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https://www.nature.com/articles/s41467-020-17419-7

I am a medical school graduate and I really want to learn AI/ML for computer-aided diagnosis.

I was building a symptom checker and I found the material. It clarifies the drawbacks of associative models which are performing differential diagnosis. And it suggests counterfactual(causal) approach to improve accuracy.

The thing is I couldn't understand what the formulas mean in the article, e.g.:

$$P(D \mid \mathcal{E}; \theta )$$

I really want to know what | and ; are doing here, what do they mean, etc.

I would really happy if someone can directly answer or just provide me some references to get general idea quickly.

Here comes the most tricy part...

enter image description here

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    $\begingroup$ This is a notation of a conditional probability, which means: The probability of $D$ given $\mathcal{E}$ with some parameters $\theta$. In other words, the likelihood of the event $D$ if event $\mathcal{E}$ happens. $\theta$ denotes parameters for the model (e.g., for neural network). $\endgroup$ Aug 16 at 17:21
  • $\begingroup$ Thanks, @ArayKarjauv, I really want to know what the tau, sigma, and the division meant. $\endgroup$
    – Nikita
    Aug 16 at 17:37
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According to the provided article,

$$ \begin{equation} \tag{1} P(D| {\mathcal{E}};\ \theta ) \end{equation} $$

is a probability of disease $D$ given findings $\mathcal{E}$, and a model $\theta$ that is used to estimate this probability.

$D$ represents a disease or diseases, and findings $\mathcal{E}$ can include symptoms, tests outcomes and relevant medical history.

The Sheffer stroke symbol (vertical bar) | in the conditional probability notation is read "given that", whereas the semicolon symbol ; tells that we use a model (or parameters for the model) to calculate this probability. For instance, we can define this probability as $P(D| {\mathcal{E}};\ \theta ) = M_{\theta}(\mathcal{E})$, where $M$ can be a nueral network with parameters $\theta$ that takes $\mathcal{E}$ as input and returns the probability of $D$.

As we read the article further, we see that Equation 1 is nothing more than the posterior probability from Bayes' theorem:

$$ \begin{equation} \tag{2} P(D| {\mathcal{E}};\ \theta )=\frac{P({\mathcal{E}}| D;\ \theta )P(D;\ \theta )}{P({\mathcal{E}};\ \theta )}. \end{equation} $$

where $P({\mathcal{E}}| D;\ \theta )$ is a likelihood of findings $\mathcal{E}$ given that we have the disease $D$, $P(D;\ \theta )$ is the prior probability of the disease $D$, and $P({\mathcal{E}};\ \theta )$ is a likelihood of findings $\mathcal{E}$.

As the article suggests, Theorem 2 is related to the Noisy-OR model, which itself is a large area of research. I encourage you to read the references provided in the article to learn more about approaches used by authors. If you have further questions regarding this theorem, I suggest you to open another question.

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