# In courts, should we assume that $\Pr(Y=y)$ is constant, for all $y \in \mathcal{Y}$?

In the case of supervised classification, we wish to predict the label of unseen observation $x\in\mathcal{X}$ by assigning it to some label $y \in \mathcal{Y}$. Specifically, we want to find label $y^*$ as follows: $$\begin{split} y^* &= \underset{y\in\mathcal{Y}}{\text{arg max}\;} \Pr(Y=y|X=x)\\ &= \underset{y\in\mathcal{Y}}{\text{arg max}\;} \Pr(Y=y)\Pr(X=x|Y=y) \quad\text{(by Bayes theorem)}\\ \end{split}$$

Of course, if the distribution of labels is uniform, then $\Pr(Y=y) = 1/|\mathcal{Y}|$ is constant for all $y \in \mathcal{Y}$. In such case, we can simplify the above, by dropping $\Pr(Y=y)$, into: $$\begin{split} y^* &= \underset{y\in\mathcal{Y}}{\text{arg max}\;} \Pr(X=x|Y=y)\\ \end{split}$$

And that's essentialy what any supervised classification learning algorithm aims to find. E.g. SVM, NB, etc, essentially find classification models that necessarily imply some definition of those probabilities.

Now, my question is: suppose that a suspect $x$ is to be classified whether he/she is guilty, or not guilty. Suppose that $\Pr(Y=\text{guilty}) =0.6$. Should we use this knowledge when judging on suspects? Or, alternatively, should we ignore such probability and assume that $\Pr(Y=\text{guilty}) = \Pr(Y=\text{not guilty}) = 0.5$?

## My attempt:

I would imagine that dropping $\Pr(Y=y)$ is recommended in legal systems, such as courts. For example, if "theft" is a highly common crime, e.g. $\Pr(Y=\text{theft}) = 0.8$, then we must not tend to rule that suspect $x$ is a theif simply cause others tend to be thieves. In other words, we should assume that $\Pr(Y=y)=1/|\mathcal{Y}|$ for any crime $y$. Instead, all judgements against suspect $x$ should be solely based on maximizing $\Pr(X=x|Y=y)$, under the assumption that $\Pr(Y=y)$ is constant.

Any thoughts?

• What about $Pr(Y = \text{not guilty}) = 0.99$ : - ) May 18, 2017 at 7:16
• Aug 30, 2022 at 18:35

This is actually an example where we have strong a priori knowledge that should be considered. If by time travel, or by traveling to some other, parallel universe, you would be relocated to some kind of Mad Max reality, then maybe you should revise your prior and until learning more about how the society in the other universe functions, you possibly could assume a priori that you really do not know if you should trust, or not, it's inhabitants and assume a priori that they may be dangerous criminals with $50/50$ chance.
• Thank you very much. Just to make sure if I'm on the right track, here is my thought about the best strategy, and would be thankful if you could share your thoughts about it: My understanding: I guess the best way is: start by measuring $\Pr(Y=y)$ empirically, then apply a correction to have a bias in such a way that the optimization objective is maximized. The optimization objective is probably some realistic quantification of the risk of misclassification. May 19, 2017 at 6:15