Rationality, Expected Value, and Pascal's Wager




There's been a lot of talk about probability in the context of Christian belief these days. From academic textbooks and journal articles, to message boards and blogs (including this one), material abounds on the probability of God's existence, the probability of miracles generally, and the probability of the resurrection of Jesus given certain facts of history in the light of Bayes' theorem. The guiding assumption behind all this, though not always explicitly stated, is that accepting the more probable hypothesis results in our holding the most rational belief. And most of us want to be rational.
 
In a recent post here it was argued that the probability of the resurrection of Jesus, given the specific evidence and general background knowledge bearing on the case, is high relative to competing hypotheses. At the least, I suggested, the resurrection should be considered not improbable (hence not irrational to believe). But some people objected that without specific calculations that suggestion is too vague. So now we will "plug in" some numbers. Recall again that per Bayes' Theorem, the probability of a hypothesis H, given evidence E and background knowledge K, equals the conjunction of its explanatory power and prior probability:
 

             P(E│H & K) x P(H│K)
P(H│E & K) = -----------------------------
                                   P(E│K)

If we estimate .25 to indicate a relatively low prior probability (low for me, high for most skeptics) of the truth of the resurrection as a hypothesis, as well as .25 for the probability of our having the particular sort of evidence for it that we do have (high for me, low for most skeptics), and .4 to mean somewhat modest "predictive power" of the hypothesis, we have: 
 

             .4 x .25
P(H│E & K) = ----------- = .4
                           .25 

So the resurrection in this scenario would be slightly improbable. Does this mean that we should be committed to believe with precisely 40% confidence that the resurrection actually occurred, or to withhold belief until probability exceeds .5? Not necessarily. 
 
A rational approach would surely consider not only the probability of hypotheses being true or false, but the consequences that would follow from those hypotheses being true or false. There would seem to be nothing especially rational about my taking a shortcut to work by crossing a bridge high over an icy lake where the probability of its collapsing sometime within the next year is "only" .3, for example; or moving into a neighborhood where a full 60 percent of the residents have never been physically assaulted because it's closer to my favorite shopping center. A good and prudent soldier does not lay down his arms whenever the probability of his survival dips below .5 in the heat of battle (indeed, he is far too busy to bother with such calculations, knowing that he maximizes his probability of survival by continuing to fight). A truly rational outlook, then, knows the difference not merely between true and false, or between probable and improbable, but between wisdom and folly.  
 
What this means it that a rational approach should take into account the expected value of a given decision, in addition to the probability of its success. Whereas expected value is a sophisticated statistical concept, it can be defined informally for non-mathematical sorts like me as "the weighted average of the values that X can take on, where each possible value is weighted by its respective probability"[1] – and where X is a random variable, meaning a variable with different probabilities corresponding to different possible outcomes. For example, a business owner might use an expected value approach to determine the more promising of two locations for a new plant, the one with the highest expected payoff given its probability of success and its profitability if successful. 
 
Now let us imagine an admittedly arbitrary but finite "payoff" scale of utils[2] to represent the level of happiness or satisfaction that believers and skeptics should expect to receive given the truth value of the resurrection – and given the traditional theological position that the resurrection of Jesus ensures eternal reward for believers, eternal judgment for unbelievers. Let's suppose that 1,000,000,000 here represents the ultimate prize of enjoying eternal life in the kingdom of heaven, in fellowship with Jesus Christ, along with a host of angels and redeemed believers. This signifies the reward of believers given that the resurrection hypothesis is true. 1, on the other hand, is the “loneliest number,” here meaning the despair of eternal judgment upon sin in the kingdom of darkness, in subjection to Satan, along with a host of demons and desolate unbelievers. In other words this is the complete, or virtually complete, absence of hope, joy or satisfaction – the reward of nonbelievers given that the resurrection hypothesis is true. In between are values representing various less extreme levels of expected satisfaction, depending on possible outcomes and their probabilities. In this way each decision outcome can be assigned an expected level of reward on the scale.
 
For the somewhat conservative (at least for a believer like me) probability estimate for the resurrection mentioned above, this leaves the following:
 
Posterior Probabilities:
 
P(R), probability that the resurrection hypothesis is true = .4
P(~R), probability that the resurrection hypothesis is false = 1 - .4 = .6 
 
Expected Value of Outcomes:
 
For believers, expected value of R = 1,000,000,000 x .4 = 400,000,000.  Expected value of ~R for believers = +/- 500,000 x .6 = 300,000. As mentioned above, the 1,000,000,000 represents the maximum level of satisfaction a soul can enjoy in principle. The admittedly arbitrary number of 500,000 for the believer means here something like: "life on earth is not too terrible, and still has its rewards – but it's not anything like what eternal life will be." (The important thing to keep in mind here is that this latter number is roughly equal for both believers and unbelievers.) So on our subjective-arbitrary scale the total expected value for believers = 400,000,000 + 300,000 = 400,300,000.
 
