I used to think that the fine-tuning argument was the strongest argument out there for the existence of a creator deity. I was especially impressed by the apparent magnitude of the fine-tuning – Steven Weinberg has stated that the value of the cosmological constant was fine-tuned to one part in 10¹²⁰.

If one takes a naïve (and as we’ll see, incorrect) Bayesian approach to assessing this as evidence, then it looks like this should serve as an incredible amount of evidence for the existence of a God, enough to totally overwhelm all other possible considerations. Why? Because if there is a God, then we expect fine-tuning, while if not, then the fine-tuning looks incredibly unlikely. Given this, the God explanation should receive a credence bump proportional to 10¹²⁰ upon updating on the observation of fine-tuning.

As a quick aside before diving into the numbers, there is a lot of dispute about whether or not there even is fine-tuning in our universe. For the purposes of this post, I’m going to ignore all of these disputes, and pretend that there is a strong consensus on this matter. I’ll use Weinberg’s estimate of 10^-120 for the fine-tuning of the cosmological constant. I know that this is controversial, but the point I’m making will stand for even this insanely tiny value.

Okay, so let’s first present a formal version of the fine-tuning argument for God.

F = “The universe is fine-tuned for life.”
G = “A creator deity fine-tuned the universe for life.”

O(G | F) = L(F | G) · O(G)
L(F | G) = P(F | G) / P(F | ~G) ≈ 1 / 10^-120 = 1200 dB

So O(G | F) = 10¹²⁰ · O(G)

This uses the odds formulation of Bayes’ rule – look it up if you’re unfamiliar.

This argument says that your credences should be adjusted by a factor of 10¹²⁰ upon observing the fine-tuning of the universe. In other words, to not be virtually certain that there exists a creator deity that rules the universe after updating on fine-tuning, you’d have to have initially had a credence on the order of 10^-120.

Let me point out that 10^-120 is a really really small number. It’s virtually impossible to imagine any good reason why you would be justified in having a prior credence on this order of magnitude. Nobody should be that sure about anything. Evidence of a strength of 1200 dB is analogous in strength to a noise that is a quadrillion times more intense than the threshold of human hearing.

So what’s wrong with this argument? In fact, it fails at the first step. In calculating the strength of the evidence, we only considered two possible hypotheses: either God or, if not, then random coincidence. But there are many other options that we have to factor in as well, most famously the multiverse hypothesis.

But even if there are other hypotheses out there, shouldn’t they just all share the benefit of the credence boost? The existence of another hypothesis that made the same prediction shouldn’t count as a penalty, right?

Wrong! Probabilities have to add up to 1, and you can’t have multiple mutually exclusive competing hypotheses that you have virtual certainty about. Whatever happens when you add other hypotheses must be more subtle than that. So let’s calculate using Bayes’ rule!

O(T | E) = L(E | T) · O(T)
L(E | T) = P(E | T) / P(E | ~T)

For each theory T we consider, we have to take into account all other theories in the denominator of our likelihood function L. We’ll want to keep in mind the following identity:

P(B & C) = 0
implies
P(A | B or C) = [P(A | B) P(B) + P(A | C) P(C)] / [P(B) + P(C)]

So, for instance, let’s divide up our explanations of the fine-tuning F into three disjoint categories: (1) random coincidence C, (2) a deistic God G, and (3) all other explanations that are mutually incompatible.

L(F | X) = P(F | X) / P(F | ~X)
= P(F | X) (1 – P(X)) / ∑_Y≠PP(F | Y) P(Y)

P(F | C) ≈ 10^-120
P(F | G) ≈ 1
P(F | O) ≈ 1

L(F | C) ≈ 10^-120
L(F | G) ≈ P(~G) / P(O)
L(F | O) ≈ P(~O) / P(G)

O(C | F) = 10^-120 · O(C)
O(G | F) = P(G) / P(O)
O(O | F) = P(O) / P(G)

In the end, what we find is that the “Coincidence” hypothesis has been down-voted completely out of existence, leaving only the “God” hypothesis and the “Other” hypothesis.

And importantly, our final credence in either of these hypotheses is not on the order of magnitude of 1 – 10^-120. The final balance depends entirely just on the ratio of prior credences in the two explanations.

Trial Run

Let’s look at two individuals updating on the observation of fine-tuning.

Atheist
P(G) = .01%
P(O) = 50%

Deist
P(G) = 99%
P(O) = 1%

(The exact details of these numbers aren’t that important, just that they’re somewhat qualitatively accurate.) Their final credences will be:

Atheist
P(G | F) = 0.02%
P(O | F) = 99.98%

Deist
P(G | F) = 99%
P(O | F) = 1%

And we see that nobody ends up significantly updating their religious beliefs on the evidence of fine-tuning. The deist held a worldview in which the random coincidence hypothesis was already ruled out, so the observation of fine-tuning doesn’t change anything for them. And the atheists were initially fairly agnostic about whether or not the universe was fine-tuned, but were very confident in the existence of other explanations besides God. As such, the observation of fine-tuning served as a minor increase in their belief in God (+.01%), while they become extremely confident that there must be some other explanation.

