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Meditation 1078
Belief Theory

by: Bill Black

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I am an actuary. What is an actuary? Actuaries create models to analyze just about anything using mathematics, logic, and the scientific process. Many professions with this type of background apply these tools to model natural phenomena, such as engineering, physics, chemistry, etc… Actuaries apply these tools to social phenomena, such as economics, markets, business risks, etc…

As you can imagine, the resulting models are often incomplete in a definitive sense because much of the outcome is influenced by factors that are not so rigidly defined. For those factors, we make assumptions or model alternatives resulting in a distribution of possible outcomes.

As a result, everything I think about becomes subject to mathematical analysis, including my beliefs. So I wanted to share this analysis with others to see if it provides some insight into our tendency or reluctance to believe in God.

I want the model to be unbiased, such that it only provides a framework for the analysis. Therefore, some quantities or assumptions must be provided by the user. In some cases, the model simply becomes  Pascal’s Wager, but that seems to be appropriate, since this was the conclusion that Pascal reached (i.e., the model works for Pascal). But, unlike Pascal, I leave the subjective inputs as variables for the user to decide what quantities make sense to them.

The mathematics are relatively simple and mostly conceptual. So understanding the model does not require advanced mathematical knowledge. The model uses some concepts from calculus, statistics, and utility theory, but only at a conceptual level. So be brave, you may actually find it entertaining that people have devised mathematics for such things.

On the other hand, if you are one of those unique individuals that actually enjoys math, you can take the concepts further. If you’re in this group, please bear with my brief introductions. Face it; most people are bored by the things that thrill you.

For our belief in God, we will use probability theory. Probability is the likelihood of a possible result. I flip a coin. It lands either heads or tails (two possible results). If the coin is fair, there is a 50/50 chance of either result. So, what is the probability that God exists? Let’s call it Pr(God).

As with most probability problems, first we need to define the possible outcomes. The simplest division of the universe is just like flipping the coin: God either exists (Pr(God)=1… i.e., 100% certain) or God does not exist (Pr(God)=0). Well, we already knew that, so to further the analysis let’s consider these two states in a  Bayesian framework.

In a Bayesian framework, we take each possibility as a given fact and then analyze that case. For example, if we throw a six sided die, we know there are six possible outcomes (1, 2, 3, 4, 5, 6). Assuming the die is fair, the probability of rolling a 1 is 1/6, but only if we are given no additional information about the result.

Now consider splitting the possible outcomes into two exclusive states: “less than 4” and “greater than 3”. If we are given additional information, such as the state of the outcome is “less than 4”, now the probability of having rolled a 1 is 1/3. In other words, this additional information reduced the number of possible outcomes. In English, we say “the probability of rolling a 1 given that the number rolled is less than 4 is 1 out of 3 chances.” In mathematical symbols we write Pr(X=1|X<4) = 1/3, where X represents the value rolled on the die.

OK, so here’s our framework so far. We have these two exclusive universes. In one universe, God exists. In the other universe God does not exist. We take this as known information. Up until this point the model is quite objective, but now we need something to analyze in each universe. Let’s call this quantity the value of existing right now. Obviously this is quite subjective and we turn to utility theory for this analysis.

Utility essentially means the value of something, but the utility I derive may differ from the utility you derive. For example, let’s say you have a favorite t-shirt. The value of your t-shirt in the used clothes market is around 25 cents. So this is the value that essentially everyone assigns to your t-shirt. But, maybe this is the t-shirt you were wearing the day you proposed to your wife, or the day your first child was born, or the day you learned of the death of a loved one. To you, the t-shirt is priceless.

Utility let’s us consider the value of something without necessarily assigning a price to it. In other words, we do not need to worry about the units (Dollars, Euros, Yen, etc…). We simply recognize that there is some relative value… larger or smaller. For example, your t-shirt is priceless to you, but it is worth very little to  me. So the utility you derive is greater than the utility I derive from the t-shirt.

We are interested in the utility that one derives from their own present existence. This is completely subjective and must be determined by the user (i.e. you or me or the “one”). So this needs to be a variable. Let’s label this variable U.