For nonbelievers, expected value for R = 1 x .6 = .6. (This is a level of satisfaction slightly lower than the lowest on our scale, only because the lowest whole number on the scale is multiplied by a probability of less than one, so we can round this up to one). Expected value of ~R for unbelievers = +/- 500,000 x .6 = 300,000 (the same as believers). Again, 1 represents the lowest possible level of happiness a soul can experience in principle. And for nonbelievers, the 500,000 means something like: "life on earth is not too terrible, and still has its rewards – it may not be paradise, but it could always be worse." So for the same arbitrary scale the total expected value for nonbelievers = 1 + 300,000 = 300,001.
 
The basic idea here is that in terms of the extreme consequences at stake, it would be more rational to accept even a slightly improbable position of faith in Christ because of the substantially higher potential reward. It's true that faith in Christ is costly, in that Jesus calls us to take up a cross, deny self and follow him (Luke 9:23). We are to invest our lives in heaven, not on earth (Matt. 6:19-21). But faith also pays compensating dividends in the way of fruit of the Spirit and a deep sense of purpose and calling. Circumstantially, there is no appreciable difference between the life experiences of believers and nonbelievers. Rain falls on the just (or justified) and the unjust (Matt. 5:45). Since life on earth has its ups and downs, pains and pleasures, joys and heartaches, etc., for everyone, the venture of faith, while costly, is relatively low-risk. 
 
Of course all this is essentially a restatement of Pascal’s Wager. Pascal developed his “wager” as something akin to the risk-reward principle that operates in all of life, from business and finance to romance. Given the relative scarcity of happiness on earth, the abundance of joy in the eternal kingdom of heaven, and the considerable prospect that the Christian gospel is true, it seems to me, as it did to Pascal, that to repent and believe in Christ would be the wisest investment one could ever make. I would say more about Pascal and his development of the wager, but unfortunately I am, like the rest of us, running out of time. 


[2]In microeconomics, happiness is measured by a concept called utility. The standard unit of measurement that microeconomics uses to measure utility is called the util…. The util has no concrete numerical value like an inch or a centimeter. It is merely an arbitrary, subjective and convenient way to assign value to consumer choices and to measure the consumer utility or utils of one choice against another choice.” – Marc Davis, “Microeconomics: Assumptions and Utility,” Investopedia, www.investopedia.com/university/microeconomics/microeconomics2.asp.

Comments

Don McIntosh said…
I'm not exactly a mathematician or statistician (far from it), so if anyone notices errors please let me know and I'll edit the post. Thanks.
Anonymous said…
DM: Does this mean that we should be committed to believe with precisely 40% confidence that the resurrection actually occurred, or to withhold belief until probability exceeds .5? Not necessarily.

If the probability is 0.4, then we can conclude the probability is 0.4. Sure, once it is higher than 0.5 we can label it as "probable", but even then if the probability is 0.6 then that meansd the probability is 0.6. It is still not rational to have a conviction that it actually happened.

Your bigger problem is that all your figures are essentially made up. In my view, this sort of approach is meaningless when it comes to miracles or any sort of one off event, as we have no way of estimating probabilities.

Pix
Joe Hinman said…
the problem of numbers being made up is true of any use of Bayes. I am not commenced Bayes can be used for things like the resurrection or for any historical events. I am totally against applying it to God, it violates the principle of mystical theology.

it works well for things like submarine detection and finding missing children. It doesn't work well for things that are beyond our understanding.
Anonymous said…
JH: the problem of numbers being made up is true of any use of Bayes.

If you mean any use in this context (and from the rest of your comment it sounds like you do) then I agree.

Pix
Joe Hinman said…
any use of Bayes requires a prior distributions which is essentially a guess.
im-skeptical said…
There is no guessing about the prior probability of resurrections. It is 0. Even the so-called "witnesses" in the NT didn't see that happen.
Don McIntosh said…
Pix, I only argue about probability because so many atheists assert that the probability of theism or miracles is "vanishingly low," and they use devices like Bayes' theorem in an attempt to prove the point. I don't think probability calculus is the best way to answer metaphysical and historical questions. I'm only trying to speak the language certain atheist friends seem to prefer.

In other words your point about assigning specific probabilities to singular events like miracles is well taken.

