The Importance of Probabilistic Reasoning
How do you determine if something is true? It could be anything from a claim about history or why your car is making a funny noise to what is most likely to happen in the future. Every single day of our lives, whether we realize it or not, we use probabilistic reasoning. As much as we would like it, it’s usually impossible to prove anything to a point of being irrefutable. This makes some people very uncomfortable as we humans love certainty. Nonetheless, we reason this way all the time probably without realizing it.
The reason I’m bringing this up is twofold. One is that this type of reasoning allows us to make complete sense out of a lot of discussions about religion, science, history, politics and many other fields. Words like “faith”, “belief”, “truth”, “certainty” and “trust” get thrown around a great deal in discussions about religion and I think we should be aware of what we are really saying when we use certain terms. Second is that there is a very common error in reasoning made that I intend to expose in this article. This error is made and accepted by both those attempting to prove and refute claims. For example, if an atheist is having a discussion with a religious believer, both sides often make this same error.
But before I go into detail, let’s look at a short example.
“You need faith to believe that the sun will come up tomorrow”
Claims like these are often made in discussions about religious belief. The attempt here is to illustrate that the non-believer uses faith just as much as the believer. You need faith to believe that the sun will come up. You need faith that the lights will come on when you flip the switch. You have faith that your spouse actually loves you. The claim is followed with statements about not being able to know the future or the possibility that a light bulb is blown or that your spouse is secretly deceiving you. Therefore, you must have faith in these things and this makes you, the non-believer, no different than the person who uses faith to believe in their god of choice.
But is it really faith that we need here? No.
This is because we are dealing with probabilities. Is it possible that the sun will not come up tomorrow? Wow, I suppose that there is some very strange, remote possibility that it will not. Maybe I’m in the Matrix and the robots could reprogram it so that the sun will not come up. Or maybe the Earth will get hit by a gamma ray burst or stray planet and cease to exist. Or maybe the sun itself will get swallowed by a rogue black hole traveling at near light speed through the universe. So I suppose that there’s a chance that it will not come up tomorrow.
But how high is that probability? I provided several means by which the sun may not rise tomorrow, but the likelihood of any one of them actually happening is vanishingly small. So small, in fact, that I needn’t be concerned about any of them. I’m going to make a list of each one and assign them a probability. In fact, a MUCH higher probability than they even deserve. We could actually calculate some of these based on past observations, but let’s go with these just for illustration.
- The sun rises as usual: 99.9999997%
- The Reprogrammed Matrix: 0.0000001%
- Earth hit by Gamma Ray Burst: 0.0000001%
- Sun swallowed by black hole: 0.0000001%
If you were to bet your life on it, which one are you going to count on? You see, these four hypotheses are in competition with each other. They don’t exist in vacuums. Could the sun be swallowed by a black hole? Sure, but there’s a much better and likely hypothesis there and the probability of that hypothesis being true is so high that it would be absolutely ludicrous and irrational to believe the others.
That’s a future claim. But what about a current claim: does my wife love me? Some would say I can’t know for sure and that I resort to faith, just like a Christian does to believe in the resurrection of Jesus. But let’s approach this the way I did the previous situation. This time, however, I’m going to incorporate actual evidence. Here is the evidence:
- We have been together for over 10 years
- We never have arguments
- She has never shown any measure of dishonesty in other matters
- I have zero evidence that she has been unfaithful
- She makes sacrifices for my well-being and happiness (and vice versa)
- She continually provides gestures of love and caring
- She tells me so every single day
- We have very good communication and talk a lot
- We work great as a team
- She’s been with me through some very difficult times
- We do many things together and she appears to have fun
- The divorce rate is very high in the US
- There are marriages that have lasted a long time that still fail
I’m sure I could come up with more, but those are off the top of my head. Based on this evidence, what is the probability that my wife really does love me? Well there are two pieces of evidence against that probability: the high divorce rate and the fact that even long-lasting marriages can fail. But these two are true for absolutely any relationship and therefore do not lower the probability of my wife loving me. So we are left with all those good things (and the many I didn’t list). So far it looks like a near 100% probability that my wife loves me. Yay!!