What is the expected value of U? In statistics, the expected value of a random variable is the average or mean of the variable. This is calculated as the weighted average, where the weightings are equal to the probability of each outcome.

For example, say I take 5 math tests and my scores are 80 on two tests, 90 on two tests, and 100 on one test. If I randomly pick one test from the group, the probability of selecting the test with a score of 100 is 1/5. Likewise, the probability of selecting one of the 90s is 2/5 and the probability of selecting one of the 80s is 2/5. So the expected test value = 100 × 1/5 + 90 × 2/5 + 80 × 2/5 = 88 (not too bad).

Often we just think about this as the average, where we sum up all of the values and then divide by the count. This is the same concept, but calculated more generally for any randomly distributed variable. What this “average” tells us is that we could replace all of the different values with this one amount and we still get the same total. From the example, my total test score is 100 + 2 × 90 + 2 × 80 = 440 = 5 × 88.

Now let’s put all of these concepts together to write a formula for the expected value of U. From our Bayesian framework of the universe, we have two given states (God or NoGod). Now consider the utility derived from existing in a given state at the present moment:

U|God = the utility derived from being here right now given that God exists.
U|NoGod = the utility derived from being here right now given that God does not exist.

And the expected value of U = U|God × Pr(God) + U|NoGod × Pr(NoGod).

That’s a really simple formula, but sort of a mouthful to read. So let’s assign some symbols to the various concepts. We will want to use some Greek symbols to make it look cool, so let:

μ = the expected value of U (read “mu”, the twelfth letter of the Greek Alphabet)
ω= U|God (read “omega”, the last letter of the Greek Alphabet)
α = U|NoGod (read “alpha” the first letter of the Greek Alphabet)

Now Pr(God) and Pr(NoGod) are associated with binary states. In other words, it is one or the other, but until the reality is known they are binomial probabilities. Actuaries like to use the symbols p and q (English Alphabet) for these, so let:

p = Pr(God)
q = Pr(NoGod)

Substituting these symbols into our formula (and dropping the “×” for implied multiplication of variables), we have:

μ = ωp + αq

Yea! Our formula is tiny, easy to read, and it seems intuitive. One’s expected value of being (μ) can be thought of as a weighted average of two things. Thing1 (ω) is the value of being, if God exists and Thing2 (α) is the value of being, if God does not exist. The weighting of the two is based on our tendency to believe in God (p) and our tendency to believe there is no God (q).

Now I’m going to make a general assertion about μ:
A sane person will form beliefs in such a way that maximizes μ.

In other words, our beliefs enhance our self-image. Whatever that self-image is, it is our value of being.

For now, let’s also assume that both α and ω are positive. In other words, we are dealing with individuals who generally have a positive outlook on life. These individuals find value in their existence whether or not God exists.

The Positive Spiritualist (ω >> α)

For the spiritual, we know that they tend to believe in God. In other words, p is greater than q (we write this p > q). Therefore, in order for µ to be maximized, it follows that ω > α. In other words, they derive more purpose from being in a universe where God exists. OK, so we probably could have said that before we started this whole thing, but let’s think about that mind-set more deeply. Why would that be the case? What is the motivation to see it this way?

My positive interpretation is that there is more to our being than meets the eye. Because God exists, there is another dimension to our existence that lies beyond our day-to-day understanding of the physical world. Another way to say this is that there is our physical existence plus some additional value derived from a spiritual existence. So let’s manipulate our equation a little to reflect that idea.

We know that p and q are probabilities of two possible outcomes. In other words p + q = 1. It is 100% certain that either p or q will materialize as the eventual reality. So we can write our equation:

μ = ωp + αq = ωp + α(1 – p) = ωp + α – αp = α + (ω – α)p

We can interpret the coefficients, as follows:

α = our baseline, the value of existing in the physical realm of our day-to-day lives.
(ω – α) = the differential additional value derived from our spiritual existence.

Now imagine an extreme case where ω is much, much greater than α (written, ω >> α). Conceptually we can think of ω becoming infinitely valuable (written ω → +∞), while our base line α is held constant.

Calculus provides a means for thinking about extreme cases via the limit, we write this concept:

Sorry - if you want to see this formula, you are going to need to turn on graphics.

