# What does all the formula and pictures mean?

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...

• 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). Commented Aug 16, 2021 at 17:21
• Thanks, @ArayKarjauv, I really want to know what the tau, sigma, and the division meant. Commented Aug 16, 2021 at 17:37

According to the provided article,

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

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:

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

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.