But could she be deceiving me? Maybe she has been dishonest for over 10 years. Maybe she’s really having an affair and has hidden it extraordinarily well. Perhaps she’s in it to win a very long term bet. How do I know that she’s not being deceitful? Well, I can’t for certain because she may just be the greatest liar in all history. I have to concede that possibility.
But just because I have to concede that as a possibility, does that mean I should believe it? Of course not. First of all, if she truly didn’t love me, then there has to be some explanation for all of those other good things. Could you come up with some? Sure, but you have to account for the probability of your explanation as well. And if that probability is low then it lowers the overall probability that she doesn’t love me.
Let’s put it another way: what would our relationship look like if she truly didn’t love me and what would our relationship look like if she did? Remember, these two hypotheses are in competition with each other. We have to consider the evidence for BOTH and compare them.
Our relationship looks absolutely nothing like it would if my wife didn’t truly love me and it looks exactly as it should if she did. We could say that the evidence for her love is 100% probable if she did love me. It is exactly what we would expect. But all of this evidence is perhaps only 0.001% likely if she did not. I therefore do not need faith to believe that she does. I can believe it as fact, or at least a close to fact as we can ever hope to get outside of mathematics.
The Big Problem
I mentioned a big problem that I’ve seen in informal discussions and debates between believers and non-believers. The problem is that very often a believer is able to come up with a hypothetical explanation for an extraordinary claim and because they can come up with it and that it’s plausible, they are granting it a probability equal to the natural explanation offered by the non-believer. Quite often, the non-believer concedes and allows this possibility and the conversation moves on to another topic.
But that conversation isn’t over.
Again, hypotheses do not exist in a vacuum all by themselves. Their mere existence does not prove truth. They must be compared to other hypotheses. This is the only way we can tell which is the more likely to be true.
Here’s an example. If a secularist and a Creationist are debating origins and the secularist brings up the presence of fossils, the Creationist may say something like “Those fossils were put there by Satan to deceive us and guide us away from the truth that God created the world.” This is a hypothesis about past events. This individual is asserting that this is what actually happened in Earth’s history.
Now, we could just accept this hypothesis and say, “Well, yeah, I guess if Satan does exist he could have done that”, but this is giving equal credibility to this hypothesis and this is The Big Problem. We aren’t done with this conversation yet.
First there is something called “prior probability”. This is basically a starting probability for any given claim based on relevant background knowledge. For example, if someone claims to have been abducted by aliens, before we even examine the evidence, we have to assign this a very low probability since according to our background knowledge, thus far we haven’t found any alien abduction stories to be reliable. Let’s say we start with a one in a million shot (that may even be too high). This is a probability of 0.0000001%. That’s extremely low, yet it is reasonable to approach this sort of claim with that much skepticism.
But we aren’t finished yet. That probability is just our starting point based on what we know about alien abduction claims. This person may have evidence of their abduction. So we now have to examine the actual evidence for the claim, but we have to do this in two ways. First we examine the evidence and determine how likely it is to exist given the hypothesis (that someone was abducted by aliens). Then we examine the evidence to see how probable it is given by the opposite (that they were not abducted by aliens). This is what I mean when I say hypotheses don’t exist in a vacuum. We must compare them.
If the evidence is good enough it can overcome that extremely low prior probability. But what does “good enough” mean? It means that a) the evidence is exactly what we would expect if the hypothesis were true and b) no other explanation makes sense. Note that both of these have to be true for a hypothesis or claim with such a low prior probability. Because even if the evidence is explainable by a hypothesis, if there is a counter-hypothesis that also makes sense of the evidence and this one has a higher prior probability, then it is the counter-hypothesis that is the more likely.