At the same time, let p → 0. Now p is a probability and must be between 1 and 0, so p approaches zero necessarily from the right (i.e., from values greater than 0, we write p→0+). We have:

Sorry - if you want to see this formula, you are going to need to turn on graphics.

Now, the quantity ωp at this extreme is quite interesting, since it is indeterminate (essentially 0/0). All we can say is that it is positive (or at worst zero), no matter how small p gets. In other words, if ω is so immense that it becomes infinite, then it does not matter how unlikely it is that God exits. The most remote probability continues to add immense value to one’s existence and, as a result, one must believe in God.

This is the same conclusion that Pascal reaches in his wager analysis. In summary, the potential gain is so great, that you should take the bet (i.e., believe in God) regardless of how small the possibility is of winning. Accordingly, he concludes that everyone should believe in God, but that is clearly not the case… There must be other interpretations.

The Positive Atheist (α >> ω)

For the atheist, it is clear that α must be greater than ω, which will lead to exactly the opposite conclusion. Making similar manipulations as above, we arrive at the equation:

μ = ωp + αq = ω(1-q) + αq = ω - ωq + αq = ω + (α – ω)q

And we can interpret the coefficients, as follows:

ω = our baseline, the value of existing in a universe created by God.
(α – ω) = the additional value derived from existing in a universe without God.

So how does this additional utility arise? Why would that be the case? What is the motivation to see it this way?

My positive interpretation is that there is nothing beyond human understanding. The human mind has unlimited capability to understand the universe and can solve any problem. Essentially, Mankind is God, the Supreme Being of the universe.

Once again, when this concept has extreme value to the individual, such that α is essentially infinite, then we end up with the extreme case analysis:

Sorry - if you want to see this formula, you are going to need to turn on graphics.

Inferring that one should never believe in God when α has unlimited value.

The Positive Agnostic (α ≈ ω)

The agnostic point of view arises naturally when α and ω are approximately equal in value (α ≈ ω). Let’s say that both are approximately equal to the value k. Then manipulating the original equation, we have:

μ = ωp + αq = kp + kq = k(p+q) = k

The probability that God exists is completely removed from the equation. It becomes irrelevant.

So how does this happen? Why would α = ω?

I think both of my positive interpretations for large α or large ω still apply. However, in this case one is not significantly larger than the other. They may both be very large or they may both be very small. Let’s continue to work from the positive point of view and assume that they are both very large such that this individual is just as positive as our spiritualist and atheist individuals.

For large α, my interpretation is that the individual finds great value in the human mind’s capability to understand everything. For large ω, my interpretation is that the individual finds great value in there being more to existence than the human mind experiences and understands. These seem to be opposing points of view, but given that both states are mutually exclusive, one could place great value on both states, leading to positive agnostic beliefs.

However, this is a precarious position of equilibrium, especially if both α and ω are large. If one state becomes measurably more desirable than the other, the equilibrium is lost and the individual would quickly gravitate to spiritualist or atheist beliefs. So I would expect that highly positive agnostics rarely find they are completely agnostic, but probably bounce back and forth between spiritual and atheist beliefs.

Likewise, positive individuals who tend to be spiritual are probably not always spiritual, but sometimes move toward being agnostic. When the equilibrium fails they quickly move back toward spiritual. On the other hand, the positive atheist probably moves toward agnostic at times, but snaps back to atheist when the equilibrium is lost.

Negative Cases

The approach so far has been from the positive perspective, but there are negative interpretations for both states as well. Under these interpretations, one may be strongly repelled from a state as opposed to being strongly drawn toward a state. These scenarios are just the flipside of the positive scenarios. So α < 0 should correspond to ω > 0, but for a negative reason. Accordingly, ω < 0 would be the negative equivalent of α > 0.

So my negative interpretation of the universe without God (α < 0) is “what you see is what you get” (WYSIWYG, say wiz-e-wig). This is the “death and taxes” point of view. One may say “Since there is no God, there is nothing beyond my mundane, day-to-day life and one day it will be over.”

And my negative interpretation of the universe with God (ω < 0) is then that there is limited human capacity to improve things. This is the “God hates us” point of view. One may say “Since there is a God and bad things still happen to me, God must want me to suffer and there is nothing I can do about it.”