Fine-tuning would only serve as strong evidence for you if you were initially very sure that there was a God, but agnostic about if God would have designed the universe to accommodate human life, or if its design was purely random coincidence. Even in this case, the bump in credence you’d receive would be nothing like the massive update that seems apparent from a naïve (and wrong) application of Bayesian reasoning.

Scope insensitivity is a cognitive bias that involves a failure to internalize the true scale of quantities. Some of the most striking and frankly depressing examples of this phenomenon involve altruistic behavior, where people care just as much about a cause regardless of how many lives are concerned. In some cases, increasing numbers of affected people result in decreasing willingness to pay.

This issue arises when quantitative metrics don’t line up with our intuitive metrics – 10 billion doesn’t feel 1000 times larger than 10 million. A solution that might be sometimes possible is to adjust the numerical scale you are dealing with to try to get the true scale to match the intuitive scale.

This is a large part of what I think is great about the notion of evidence as noise.

Humans have scope insensitivity with respect to very large and very small probabilities. 99.99% doesn’t feel that different to us from 99.9999%. But they are extremely different. The amount of evidence required to push you from 99.99% to 99.9999% is the same as the amount of evidence that would have pushed you from 9% to 91%. There is a big difference between 99.99% and 99.9999% in terms of the state of knowledge represented.

The problem is that as the probability approaches 100%, the number looks to us like it is barely budging. This can be fixed by making our scale logarithmic. We do this by first converting our probabilities to odds ratios (so 50% becomes 1:1 odds, 75% becomes 3:1 odds, etc), and then taking a logarithm. This is exactly analogous to the decibel scale for noise, so this is called the decibel (dB) scale for evidence.

Probability of A = P(A)
Odds of A = P(A) / P(~A)
Decibel strength of A = 10 · log₁₀(P(A) / P(~A))

Very strong evidence is very noisy, and weak evidence is silent, barely affecting our beliefs. This is also nice because Bayes’ rule becomes additive:

Posterior Odds Ratio = Likelihood Ratio · Prior Odds Ratio
O(T | E) = L(E | T) · O(T)
becomes…
O_dB(T | E) = L_dB(E | T) + O_dB(T)

If your evidence E is equally likely whether or not the theory T is true, then L(E | T) = 1 and L_dB(E | T) = 0. Thus you add 0, and end up with the same odds as you started with.

Theories that are very high or very low in credence are very noisy, while those that are around 50% are silent.

Now what’s the difference between 99.99% and 99.9999%?

99.99% = 9999:1 = 40 dB
99.9999% = 999999:1 = 60 dB

A 20 dB difference in strength of belief is a lot easier to wrap your head around than a 0.0099% difference!

In addition, equally strong evidence always looks equally strong when expressed in dB, while it can look increasingly weak when expressed in probabilities.

For example, imagine that somebody comes up to you and claims to be able to read your mind. To test them, you decide to ask her to tell you what number between 0 and 10 is in your head right now. If she gets this right, then this counts as 10 decibels of evidence for her psychic abilities.

L(correct | psychic) = P(correct | psychic) ÷ P(correct | not psychic)
≈ 100% / 10% = 10

10 log₁₀(10) = 10 dB

So if your previous belief in her psychic abilities was at -50 decibels (100,000:1 odds against), then it should now be at -40 decibels (10,000:1 odds against).

The same calculation would tell you that another successful test would nudge you another +10 dB, from -40 to -30. Extrapolation seems to indicate that you should be pretty much agnostic as to whether or not she is psychic after three more such successful tests, and strong believers after only eight total tests.

Initial strength of belief = -50 dB
First test gives evidence of +10 dB
New strength of belief = -40 dB
Four more tests give total evidence of +40 dB
New strength of belief = 0 dB
Three more tests give total evidence of +30 dB
Final strength of belief = +30 dB (99.9%)

This example actually gets things wrong in a very important way. Eight tests like those that I described is probably not sufficient to establish psychic abilities. This is a little off topic, but is useful to go into as a demonstration of how naive usage of Bayes’ rule can lead you off the rails.

Where we went wrong was in the very first step, in calculating the decibel strength of the evidence.

L(correct | psychic) = P(correct | psychic) ÷ P(correct | not psychic)
≈ 100% / 10% = 10

The presumption behind this calculation is that if she were psychic, then she would almost definitely be able to get the number right (≈ 100%), but if not, then she would have a random shot (10%). But “psychic” and “random” are not the only two theories! For instance, maybe the apparent psychic has actually just figured out a masterful method for reading subtle facial movements to guess at the number being guessed, rather than actually being able to look into your mind.

The face-reading hypothesis seems unlikely, but probably less so than true mind-reading abilities. Let’s give it a decibel score of -20 (corresponding to an initial credence of about 1%). This should barely factor into our initial calculation, so let’s suppose that +10 dB is the actual strength of evidence for psychic abilities.

