Question $3$

CS$280$ students: Answer parts $($a$)$ to $($c$)\backslash ((10+10+12)\backslash ).$

CS$491$ students: Answer all parts $\backslash ((8+8+8+8)\backslash ).$

Suppose you are creating a deep learning model that outputs Yes when a given picture of an animal contains a visible eyeball, and No when it does not. Your model has a true$-$positive rate, which is the probability that the model correctly outputs Yes for an image that has an eye, and a true$-$negative rate, which is the probability that it correctly outputs No for an image that does not have an eye. You pull training pictures from a large database of images, and get an image that has an eye with probability $\backslash (\backslash$frac$\{47\}\{100\}\backslash )($otherwise the image does not have visible eyes$).$

Answer the following questions. Treat every question separately $-$ assume that the conditions stated in each part persist only for that part.

Part d

$[8$ marks$]$

You want to build a better version of your model. You decide that the best way to do this is to train the model to improve the true$-$negative rate. You go back to the old training data with $\backslash (\backslash$frac$\{47\}\{100\}\backslash )$ probability of getting an eye image. Your goal is$,$ when your model outputs Yes, the image actually has an eye with probability $\backslash (\backslash$frac$\{99\}\{100\}\backslash ).$ Assuming the true$-$positive rate is fixed at $\backslash (\backslash$frac$\{96\}\{100\}\backslash ),$ what is the needed true$-$negative rate to achieve this?