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Method 1

I am given a classifier for some disease that takes as input patient characteristics and has some sensitivity and specificity.

Hence the classifier is a function c(patient characteristics) = 1 or 0

I can then use Bayes rule to convert:

P(disease | c(patient characteristics) = 1) = P(c(patient characteristics) =1 |dz) P(dz) / P(c(patient characteristics) = 1)

Using classifier sensitivity and specificity to write P(c(patient characteristics) = 1 |dz)/P(c(patient characteristics) = 1).

So, even if all I have is a classifier (which made a decision), I can get some probability estimate.

Method 2

The better approach is to develop an estimator directly for

P(disease | patient characteristics)

Eg using logistic regression or just never binarizing classifier output in the first place.

Classifiers are heavily criticized in medicine, and I agree that it's a poor choice to make a decision without patient and physician utilities, but why can't we just use the first method to convert the classifiers to probabilities?

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Don't overcomplicate your use of probabilistic reasoning by bringing in extraneous considerations. Regardless of the nature of the particular event (whether it involves patient characteristics or the outcome of a medical test) you can apply Bayes' rule to obtain the posterior probability that the patient has the disease. For any arbitrary event $\mathcal{A}$ you have:

$$\underbrace{\mathbb{P}(\text{Disease} | \mathcal{A})}_{\text{Posterior}} = \frac{\mathbb{P}(\mathcal{A} | \text{Disease} ) }{\mathbb{P}(\mathcal{A})} \cdot \underbrace{\mathbb{P}(\text{Disease})}_{\text{Prior}}.$$

Every event $\mathcal{A}$ that is statistically related to the event $\text{Disease}$ gives information that updates the probability of the latter according to Bayes' rule. Whether or not a medical test or a set of patient characteristics constitute better information depends on the data.

Update: Your additional edits to your question make it clear that you are also asking about the difference between conditioning on a vector of patient characteristics $\boldsymbol{x}$, versus a binary classifier $c(\boldsymbol{x})$ that maps the patients into two groups. The latter entails a loss of information, so you are correct that you are better off modelling based on the unclassified characteristics directly.

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  • $\begingroup$ Thank you I edited to be more explicit, but I suspect your answer will be the same. $\endgroup$ – user0 Apr 27 '18 at 14:23
  • $\begingroup$ I have added an update to my answer to add more information responding to your edit. Hope this is helpful. $\endgroup$ – Reinstate Monica Apr 28 '18 at 3:40
  • $\begingroup$ Thank you, yes--your original answer actually pretty much answered in saying that method 1 is feasible, if I understand correctly. My question was really whether method 1 was reasonable (even if not optimal) since many times in medicine we are given a classification, or some result of a medical test, which I suppose is also a classification, but no posterior. So I was just hoping to double check that even with a very complicated classifier the same probabilistic result held $\endgroup$ – user0 Apr 28 '18 at 23:59
  • $\begingroup$ Yep, still feasible. $\endgroup$ – Reinstate Monica Apr 29 '18 at 2:13

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