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Question

(c) Finally, show that the hyperplane in (6.19) can be rewritten in the form wT(x−x0)=0,(6.21)w^T(x - x_0) = 0, \quad (6.21) w=Σ−1(μi−μj),x0=μi+μj2−(μi−μj)∥μi−μj∥Σ2log⁡πiπj.(6.22)w = \Sigma^{-1}(\mu_i - \mu_j), \quad x_0 = \frac{\mu_i + \mu_j}{2} - \frac{(\mu_i - \mu_j)}{\|\mu_i - \mu_j\|_\Sigma^2} \log \frac{\pi_i}{\pi_j}. \quad (6.22) What is the interpretation of ww and x0x_0? What is the effect of the priors {πi,πj}\{\pi_i, \pi_j\} on x0x_0?