly, describe an interpretation of the parameter αz_i.

5. Briefly describe one biological reason why this model may be inappropriate or sub-standard.

Problem 2

Part A:

You have recently learned about Gaussian processes. You think they are the coolest things ever. So you decide to represent the above model as a Gaussian process. What is the mean function m(.) and a kernel function K(.,.) that represents the above model as a Gaussian process of the form y GP(m(.), K(.,.)). In other words, find m and K such that the GP model matches the marginal distribution of your friends proposed Bayesian model.

Part B:
You realize that in your haste the above GP model ignored the parameters α that were the most important part of your analysis: you marginalized over them (oops!). Still, you think GPs are the best so you decide to instead use Gaussian Process Regression. Find m and K that represents your friends model as the following Gaussian Process Regression model

y_i ∼ N (α(z_i), σ²)
α ∼ GP(m(.), K(., .))

Attachment:- Data Mining.rar