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Bayes' Theorem
Bayes' Theorem
We have shown the potential problems of NHST. One solution to the problems is to use the Bayesian method that is built on Bayes' theorem. The Bayes' theorem is stated mathematically as:
where A and B are events and P(B)≠0. For example, A could be a hypothesis and B could be data. In the formula above,
Pr(A) is the probability of observing only A (without B)
Pr(B) is the probability of observing only B (without A)
Pr(A|B) is the probability of observing A given the B is true
Pr(B|A) is the probability of observing B given the A is true.
In terms of hypothesis testing, we are interested in P(H0|D) and P(H1|D), the probability that either out null or alternative hypothesis is true given the data we collect. Suppose there are only one null hypothesis and one alternative hypothesis. Then the probability that the null hypothesis is true given the observed data is
and the probability that the alternative hypothesis is true given the observed data
Clearly, Bayes' theorem provides a way to directly tangle the probability of the hypotheses, which is often the focus of a study. The Bayesian interpretation of the formula is as follows. P(H0) is called prior that presents one's belief about the probability that the hypothesis H0 is true before collection of data and/or conducting a study. With the collected data D, one can update the probability about the same hypothesis to get the posterior Pr(H0|D).
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EXAMPLE 1
Suppose a crime has been committed and blood is found at the crime scene that is directly related to the crime. Statistics shows that the type of blood is present in 1% of the population.
The prosecutor may state: "There is a 1% chance that the defendant would have the crime blood type if he were innocent. Thus, there is a 99% chance that he is guilty."
However, the defender may argue: "The city of crime scene has 800,000 people. The blood type would be found in about 8,000 people. The defendant only has a chance of 1 in 8,000 to be guilty. Thus, the defendant is innocent."
There are problems in both of the statements. The first statement is known as the prosecutor's fallacy and the second one is known as the defender's fallacy.
In this example, the null hypothesis is that the defendant is innocent (I) and the alternative hypothesis is that the defendant is guilty (G). Now we want to know the probability that the defendant is guilty or innocent given the evidence of the found blood (E). From the statistics, we know that p(E|I) = 1/100. The prosecutor seems to believe p(G|E)=1−p(I|E)=1−p(E|I)=99/100, which is not true. But note this is a typical mistake when employing NHST.
If there is no any evidence (or information) about who is the criminal, then everyone in the city has the equal probability to be guilty so that
Those probabilities are prior probability in Bayesian terms. Now we need to find that conditional probability of I given the evidence E, otherwise written as P(I|E). Based on the statistics, we know p(E|G)=1 and p(E|I)=1/100. This is because if the defendant committed the crime, the probability that his/her blood type would be found is 100%. However, even he/she is innocent, there is still a 1% chance that her/his blood type would be found.
Using Bayes' theorem, we know that
Thus, the defendant is very likely to be innocent based on the current limited information. This is equivalent to the argument of the defender.
To investigate the two kinds of fallacies, we can also get
Clearly, the odds of guilty to innocent conditionally on the evidence found falls back to the odds of prior probability - the probability without considering the current evident at all. The prosecutor believes that the defendant has equal chance (50%) to commit the crime. Thus, the prior odds is 1 and the posterior odds is 100. Then the prosecutor can conclude that the defendant has 99% chance to be guilty.
The defender assumes that the p(G)/p(I)=1/799,999. Thus, the posterior odds are
100/799,999≈1/8,000
However, if the defender is arrested, there must be more evidence than the blood type only. Then, the prior odds could be much larger. As shown above, even the prior odds is 1, there is a 99% probablity for the defendant to be guilty.
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Concept Review
In essence, the idea of NHST assumes a "right" and "wrong", a "guilty" and "innocent", an "is" and "isn't", or a "true" or "false" based on how often (the frequency) something happens. Based on the data or evidence we collect, we make a judgement about whether our null hypothesis is true (we accept the null) or false (we reject the null). As mentioned in previous sections, there are problems with assuming that an alternative is true when the null is false.
In order to move away from the constraints of "right" and "wrong", we look to Bayes' Theorem. Rather than deciding on a "right" or "wrong", Bayes' Theorem seeks to update our current beliefs based on the data/evidence we collect. We may never know with certainty what the "correct" answer is to the question we are asking, but the more data we collect, the closer to the actual truth we assume we will be.
However, it is important to remember that any method we use has unique advantages/disadvantages, and no method is perfect on its own yet.
References:
References:
Zhang, Z. & Wang, L. (2017). Advanced statistics using R. [https://advstats.psychstat.org]. Granger, IN: ISDSA Press. ISBN: 978-1-946728-01-2. https://advstats.psychstat.org/book/hypothesis/bayes.php
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