Conditional Probability of Having Covid-19 Given Some Symptoms What is the likelihood of being infected with covid-19 if I have 3 of the main symptoms?
I'm trying to formulate this as a bayesian statistics problem, would appreciate any suggestions.
For example the symptoms can be fever, muscle ache, diarrhea.
This table from a 2020 study shows the percentage of infected patients (total 99) that show a particular symptom
Fever                (83%)
Cough                (82%)
Shortness of breath  (31%)
Muscle ache          (11%)
Confusion            (9%)
Headache             (8%)
Sore throat          (5%)
Rhinorrhoea          (4%)
Chest pain           (2%)
Diarrhoea            (2%)
Nausea and vomiting  (1%)

In other words given a probability $S_i$ for each symptom and showing k number of symptoms, what is the likelihood $P$ that I'm infected?
Using $P(A|B) = P(B|A) \cdot P(A) / P(B)$, I think the main formulation is:
$$
P(c19 | [fever, cough, \cdots]) = P(c19 | S_1) \cdot P(c19 | S_2) \cdots P(c19 | S_k) / X
$$
Here $X$ combines all $P(B)$ for each symptom, but what is P(B) for each symptom? and do the probabilities simply multiply?
 A: You are asking about conditional independence: $S_1 \; {\rm INDEP} \; S_2 \mid C19$. The way you write the joint probability, as a product over the probabilities of each feature - that model assumes conditional independence.
You can check whether this assumption holds by comparing the joint distributions of pairs of input variables, for each of the possible outcomes of $C19$.
A: Many clinics and medical offices scan for Temperature when you come in the door as
a way to detect people who might have C-19 in order to take special precautions if necessary. I have no idea what the actual probabilities are, but here is how Bayes' Theorem would be used in order to find P(C-19 | Fever), denoted $P(C|F)$ below.
$$P(C|F) = \frac{P(CF)}{P(F)} = \frac{P(C)P(F|C)}{P(CF)+P(C^cF)}\\
=\frac{P(C)P(F|C)}{P(C)P(F|C)+P(C^c)P(F|C^c)}.$$
So in order to find $P(C|F),$ you need to know all of the probabilities in the last expression. Right now, where I live, just knowing the prevalence $P(C)$ seems
difficult. And if $P(F|C^c)$ gets too large (that is, lots of people have fever
for reasons unrelated to Covid-19), then the temperature scans as people come in
the door become useless as a quick screen for Covid-19.
However, if you had information to be sure $P(C|F) > P(C),$ then you'd know temperature scans are useful, and maybe you can get an even larger probability of $C$ given a longer list of symptoms.
A: Here's the way I would do it (check it's correct!)
If you have more than one symptom you apply the Bayesian theorem recursively using the previous result as the new prior probability. Here's an example (works in R) - I made up some starting values without too much thinking about them:
p_fever_C19 = 0.83   # P of fever given you have C19
p_fever_noC19 = 0.01 # P of fever given you DO NOT have C19
p_C19 = 0.0001       # P of C19 without any information about symptoms (i.e. P of C19 in the general population) 

If you have fever your probability of having C19 is:
p_C19_given_fever = (p_C19 * p_fever_C19) / 
                    ((p_C19 * p_fever_C19) + ((1-p_C19) * p_fever_noC19))

Let's say now you also have cough, instead of p_C19 use the previous p_C19_given_fever
p_cough_C19 = 0.82
p_cough_noC19 = 0.1

p_C19_given_fever_cough = (p_C19_given_fever * p_cough_C19) / 
                          ((p_C19_given_fever * p_cough_C19) + ((1-p_C19_given_fever) * p_cough_noC19))

You also have short breath:
p_short_C19 = 0.31
p_short_noC19 = 0.2

p_C19_given_fever_cough_short = (p_C19_given_fever_cough * p_short_C19) / 
                                ((p_C19_given_fever_cough * p_short_C19) + ((1-p_C19_given_fever_cough) * p_short_noC19))

.. and so on for other symptoms. For this example the results are:
p_C19_given_fever
0.008232494
p_C19_given_fever_cough
0.06372898
p_C19_given_fever_cough_short
0.09543484

