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2. Logistic regression and LDA. The LDA model posits the (joint) distribution of (X, Y) is Y~ Ber(π), X|Y=y~ N(µy, Σ), + where € (0,1) is the class proportion, and μ0,₁ € Rd, and Σ € Sdxd are the means and (co)variance of the class conditionals. Show that the predictive distribution of Y in the LDA model satisfies the logistic regression model; i.e. 1 1 + exp(Bo + BTx) P{Y = 1 | X = x} = . where Bo and 3 are functions of T, 0, 1, and Σ.

2. Logistic regression and LDA. The LDA model posits the (joint) distribution of (X, Y) is Y~ Ber(π), X|Y=y~ N(µy, Σ), + where € (0,1) is the class proportion, and μ0,₁ € Rd, and Σ € Sdxd are the means and (co)variance of the class conditionals. Show that the predictive distribution of Y in the LDA model satisfies the logistic regression model; i.e. 1 1 + exp(Bo + BTx) P{Y = 1 | X = x} = . where Bo and 3 are functions of T, 0, 1, and Σ.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. Logistic regression and LDA. The LDA model posits the (joint) distribution of (X,Y) is Y~ Ber(π), X|Y=y~ N(µy, Σ), where € (0, 1) is the class proportion, and µo, µ₁ € Rª, and Σ € Sdxd are the means and (co)variance of the class conditionals. Show that the predictive distribution of Y in the LDA model satisfies the logistic regression model; i.e. P{Y = 1 | X = x} = 1 1+ exp(30 + 3x) where Bo and 3 are functions of π, μo, μ₁, and Σ. Hint: Recall the pdf of a N(u, E) random vector is p(x) = 1 - (2π) ¾ det(T) ³ exp(−½(x − µ)¹¶−¹(x − µ)).
Transcribed Image Text:2. Logistic regression and LDA. The LDA model posits the (joint) distribution of (X,Y) is Y~ Ber(π), X|Y=y~ N(µy, Σ), where € (0, 1) is the class proportion, and µo, µ₁ € Rª, and Σ € Sdxd are the means and (co)variance of the class conditionals. Show that the predictive distribution of Y in the LDA model satisfies the logistic regression model; i.e. P{Y = 1 | X = x} = 1 1+ exp(30 + 3x) where Bo and 3 are functions of π, μo, μ₁, and Σ. Hint: Recall the pdf of a N(u, E) random vector is p(x) = 1 - (2π) ¾ det(T) ³ exp(−½(x − µ)¹¶−¹(x − µ)).
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