So what would evidence look like to overcome the low prior probability of alien abduction? Well, perhaps something like an iPad-like computer but instead of being electronic, it was apparently made of living, biological material. This would be something that would fit the alien hypothesis perfectly and at the same time, no other explanation could be made for it. The probability of this being only possible if they retrieved this object when being abducted by aliens is extremely high and the probability that someone on earth made it, given what we currently know about man-made biological devices, is extremely low. I think in this case we could be justified in believing the claim.
But what if, instead of this great physical evidence, the person making the claim presented merely stories of the experience? Well this is what all the other false claims have looked like. The natural explanations (hallucination, dreaming, lying) are all much more likely than an actual abduction.
Before moving on, let’s recap a little with a list of points to take away.
- Multiple hypotheses may be offered to describe phenomena
- These hypotheses must be compared by examining evidence
- Hypotheses with probabilities much higher than competing hypotheses are more likely to be true
- Just because a hypothesis is plausible does not make it probable
- It is irrational to believe hypotheses that have extremely low probabilities
- Hypotheses that have extremely high probabilities are the closest thing we have to “fact” and can be treated as such
- “Prior probability” is a starting probability of any given claim based on our background knowledge related to such claims (such as how often similar claims turn out to be true)
- Hypotheses with a very low prior probability can be overcome with evidence that can only be explained by that hypothesis. This is another way of saying “extraordinary claims require extraordinary evidence”
Believe it or not this is all mathematical and is governed by a lovely formula called Bayes’ Theorem. So if two people can agree on probabilities you can actually calculate final probabilities of two competing hypotheses. If you aren’t mathematically inclined or not interested, you may skip to the next section. Here is the complete Bayes’ Theorem in equation form:
P(h|e.b) = [ P(h|b) * P(e|h.b) ] / ( [ P(h|b) * P(e|h.b) ] + [P(~h|b) * P(e|~h.b) ] )
Whoah… that’s pretty scary looking. It’s pretty basic math, however. Multiplication, addition and division. The rest is just a notation we can use to describe terms. In fact, I’m going to rewrite it with different terms just to see how basic the formula is.
P = [ x * y ] / ( [ x * y ] + [ z * q ] )
So there are only four numbers needed for the calculation and the fifth term (P) is the result. So what are these terms?
P(h|e.b) = this is our final probability of a hypothesis ‘h’ being true given (that’s the | ) the evidence ‘e’ and the background knowledge ‘b’.
P(h|b) = The prior probability of the hypothesis given the background knowledge. This is the starting probability before we even look at evidence. It’s the 0.0000001% in our alien abduction claim.
P(e|h.b) = This is the probability that the evidence would exist if the hypothesis is true
P(~h|b) = This is the prior probability of our hypothesis NOT being true. This is just 1 minus the prior probability. It would be 99.999999% in our alien abduction claim.
P(e|~h.b) = This is the probability that the evidence would exist if the hypothesis is NOT true
Let’s take a look at the alien abduction example numerically.
P(h|e.b) = [ 0.00000001 * 1 ] / ( [0.00000001 * 1] ) + [99.9999999 * 0.0000000001] )
Ok, so let’s break this down. The 0.00000001 is our VERY low prior probability that any alien abduction claim is true, given what we already know about these sorts of claims (our background knowledge). This means that we are very skeptical of this claim right off the bat, however, we’re willing to consider the evidence.
Remember, our evidence was that the person making the claim also brought back an alien artifact: an iPad that functions biologically instead of electronically. In our case, P(e|h.b) is equal to one. One means it is 100% likely that this is exactly the kind of evidence we would have if someone had actually been abducted by aliens.
The 99.9999999 represents the prior probability that the claim is NOT true. This is just one minus the prior probability of it being true.
Finally, the 0.0000000001 (nine zeroes) represents the probability that the evidence (the biologically operated iPad) would exist if he had NOT been abducted by aliens. This very low probability is basically saying there is no good explanation for the existence of this evidence if his theory were not true. If he hadn’t been abducted by aliens, then we have no idea how he came across this object and there’s literally no explanation for it. However, because there is the tiniest possibility that some human made it and there is high technology on our own planet that we aren’t aware of, we have to give it *some* probability.
