Question

Consider the exact same setting as last question: A set of sample data (xᵢ,yᵢ) are generated by a mixture distribution FXY​ as follows: Y=1,2,3 equally likely, and the condition distribution X∣Y satisfies: X∣Y=1∼N(0,1);X∣Y=2∼N(1,2);X∣Y=3∼N(3,3). Except this time we use QDA to fit the classifier g(⋅), so there are three variance estimates instead of a single one. What does Σ₃ (the QDA estimated variance for class 3) converge to?

Answer 1

In QDA, the estimated variance Σ_3 for class 3 converges to the true variance of the normal distribution N(3,3) for that class, which is 3.

Step-by-step explanation:

The student is asking about the convergence of Σ_3, the estimated variance for class 3 when using Quadratic Discriminant Analysis (QDA). In the given scenario, class 3 follows a normal distribution N(3,3). The QDA method, which allows for separate covariance matrices for each class, will estimate Σ_3 based on the sample data for that class. With a sufficiently large sample size, the estimated variance Σ_3 will converge to the true variance of the class 3 distribution, which is 3 as specified by the condition X|Y=3∼N(3,3).