Yes, it’s still true. The basic principles of mathematics and probability have not changed. Thus, it is surprising to read an article saying that requiring extraordinary evidence for extraordinary claims doesn’t make sense. At least, it’s surprising until you realize the person making the claim is probably an evangelical Christian with an extraordinary claim to promote. So, how should we respond? I’ll provide a short answer and a long answer.
The Short Answer
Bayes’ Theorem, which has been proven to be formally valid, tells us that extraordinary claims (things with very low prior probabilities) do indeed require extraordinary evidence. For example, imagine my friend tells me he won the World Series of Poker. I would first be struck by the improbability of such a claim considering the Main Event has over 6,000 entrants and this particular friend isn’t that good at poker. Even so, it’s not impossible and there is evidence which would overcome that initially low probability. Examples of significant probability-raising evidence would include if he had millions of dollars suddenly and if he had a WSOP bracelet. If you want to see this idea demonstrated mathematically, keep reading.
The Long Answer
This is Bayes’ Theorem:
P(h|e.b) = P(h|b) x P(e|h.b) / [ P(h|b) x P(e|h.b) ] + [ P(~h|b) x P(e|~h.b) ]
|~||Not (or the negation of)|
In English, this says, “The probability that a hypothesis is true, given available evidence and background knowledge is equal to [and then you have the equation].” To understand the equation, let’s dig into the details just a bit.
The first term you’ll notice once you read past the = symbol is P(h|b); this is the prior probability. This term is concerned with the probability that a hypothesis is true given your background knowledge. So, when we say that the virgin birth has an incredibly low prior probability, that means that based on everything we know about the world through science, history, etc. this sort of thing doesn’t generally happen. Quite simply, we understand how babies are made, and this isn’t it. Further, in cases where parthenogenesis does actually occur in other animals, the resulting offspring are always female. So, if true, this would seem to be a one-time thing. I have no idea how many humans have ever existed, but let’s say there have been 100 billion. This would make the prior probability 1/100,000,000,000. Note that this is prior to considering any evidence for the case in question, hence the term prior probability.
The second term above is P(e|h.b), meaning the probability that you would have the available evidence given your hypothesis and background knowledge. In other words, would the available evidence be expected under the hypothesis? For example, say someone claims they’ve been to the beach for the past few hours. You notice they are sunburned and in their car is a beach towel. These things fit the claim well and would merit a high number. On the other hand, the lack of sun kissed skin and a movie ticket stub from two hours earlier would not be the sort of evidence you would expect.
This covers the numerator. It is the prior probability multiplied by the likelihood of the evidence.
Now, as you keep moving from left to right, you’ll notice that the first term in the denominator is the entire numerator repeated. That’s why the theorem is sometimes shortened to P = A / A + B. So, what is B? It is basically the same as discussed above, only for ~H, rather than H. It is the probability that your hypothesis is false and the likelihood of the available evidence given a false hypothesis.
Let’s plug in some numbers using a very low prior probability and see the earlier claim in action.
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .75) = 1.2%
Here we see a low probability event where the evidence is almost nearly as well explained by negating hypotheses and the probability is slightly raised, but remains very low. Now, let’s slowly decrease the likelihood that the evidence can be explained by alternative hypotheses and watch what happens to the outcome.
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .50) = 1.8%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .25) = 3.5%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .10) = 8.3%
- P = 0.01 x .9 / (0.01 x .9) + (0.99 x .01) = 48%
We don’t see a substantial increase in probability until we get into very low ranges of likelihood for the same evidence to be observed on alternative hypotheses. In the last example, there is only a 1% chance that the evidence can be explained by alternative hypotheses. In cases of low prior probability, the evidence must be such that it basically rules out alternative hypotheses to a very high degree.
In other words, extraordinary claims really do require extraordinary evidence.