Under these interpretations, negative value is associated with one universe or the other. We can analyze extremely negative cases in a similar manner by isolating a constant baseline, but this time the baselines are reversed.

For the spiritualist, we consider: μ = ω + (α – ω)q
For the atheist, we consider: μ = α + (ω – α)p

Again, holding the baselines constant and letting the free parameter decrease without bound, we have the limit scenarios:

For the negative spiritualist:

Sorry - if you want to see this formula, you are going to need to turn on graphics.

For the negative atheist:

Sorry - if you want to see this formula, you are going to need to turn on graphics.

For extremely positive individuals, we concluded that one adds value to their existence by maintaining their beliefs regardless of the chances of being right. Now, for extremely negative individuals, one loses all value unless they are right. In other words, the negative spiritualist must stay convinced that God exists (q=0) otherwise, all is lost. Likewise, the negative atheist must stay convinced that God does not exist (p=0) or all is lost. The limits on the right must stay indeterminate for the infinitely negative cases.

As a result, I would expect that someone with an extremely negative point of view is less willing to have an open discussion about their beliefs relative to someone with an extremely positive point of view, regardless of the actual beliefs involved. An extremely negative person must stay convinced of their beliefs, since no level of doubt is affordable. Therefore, they are not likely to recognize their beliefs as beliefs and probably present them as facts.

On the other hand, an extremely positive person will continue to maintain their beliefs, regardless of the level of doubt that arises. As a result they are likely to recognize that they have beliefs, something accepted as truth without absolute proof (or, equivalently, something accepted on faith).

Again, we end up with the same relationships and groupings:

ω >> α => Spiritual

α >> ω => Atheist

α ≈ ω => Agnostic

Since α ≈ ω is still required for the agnostic result, it seems unlikely that you will find an extremely negative agnostic. This would imply that both α and ω are extremely negative values, really a hopeless situation to be avoided.


Analyzing the extreme cases under the model may offer some insight into the drive for one to believe as they do. Given the topic, there probably are actual cases at the extremes, but it also seems likely that there would be a range of values for both α and ω. You could also imagine that these parameters may correlate with other individual characteristics.

For example, if you sorted a group of individuals along a spectrum of very scientific at one end to very artistic at the other end, then I expect that scientist would generally have high α values, since they are trained to rely on human reasoning. On the other hand, artists attempt to reach something that goes beyond human understanding, which implies high ω values. As a result, I would expect to find more atheists in the scientific community and more spiritualists in the artistic community. Then you encounter the artistic  scientist (or scientific artist) and you need to find another dimension to consider, rich to poor, young to old, etc…

I’m guessing you would be hard pressed to find a young, rich scientist with spiritual beliefs (not that you could not, it just seems less likely). Conversely, there are probably very few atheists in the old, poor, artist category. Not to try to force everyone into one stereotype or the other, but such correlation lends support to the model.

It also seems likely that someone may experience a range of values for the parameters during their lifetime. A period of high individual achievement may result α greater than ω, while a period of unexpected fortune may have the opposite effect. Similarly, the loss of a loved one may result in negative α values, while an unexpected individual failure could cause negative ω.

So I’m curious how others interpret the model. I presented possible positive and negative interpretations of two universes, but I’m sure there are many other ways to attach positive or negative values to each.

As you consider the questions below, remember the universe you are analyzing is given as a prior condition (you cannot argue against it). So, while discussing the universe without God, you cannot make a reference to God (it makes no sense in that universe). Accordingly, in the universe with God, supporting reasons for a preference or dislike should reference God, as it is given prior information that God is there and this is the quality of that universe we are interested in discussing.

With that in mind:


To me, the model seems to say that one’s natural preferences incline them to believe one state or the other. Whether this is by choice or by predisposition is not addressed, but the analysis may say something about one’s nature either way. Since present circumstances tell us that either state is a possibility (i.e., the coin is not yet flipped), we are naturally drawn to believe the state we find most appealing. This concept is beautifully illustrated in the movie “Life of Pi” (yea! More math!). If you have a chance to watch that movie, I recommend it. It’s about much more than a guy in a boat with a tiger.


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