So, after running the numbers, we arrive at a final probability of:
0.990099 = 99%
In other words, even though we were rightfully VERY skeptical and assigned it an extremely low prior probability before looking at the evidence, the magnificent evidence was enough to overcome that low starting probability. It is therefore extremely probable that this person was truly abducted by aliens. There is only a 1% chance that he didn’t because that magnificent biological iPad just might have been man made but it’s very unlikely.
That math is really cool isn’t it?
In fact, Bayes Theorem is used all over the place in the sciences and history. It is one of the most important formulas ever devised. We use it every day in our own lives, but we don’t necessarily think about it mathematically. We just think about things in terms like “How often is a claim like this actually true?” or “Is this what the evidence would look like if they are right?”.
Thus ends the math part of this article. Hope it wasn’t too painful. However, if you aren’t comfortable with math, I can’t suggest strongly enough to try.
This is why religious claims fail
Many with religious beliefs struggle to understand why non-believers or even simply believers of other religions could fail to understand what is so obviously true to them. There are also those who aren’t surprised that others refuse to believe. It’s a part of their worldview and it is accurate. But regardless, religious believers have explanations, everything makes sense, they can prove it with Scripture, they have the authority of God’s True Servants or Prophets and everyone else just can’t see it, right?
Well let’s examine religious claims with Bayes’ Theorem and see what happens.
The first problem is prior probability. Imagine you are approached by a religious person and they claim their religion has the one pure truth of the universe. Notice, I’m not telling you which religion. It could be any of the thousands out there. Well these thousands of religions can’t all be true, of course. At most only one of them is and at fewest none of them are, which itself is also a hypothesis about the universe.
What is the probability that the person who approached you is correct? Well, we can actually calculate that if we know how many religions there are out there. This can only be estimated, of course. There are estimates of over 30,000 different Protestant denominations alone depending on your definition of denomination. But to give the benefit of a doubt we’ll reduce the number to 2,000 different mutually exclusive religious worldviews worldwide. This means that the probability of this person being correct about having the One True Faith is only 1 / 2000 or 0.05%. That is one half of one tenth of a percent. That’s already very highly unlikely that this person is correct just based on how many other beliefs there are.
This is our prior probability for the claim of any given religious system being true: 0.05%. As with the alien abduction example, this means that the evidence to support this truth must be VERY good. On top of that there must be no other good, valid explanation for the same evidence. That is the only way to overcome this probability.
So now for the evidence. This part becomes quite complex as there can be so many different things presented as evidence for the truth of a religion. Scripture, personal experience, prophecy, unexplained phenomena, morality, the complexity of life and the universe, answered prayers and other perceived miracles, “unity of thought”, internally consistent doctrines, age of tradition and many others. I certainly don’t have room in this article to cover all of these but its certainly worth looking at a few and you can think about the rest.
Scripture is probably the best to start with because it is so often used as evidence for the truth of religious claims. We could technically even look at individual claims in Scripture, such as the darkening of the sky for three hours at Jesus’ crucifixion, the existence of Moses, the reality of the Flood or many others.
But the example of Scripture is an interesting one in probabilistic reasoning. Remember that when we look at evidence what we’re really trying to determine is the probability that the evidence exists if a hypothesis is true. So for example, say a Pentecostal comes to you with the Bible and says that the Bible proves that they are correct. A Presbyterian then does the same thing. Then a Jehovah’s Witness. Then a Catholic. Then a Baptist. Then a Lutheran. Then a Methodist. Each one of them can use the Bible to prove their claims. In other words, each one of them is saying that the Bible is exactly the sort of evidence we should expect if they were right and they can explain why. Even on matters that are totally opposite between different Biblically based religions, such as the Trinity, works righteousness, method of baptism, speaking in tongues, snake handling, the role of Old Testament Law, homosexuality, the end times, existence of Hell, who goes to Heaven, who should take part in the Lord’s Supper and many others, each different sect can support their claim using the Bible and they are 100% convinced they are right. All religions can do it.