You seem to have missed my own point, however. Despite the difficulty of precisely calculating "the" probability of the resurrection, whatever it may be, if there is considerable evidence to support the resurrection (and I believe there is), the final probability of the resurrection is above zero in some proportion. In that case anyone examining the issue is faced with the practical questions of risk I was addressing. (Of course, you can always take the "Skeptical" route and simply assert that the *prior* probability is zero, in which case the posterior probability must be zero regardless of the evidence.)

im-skeptical said…
Or you could take the un-skeptical route and simply believe.
Joe Hinman said…
There is no guessing about the prior probability of resurrections. It is 0. Even the so-called "witnesses" in the NT didn't see that happen.

tht shows how little you know about Bayes, because that's exactly what it is. I;'ts like a gunner finding his range, you first shoot over then under then split the difference and keep narrowing,
Joe Hinman said…
skep I challenge you to a formal debate on the existence of God,
Joe Hinman said…
Skep that does not not mean i dislike you.I am glad you come here and I appreciate your comments.
Anonymous said…
DM: Pix, I only argue about probability because so many atheists assert that the probability of theism or miracles is "vanishingly low," and they use devices like Bayes' theorem in an attempt to prove the point. I don't think probability calculus is the best way to answer metaphysical and historical questions. I'm only trying to speak the language certain atheist friends seem to prefer.

I am with you and Joe here; I think those arguments are nonsense.

DM: You seem to have missed my own point, however. Despite the difficulty of precisely calculating "the" probability of the resurrection, whatever it may be, if there is considerable evidence to support the resurrection (and I believe there is), the final probability of the resurrection is above zero in some proportion. In that case anyone examining the issue is faced with the practical questions of risk I was addressing. (Of course, you can always take the "Skeptical" route and simply assert that the *prior* probability is zero, in which case the posterior probability must be zero regardless of the evidence.)

That is basically Pascal's wager.

What are the chances that I will be in a fatal car crash if I drive today? Very low. However, if we factor in the risk, suddenly the risk of going for a drive far out weighs the risk of staying at home. Conclusion? Clearly I will have a fatal car crash if I drive today, right?

Pix
im-skeptical said…
Pascal's wager really has nothing to do with the resurrection. It is a question of risk vs. reward. From Wikipedia:

Pascal argues that a rational person should live as though God exists and seek to believe in God. If God does actually exist, such a person will have only a finite loss (some pleasures, luxury, etc.), whereas they stand to receive infinite gains (as represented by eternity in Heaven) and avoid infinite losses (eternity in Hell)

OK. Let's think about this. So if you want to include the resurrection in your calculation, it's not really relevant. If there is any finite probability of your belief being true (no matter how small), the risk vs, reward equation works out in favor of belief. Probability says that the poker player should take this bet.

But there's another way of approaching this calculus. Assuming that the probability of the belief being true is low, it is still true that by making this bet, you are likely to lose. Does it make sense to bet everything you have for the very slim possibility that you will get that infinite payoff?

Let's compare this to a super-lottery, where only one person on the planet will win. The payoff is having everything you ever could want. The cost of the ticket is throwing away everything you already have in your one and only life. (And I state it this way because the pursuit of a false goal would be tantamount to wasting the opportunity you have.) The poker player's calculus says go ahead and make the bet, but that is only rational in a context where the you bet doesn't cost you everything. In other words, it makes sense when you can afford to lose, but not when you can't afford to lose. If this is the one and only life you have, you really can't afford to waste it.
im-skeptical said…
Joe,

I would like to debate, but I have to take it slow, because I've been rather busy.
Joe Hinman said…
I understand.I have my resistance work in addition to apologetic,
Joe Hinman said…
My favorite professor,m they guy who got me to be a historian was a math major at Brown when he got saved and went into church history. He thinks Bayes is stupid and he doesn't like because in his day(he retired years ago) mathematicians didn't like it. He always told me historical probability is never quantified.
Anonymous said…
IMS: Pascal's wager really has nothing to do with the resurrection. It is a question of risk vs. reward.

The second half of the post is talking about risk; that the risk from not believing the resurrection is sufficiently high that you should believe it, even if the probability is small.

Presumably the author also thinks we should believe Islam is also true because of the risk of not doing so.

By the way, if you do debate Joe, get a good definition of "God" on the table before you start. Arguing there is no "ground of being" will be very different to arguing there is no Christian God.

Pix
im-skeptical said…
He thinks Bayes is stupid and he doesn't like because in his day(he retired years ago) mathematicians didn't like it.

There's nothing stupid or mathematically incorrect about Bayes' Theorem. What's stupid is the way people apply it. For example, you can't state the prior probability of a specific (or one-of-a-kind) event. Statistical probability applies to a genre - something for which there is statistical data from which a probability for events of that kind can be calculated. People who try to use Bayes on things like the creation of the universe, or the existence of God are usually talking out of their ass.

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