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Probabilistic Christmas reasoning with Bayes

by Bruno Van de Casteele

March 16, 2014

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Donate I'm quite a big fan of the Bayes' Theorem. In a quite succinct formula, it manages not only to explain how to work with probabilistic reasoning, it also puts a number to our beliefs or hunches. Now to be honest, the quantitative result is in my opinion not the most important, it's the fact that we have to express how much we think a hypothesis is true, given the evidence and given our background knowledge. As such, it is a valuable tool for any skeptic endeavour.

So what is the Theorem all about? Let me explain the formula. (Explanation based on Richard Carrier)

Don't fear! This formula basically says "the probability (P) our hypothesis (h) is true" (left hand side), "given all that we know so far" (right hand side).
On that right hand side, above the bar, we have "how typical our hypothesis is" multiplied by "how expected the evidence (e) is, given our background knowledge, if our hypothesis is true".
Below the bar on the left you have that same term, to which the opposite is added: "how atypical our explanation is" multiplied by "how expected the evidence is if our explanation isn't true". It's easiest to understand if you consider all probabilities expressed as percentages.

As Carrier points out, the term (b) appears quite a lot. That is because any hypothesis we formulate, even implicitly, always depends on what we already know about the world. Take the following example: What is the probability Santa Claus exists? Let's just assume that we consider an existing Santa Claus to also bring gifts to good children on Christmas morning, just for the sake of simplicity.

Please note that this is very much dependent of who makes the calculations, meaning on one's background notion. A child estimating "how typical the hypothesis is" (which is called the prior probability), will come to a different conclusion than an adult. Given that a child hears and sees a lot of stories about magical creatures who can do all sorts of feats, it might assign a high prior probability, like 80%. As Carrier points out, you cannot just assign prior probabilities, you have to justify them. If you can't, take 50%.

Our child continues its estimation on "how expected the evidence is if the hypothesis is true". This is called the consequent probability, and will take into account the evidence of gifts found under the Christmas tree on the 25th of December. Let's assume that for a young child this is very impressive, so 95%.

As the second part below the bar is the inverse of the above probabilities, we can now calculate everything: above the bar we have (80% * 95%) = 76%, below the bar (80% * 95%)+(20% * 5%) which, after division, gives an impressive 99% chance that Santa Claus exists. Better be nice ...

But, as with any scientific knowledge, the result is not static. Our child learns more about the world around him, and sits down to redo his calculations based on this new knowledge.
For instance, even though the presents are quite good evidence, they can also be the result of other hypothesis (for instance parents). Furthermore, there is evidence that is not there that should be expected, like footprints or hearing the sleigh or Christmas balls.
Our child can also adapt his prior probability, like for instance the lack of observing other magical creatures flying around in the real world, or the impossibility for any being to visit all children in the world. He does need to justify it, of course.

The actual numbers don't really matter, but suppose our child assigns a much lower prior, like 40%, and reduces the consequent to a mere 30%. The result is only a 22% probability. You could still try to be nice, but not because of Santa Claus....

Now the point about Bayes' Theorem isn't, as I said, to throw some numbers around. It just puts in nice clear terms what we implicitly do every time we have to consider whether a claim is true. We might consider something "unlikely" when first hearing it, but if we elaborate on it, we see that there are probably a lot of explanations for a given evidence, reducing the consequent probability. Doors slamming and stairs creaking can have a lot of reasons, they are not necessarily caused by ghosts. And although we have to keep an open mind, it seems a priori less likely that ghosts exist.

As such, it's a nice tool for skeptics. Firstly, because of the importance of the background knowledge. Any skeptical inquiry without a lot of knowledge or experience beforehand is doomed. Knowledge and experience teach you different explanations, which is why amateur astronomers hardly never see UFOs. Secondly, it still allows that a low probability can be countered by solid evidence that cannot be easily explained by other hypothesis. If an alien parks his UFO on my lawn and makes a declaration to the assembled TV crew, I'm very probably going to believe they exist and come here. Just show me the proof.

That is why I like the Bayes' Theorem so much.

Sources: Wikipedia, Richard Carrier (in his book "Proving History"), and numerous videos on Youtube (just search for it).

by Bruno Van de Casteele

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