So since the probability of the evidence is (claimed to be) 100% likely for any given Bible based religion, a problem occurs. The numbers totally cancel each other out and we are left with the original prior probabilities. What this means is that Biblical evidence cannot be used to support one religious system over another. It doesn’t work because they would all claim that the contents of the Bible is exactly what we would expect if they were right. A believer may be tempted to show someone of a different tradition how they are wrong, but remember, their opponent can do the same thing.
So here we are, back to square one where each religion has only a 0.05% chance of being true and the Bible can’t be used as evidence because it supports all Bible based hypotheses equally well.
The same thing happens with personal experiences, such as when Mormons claim that they know the Book of Mormon is true by how they feel about it when they read it and pray about it. I also personally know a Muslim who says similar things about the Quran. So many people from so many faith traditions have made claims of profound spiritual experiences that evidenced the truth of their beliefs that none of them can be used. If you’ve ever seen the Pixar film “The Incredibles”, it’s similar to when the villain Syndrome says, “Then everybody will be Super. And when everyone is Super…. hehehehe… nobody will be.” Personal experiences aren’t special when there are so many from conflicting traditions.
A new definition for “faith”
It occurred to me that we can redefine faith in light of Bayesian reasoning. If you compare two hypotheses, one of which turns out to have a high probability of being correct while the other has a very low probability of being correct, the rational person moves on with life as if the high probability hypothesis is correct. But what about those billions of people who at this very minute, possibly even some who are reading this article, are counting on that very low probability being the right one. Often when asked why they hold to their beliefs, they will mention faith. I think that’s about as good a word as any.
A traditional definition of “faith” often used by non-believers is to believe something without evidence. This automatically ignites arguments as the faithful then decry this notion and present what they believe to be evidence. Ironically some of the faithful, even those who present evidence, will later assert that faith is the better virtue and that if we had evidence we wouldn’t need faith and what point would there be in not believing in God? It is therefore necessary that we do not have evidence because our belief and faith in God is what is most important to God in the first place. If he presented evidence of himself this requirement would be moot.
I no longer agree with the definition that implies blind faith. Perhaps for some religious believers it truly is blind faith, but after having many discussions with believers from a variety of traditions I have found that most of them are ready and willing to provide some form of evidence, whether it be Scriptural, historical, scientific, personal experience and many others. Apologetics for all faith traditions is full of presentations of evidence.
So let’s find a new definition.
In probabilistic terms, we could define faith as “believing in a hypothesis in spite of a low probability of that hypothesis being true.” Well, what constitutes “low”? That’s certainly up for debate, but I would go with something like 25% or less. At that point a hypothesis is becoming improbable. Anything 5% or lower I would say is irrational.
But I think we could refine and clarify this definition even more. The definition in the previous paragraph, while true, I think misses out by not including the idea of evidence. So without further ado, my new revamped definition of faith in terms of Bayesian reasoning:
faith – confident belief that a hypothesis is true when a competing hypothesis explains the evidence significantly better and is therefore more probable.
This isn’t too far off from Peter Boghossian’s definition of “pretending to know things you don’t know”. In fact it matches pretty well with it. When you are confident in a belief, it’s likely that you’ll use the word “know”, as many believers “know” that Jesus died for their sins or that Muhammed is the one true prophet of God. Yet the probability of these being true based on our evidence and background knowledge is very low. Yet they “know” it. Boghossian would say they are merely pretending to know it. I would say they are claiming to “know” something even though there is a low probability that they are correct.
How can we use probabilities to benefit ourselves in life?
There’s a terrific upside to all of this. Life is full of uncertainty. What starts out as a beautiful day can be suddenly turned into a disaster. This can happen for many reasons too, from car accidents, a spouse leaving, being diagnosed with a difficult illness, natural disasters, a house fire or a number of other possible life altering events.
