Bayes’ Law

# The Theory of Bayes’ Law

If you already understand Bayes’ Law, you may begin practicing by clicking on this link. If you would like to better understand Bayes’ Law, then continue reading.

## Proof of Bayes’ Law

The following video gives a proof of the short and the long form of Bayes’ Law. It also walks you through the Multiple Sclerosis example, below.

## The Multiple Sclerosis Example

Multiple sclerosis (MS) is an autoimmune inflammatory disease in which myelin sheaths around axons of the brain and spinal cord are damaged, leading to loss of myelin and scarring. MS is more common in women and the onset typically occurs in young adults. Diagnosis is difficult as the symptoms mimic many other neurological diseases. However, the McDonald test is the current standard. It uses magnetic resonance imaging to detect brain abnormalities.

Unfortunately, MRI tests are very expensive. Furthermore, the McDonald test is only able to detect MS after scarring has taken place. Also, the McDonald test has both low sensitivity and low specificity when dealing with certain ethnicities (Asians). Research is underway to discover a test that is less expensive and is able to detect MS before brain damage occurs.

» » »   « « «

The above is all true; what follows is not. Let us pretend that your professor developed a blood test for MS — the Sparrow test. Blood tests are usually inexpensive and easy to perform. In a clinical trial of n = 500 people, we found that the test had a false positive rate of 5% and a false negative rate of 1%.

According to a paper by Rosati (2001), the prevalence of MS in the United States is 49 per 100,000 (for whites). At random, I select a white person and test her for MS using the Sparrow test. Given that she tests positive, what is the probability she has multiple sclerosis?

### Solution

The problem asks us to calculate P[ D | + ]. We use Bayes’ Law to do this. First, it would be helpful to calculate the following six probabilities:

 $$P[D] =$$ $$0.00049$$ Disease Prevalence, $$49/100,\! 000$$ $$P[\bar{D}] = P[C] =$$ $$0.99951$$ $$1 - P[D]$$ $$P[+ | D] =$$ $$0.99$$ $$1 - P[- | D]$$ $$P[− | D] =$$ $$0.01$$ False Negative Rate $$P[+ | \bar{D}] = P[+ | C] =$$ $$0.05$$ False Positive Rate $$P[− | \bar{D}] = P[− | C] =$$ $$0.95$$ $$1 - P[+ | C]$$

With these calculated, we use Bayes’ Law to calculate the requested probability:

\begin{align} P[D | +] &= \frac{P[+ | D]\ P[D]}{P[+ | D]\ P[D]\ +\ P[+ | \bar{D}]\ P[\bar{D}]} \\[2em] &= \frac{(0.99)\ (0.00049)}{(0.99)\ (0.00049)\ +\ (0.05)\ (0.99951)}\\[1em] &= \frac{0.0004851}{(0.0004851)\ +\ (0.0499755)}\\[1em] &= \frac{0.0004851}{0.0504606}\\[1em] &= 0.009613441 \end{align}

And so, even if a person tests positive for Multiple Sclerosis using the Sparrow test, that person still only has a 0.961% chance of having it.