I have a question about one task: I have two classes w1 and w2 and we are working with 2D space. The density of each of the two classes are the folowing: \begin{equation} p(x = {(x_{1}, x_{2})}'| w_{1}) = \frac{1}{4}\cdot e ^{-\frac{x_{1}+ x_{2}}{2}} \end{equation} \begin{equation} p(x = {(x_{1}, x_{2})}'| w_{2}) = \frac{1}{16}\cdot x_{1}x_{2} \cdot e ^{-\frac{x_{1}+ x_{2}}{2}} \end{equation} How to find the equation of the boundary between the two classes.

  • $\begingroup$ For finding a Bayes decision boundary, you need the priors of these classes. Are they equally likely? $\endgroup$ – gunes Apr 28 '20 at 15:23
  • $\begingroup$ Yes, I think they are equally likely. $\endgroup$ – Stats8901 Apr 28 '20 at 15:46

If they're equally likely, you'll just solve $p(x|w_1)=p(x|w_2)$ to find the decision boundary, which yields the curve $x_1x_2=4$.

  • $\begingroup$ Thanks, but if they are not equal we have p(x|w1)p(w1) = p(x|w2)p(w2) which I can't solved? $\endgroup$ – Stats8901 Apr 28 '20 at 16:21
  • $\begingroup$ you can still solve it when you substitute the values. $\endgroup$ – gunes Apr 28 '20 at 16:22
  • $\begingroup$ How? Could you give some hint? $\endgroup$ – Stats8901 Apr 28 '20 at 16:26
  • 1
    $\begingroup$ $$x_1x_2=4p(w_1)/p(w_2)$$ $\endgroup$ – gunes Apr 28 '20 at 16:27
  • $\begingroup$ And this is not the equation, I guess? We must find a way to find the p(w1) and p(w2) ? $\endgroup$ – Stats8901 Apr 28 '20 at 16:31

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