Applying Bayes Theorem

note: This article presents a hypothetical situation and is not intended as medical advice.

Now that we have a basic understanding of Bayes' Theorem (please refer to these posts on conditional probability and Bayes' Theorem for context), let's extend the application to a slightly more complex example. This section was inspired by this tweet from Grant Sanderson (of 3Blue1Brown fame):

This is a classic application of Bayes Theorem - the medical diagnostic scenario. The above tweet can be re-stated:

What is the probability of you actually having the disease, given that you tested positive?

This happens to be even more relevant as we're living through a generational pandemic.

Let's start off with a conceptual understanding, using the tools we learned previously. First, we have to keep in mind testing and actually having the disease are not independent events. Therefore, we will use conditional probability to express their joint outcomes.

The intuitive visual to illustrate this is the tree diagram:

The initial information provided is as follows:

• P(D): Probability of having the disease (covid-19)
• P(P): Probability of testing positive

• *P(D|P): Our objective is to find the probability of having the disease, given a positive test

• 1 in 1,000 actively have covid-19, P(D), this implies...

• 999 in 1,000 do not actively have covid-19, P(not D)

• 1% or 0.01 false positive (given)

• 10% or 0.1 false negative (given)

The false positive is when you don't have the disease, but your test (in error) shows up positive. False negative is when you have the disease, but your test (in error) shows up negative. We are provided this information and have to calculate other values to fill in the tree.

We know that all possible events have to add up to 1, so if 1 in 1,000 actively have the disease, we know that 999 in 1,000 do not have it. If the false negative is 10%, then the true positive is 90%. If the false positive is 1%, then the true negative is 99%. From our calculations, the tree can be updated:

Now that we've filled out the tree, we can use Bayes' Theorem to find P(D|P). Here is Bayes' Theorem that we discussed in the previous section. We have Bayes' Theorem, the denominator, probability of testing positive P(P) and the second version of Bayes Theorem in cases were we do not know the probability of testing positive (as in the present case):

Then we can plug-in the denominator to get the alternative version of Bayes' Theorem:

Here's how the numbers add up:

• P(D|P) = P(P|D) x P(D) / P(P|D) x P(D) + P(P|not D) x P(not D)
• P(D|P) = (0.9 x 0.001) / ((0.9 x 0.001) + (0.01 x 0.999))
• P(D|P) = 0.0009 / 0.0009 + 0.00999
• P(D|P) = 0.0009 / 0.01089
• P(D|P) ~ 0.08264 or 8.26%

Interestingly, Andrej Karpathy actually responded in the thread and provided an intuitive way to arrive at the same result using Python.

Here's his code (with added comments):

from random import random, seed
seed(0)

pop = 10000000 # 10M people
counts = {}

for i in range(pop):
    has_covid = i % 1000 == 0 # one in 1,000 people have covid (priors or prevalence of disease)
    # The major assumption is that every person gets tested regardless of any symptoms
    if has_covid:                  # Has disease
        tests_positive = True      # True positive
        if random() < 0.1:     
            tests_positive = False # False negative
    else:                          # Does not have disease
        tests_positive = False     # True negative
        if random() < 0.01:    
            tests_positive = True  # False positive
    outcome = (has_covid, tests_positive)
    counts[outcome] = counts.get(outcome, 0) + 1

for (has_covid, tested_positive), n in counts.items():
    print('has covid: %6s, tests positive: %6s, count: %d' % (has_covid, tested_positive, n))

n_positive = counts[(True, True)] + counts[(False, True)]

print('number of people who tested positive:', n_positive)
print('probability that a test-positive person actually has covid: %.2f' % (100.0 * counts[(True, True)] / n_positive), )

We first build a hypothetical population of 10 million. If the prior of disease is 1 in 1,000, a population of 10 million should find 10000 people with covid. You can see how this works with this short snippet:

pop = 10000000
counts = 0

for i in range(pop):
    has_covid = i % 1000 == 0
    if has_covid:
        counts = counts + 1
print(counts, "people have the disease in a population of 10 million")

Nested in the for-loop are if-statements that segment the population (10M) into one of four categories True Positive, False Negative, True Negative, False Positive. Each category is counted and stored in a dict called counts. Then another for-loop is used to loop through this dictionary to print out all the categories:

has covid:   True, tests positive:   True, count: 9033
has covid:  False, tests positive:  False, count: 9890133
has covid:  False, tests positive:   True, count: 99867
has covid:   True, tests positive:  False, count: 967

number of people who tested positive: 108900
probability that a test-positive person actually has covid: 8.29

Finally, we want the number of people who have the disease and tested positive (True Positive, 9033) divided by the number of people who tested positive, regardless of whether they actually have the disease (True Positive (9033) + False Positive (99867) = 108,900) and this comes out to approximately 8.29.

Although the code was billed as "simple code to build intuition", I found that Bayes' Theorem is the intuition.

What about symptoms?

The key to Bayes' Theorem is that it encourages us to update our beliefs when presented with new evidence. But what if there's evidence we missed in the first place?

If you look back at the original tweet, there are important details about symptoms that, if we wanted to be more realistic, should be accounted for.

You feel fatigued and have a slight sore throat.

Here, instead of assuming that prevalence of the disease (1 in 1,000 people have covid-19) is the prior, we might ask what probability that someone who is symptomatic has the disease?

Let's suppose we change from 1 in 1,000 to 1 in 100. We could change just one line of code (while everything else remains the same):

for i in range(pop):
    has_covid = i % 100 == 0 # update info: 1/1000 have covid, but 1/100 with symptoms have covid

The probability that someone with a positive test actually has the disease jumps from 8.29% to 47.61%

has covid:   True, tests positive:   True, count: 180224
has covid:  False, tests positive:  False, count: 19601715
has covid:  False, tests positive:   True, count: 198285
has covid:   True, tests positive:  False, count: 19776
number of people who tested positive: 378509
probability that a test-positive person with symptoms actually has covid: 47.61

Thus, being symptomatic means our priors should be adjusted and our beliefs about the likelihood that a positive test means we have the disease (P(D|P)) should be updated accordingly (in this case, it goes way up).

Take Aways

Hypothetically, if we have family or friends living in an area where 1 in 1,000 people have covid-19 and they (god forbid) got tested and got a positive result, you could tell them that their probability of actually having the disease, given a positive test was around 8.26–8.29%.

However, what’s useful about the Bayesian approach is that it encourages us to incorporate new information and update our beliefs accordingly. So if we find out our family or friend is also symptomatic, we could advise them of the higher probability (~47.61%).

Finally, we may also advise our family/friends to get tested again, because as much as test-positive person would hope they got a ‘false positive’, chances are low. And even lower, is getting a false positive twice.

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