Answers.Body: <p>According to the provided article,</p>
  <p><span class="math-container">$$
  \begin{equation} \tag{1}
  P(D| {\mathcal{E}};\ \theta )
  \end{equation} 
  $$</span></p>
  <p>is a probability of disease <span class="math-container">$D$</span> given findings <span class="math-container">$\mathcal{E}$</span>, and a model <span class="math-container">$\theta$</span> that is used to estimate this probability.</p>
  <blockquote>
  <p><span class="math-container">$D$</span> represents a disease or diseases, and findings <span class="math-container">$\mathcal{E}$</span> can
  include symptoms, tests outcomes and relevant medical history.</p>
  </blockquote>
  <p>The Sheffer stroke symbol (vertical bar) <code>|</code> in the <a href="https://en.wikipedia.org/wiki/Conditional_probability" rel="nofollow noreferrer">conditional probability</a> notation is read &quot;given that&quot;, whereas the semicolon symbol <code>;</code> tells that we use a model (or parameters for the model) to calculate this probability. For instance, we can define this probability as <span class="math-container">$P(D| {\mathcal{E}};\ \theta ) = M_{\theta}(\mathcal{E})$</span>, where <span class="math-container">$M$</span> can be a nueral network with parameters <span class="math-container">$\theta$</span> that takes <span class="math-container">$\mathcal{E}$</span> as input and returns the probability of <span class="math-container">$D$</span>.</p>
  <p>As we read the article further, we see that Equation 1 is nothing more than the <a href="https://en.wikipedia.org/wiki/Posterior_probability" rel="nofollow noreferrer">posterior probability</a> from <a href="https://en.wikipedia.org/wiki/Bayes%27_theorem" rel="nofollow noreferrer">Bayes' theorem</a>:</p>
  <p><span class="math-container">$$
  \begin{equation} \tag{2}
  P(D| {\mathcal{E}};\ \theta )=\frac{P({\mathcal{E}}| D;\ \theta )P(D;\ \theta )}{P({\mathcal{E}};\ \theta )}.
  \end{equation}
  $$</span></p>
  <p>where <span class="math-container">$P({\mathcal{E}}| D;\ \theta )$</span> is a likelihood of findings <span class="math-container">$\mathcal{E}$</span> given that we have the disease <span class="math-container">$D$</span>, <span class="math-container">$P(D;\ \theta )$</span> is the prior probability of the disease <span class="math-container">$D$</span>, and <span class="math-container">$P({\mathcal{E}};\ \theta )$</span> is a likelihood of findings <span class="math-container">$\mathcal{E}$</span>.</p>
  <p>As the article suggests, Theorem 2 is related to the <a href="https://www.sciencedirect.com/science/article/pii/S0888613X01000391?via%3Dihub" rel="nofollow noreferrer">Noisy-OR model</a>, 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.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2021-08-16T19:32:53.870Z
Answers.LastActivityDate: 2021-08-16T19:32:53.870Z
Answers.UserDisplayName: Aray Karjauv
Body: <p><a href="https://www.nature.com/articles/s41467-020-17419-7" rel="nofollow noreferrer">https://www.nature.com/articles/s41467-020-17419-7</a></p>
  <p>I am a medical school graduate and I really want to learn AI/ML for computer-aided diagnosis.</p>
  <p>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.</p>
  <p>The thing is I couldn't understand what the formulas mean in the article, e.g.:</p>
  <p><span class="math-container">$$P(D \mid \mathcal{E}; \theta )$$</span></p>
  <p>I really want to know what | and ; are doing here, what do they mean, etc.</p>
  <p>I would really happy if someone can directly answer or just provide me some references to get general idea quickly.</p>
  <p>Here comes the most tricy part...</p>
  <p><a href="https://i.sstatic.net/B4TCL.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/B4TCL.png" alt="enter image description here" /></a></p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-08-16T17:21:18.807Z
Comments.Text: 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).
Comments.UserDisplayName: Aray Karjauv
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-08-16T17:37:31.870Z
Comments.Text: Thanks, @ArayKarjauv, I really want to know what the tau, sigma, and the division meant.
Comments.UserDisplayName: pullidea-dev
Tags: <neural-networks><machine-learning><algorithm><bayesian-networks>
Title: What does all the formula and pictures mean?
UserDisplayName: pullidea-dev
ThreadText: # What does all the formula and pictures mean?
  
  (Asked by: username_0 on 2021-08-16)
  
  <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](https://i.sstatic.net/B4TCL.png)](https://i.sstatic.net/B4TCL.png)
  
  **Tags:** neural-networks, machine-learning, algorithm, bayesian-networks
  > **Comments:**
  > **username_1**: 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).  
  > **username_0**: Thanks, @ArayKarjauv, I really want to know what the tau, sigma, and the division meant.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_1 on 2021-08-16. Score: 2 | Accepted answer
  
  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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Posterior_probability) from [Bayes' theorem](https://en.wikipedia.org/wiki/Bayes%27_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](https://www.sciencedirect.com/science/article/pii/S0888613X01000391?via%3Dihub), 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.
  
  
  ---
  
  