It seems as though we have no control of our lives when these sorts of catastrophes can befall us. In fact, it is because of this uncertainty or even in the midst and emotions of these disasters that we may look to the Divine for answers. After all, the religions of the world offer certainties. They offer answered prayers. They offer protection from God. Best of all they offer a life of happiness. Sounds like a pretty good deal to me, right? I know that I spent many years feeling safe and secure under the security blanket of my religion.
But can thinking in probabilities help us? It sure can and probably in ways you never thought of.
Let’s look at the example of physical health. A person can exercise regularly, eat a good diet, avoid bad habits and still die in their 40’s. It’s scary. Then there’s that 95 year-old guy who eats bacon and biscuits every day and drinks bourbon every night. So what’s the point in all this good health when you can still keel over before you’re 50?
Well, we have an unfortunate habit of remembering these exceptional situations. But that’s precisely why we remember them. They’re exceptional. We don’t remember the thousands more whose healthy lifestyles benefit them in so many ways and into a very ripe old age. We also don’t remember the smokers with lung cancer and the numerous who suffer from heart problems, diabetes and obesity from eating a poor diet.
When we live a healthy lifestyle we are tilting the probabilities in our favor. We are increasing the probability that we will live longer and better.
When we drive less, drive defensively and maintain safe and assured clear distance from other drivers we decrease the probability of being in a car accident.
When we deliberately choose to live away from Tornado Alley, the East Coast and flood plains we decrease the probability of being involved in natural disasters. When we choose to live in areas with lower crime rates we decrease the probability of breaking and entering and robbery. In fact, if we live a more modest lifestyle without a lot of visible luxury items, we decrease the probability that a thief is even interested in our property.
When we treat others well we increase the probability that we will be treated well in return. When we take time for others we increase the probability that they will take time for us. In fact, this can have a feedback loop effect.
When we get an education we increase the probability of living comfortably and better understanding the world. When we learn psychology we increase the probability that we will understand people better and that we will better understand ourselves and avoid the cognitive errors and biases we are all prone to. When we learn many skills we decrease the probability that we will ever be without some source of income.
When we spend less money on things we don’t need, we increase the probability of financial independence and decrease the probability of being stressed about money. When we drive less we spend less. When we ride our bicycle instead of drive we increase our probability of being healthy and decrease our spending, at the same time decreasing the chances of being in a car accident. When our home isn’t filled with and surrounded by luxury items attractive to the criminal, we decrease our chances of being robbed while at the same time increasing our savings. And if we’ve been nice to others and developed a strong community of friends and family, we increase our probability that during times of difficulty there will be others there to help and support us.
See how these probabilities relate to each other and begin to stack? This is remarkably powerful. Now, could the rug still be pulled out from under us? Sure. Disasters can still happen. It’s an unfortunate part of reality, which is why we should make the best of every day we have and do our best to make sure we get as many days in our lives.
When we are truly honest with ourselves or when our beliefs and worldviews are challenged by someone else we must acknowledge that there are other competing worldviews and hypotheses. We humans are great at convincing ourselves of how correct we are in our views. We have a habit of being entirely too confident when we have little reason to be, or even becoming confident of ideas that are probably not true at all.
In our lives we rarely deal with certainties. Most of the time it’s probabilities but this doesn’t mean that we also constantly rely on faith. When we are attempting to ascertain truth probabilistic reasoning based on background knowledge and evidence will get us closer to truth. We also don’t have to give equal credit to a hypothesis just because it is plausible.
We can also make efforts at tweaking the probabilities in our own lives. They aren’t guarantees, of course, but it is a powerful way of leading a happy, successful and fulfilling life.
This article was mostly a highly condensed version of Dr. Richard Carrier’s “Proving History” where he reviews historical methods and reduces them all to Bayes Theorem. He then shows how it may be applied to history, in particular attempting to find the historical Jesus. This book is a prelude to Carrier’s magnum opus, “On the Historicity of Jesus”, a book where he very thoroughly demonstrates using Bayes Theorem and probabilistic reasoning that Jesus likely never existed. It’s one of the most fascinating books I’ve ever read.