Now P_dB(psychic) goes from -50 dB to -40 dB, and P_dB(face-reading) goes from -20 dB to -10 dB. They have both gotten more likely, because they both successfully predicted the outcome! And now for the second test, face-reading should have a bigger effect on the calculation! I’ll skip the algebra and just present the new strengths of evidence for the second test:

L(correct | psychic) = 7 dB
L(correct | face-reading) = 10 dB

Notice that the evidence is now weaker for the “psychic” hypothesis, because it has a more likely competing hypothesis. The evidence is still equally strong for face-reading, on the other hand, because its competing hypothesis (that she is psychic) is still very weak.

So we update again!

Psychic: -40 dB to -33 dB (.05%)
Face-reading: -10 dB to 0 dB (50%)

Now the face-reading hypothesis is 50% – apparently equally likely to be true and false! This will sway the strength of the evidence for the ‘psychic’ hypothesis even more on the third trial:

L(correct | psychic) = 3 dB
L(correct | face-reading) = 10 dB

Now with such a likely alternative explanation, the evidence is even weaker than previously for the psychic hypothesis. After our third trial, our beliefs will update as follows:

Psychic: -33 dB to -30 dB (.1%)
Face-reading: 0 dB to 10 dB (90%)

As you can see, the face-reading hypothesis takes off, while the psychic hypothesis ends up staying stuck around .1%.

I’ll talk more about this in a post tomorrow, in which I show how the exact same simple error in our first argument is being made in fine-tuning arguments for God!

The method of reasoning illustrated here is somewhat reminiscent of Laplace’s “principle of indifference.” However, we are concerned here with indifference between problems, rather than indifference between events. The distinction is essential, for indifference between events is a matter of intuitive judgment on which our intuition often fails even when there is some obvious geometrical symmetry (as Bertrand’s paradox shows).

E. T. Jaynes
Prior Probabilities

I’ve previously written praise of the principle of maximum entropy as a prior-setting method that is justified on the basis of a very minimal and highly intuitive set of epistemic features.

But there’s an even better technique for prior-setting, one that is justified on incredibly fundamental grounds. This technique can only be used in rare times, and is immensely powerful when it is used. It’s the principle of transformation groups.

Here is the single assumption from which the principle arises:

“In problems where we have the same prior information, we should assign the same prior probabilities.” (Jaynes’ wording)

This is simple to the point of seeming almost tautological. So what can we do with it?

We’ll start with one of the simplest applications of transformation groups. Suppose that somebody gives you the following information:

I = “This coin will land either tails or heads.”

Now you want to say what the following probabilities should be:

P(This coin will land tails | I) = p
P(This coin will land heads | I) = q

Intuitively, it seems obvious to us that absent any other information, we should assign equal probabilities to these. But why? Is there a principled reason for assuming that the coin is a fair coin? Or is this just a presumption that is importing into the problem our background knowledge about most coins being fair?

The method of transformation groups gives us a principled reason. It says to rephrase the problem as follows:

I’ = “This coin will land either heads or tails.”

Now, our initial problem has only changed to our new problem by replacing every “heads” with “tails” and “tails” with “heads”. Since our prior-setting procedure found that P(This coin will land tails | I) = p in the first problem, it should now find P(This coin will land heads | I) = P in this new one. This is required for any consistent prior-setting procedure! If the problem changes by just switching places of labels, then the priors should change in the exact same way. This means that:

P(This coin will land heads | I’) = p
P(This coin will land tails | I’) = q

But clearly, I = I’; the logical operator “OR” is symmetric! Which means that:

P(This coin will land heads | I’) = P(This coin will land heads | I)

And this is only possible if p = q = ½!

This is simple, but beautiful. The principle tells us that the only logical way to set our priors in this case is evenly – anything else would be either logically inconsistent, or assuming extra information that breaks the symmetry between heads and tails. It goes from logical symmetry to probability symmetry!

Finding these symmetries is what the method of transformation groups is all about. More generally, one can represent a choice between N different possibilities as the statement:

I = “Possibility 1 or possibility 2 or … possibility N”

But this is symmetric with:

I’ = “Possibility 2 or possibility 1 or … possibility N”

As well as all other orderings.

By the exact same argument as above, your prior-setting procedure is required by logical consistency to evenly distribute credences across the N procedures. So for each n from 1 to N, P(Possibility k | I) = 1/N.

The method of transformation groups can also be applied to continuous variables, where finding the right set of priors can be a lot less intuitive. You do so by noting different types of symmetries for different types of parameters.

For instance, a location parameter is one that serves to merely shift a probability distribution over an observable quantity, without reshaping the distribution. We can formally express this by saying that for a location parameter µ, the distribution over x depends only on the difference x – µ:

p(x | µ) = f(x – µ)

For such parameters, it must be the case that the prior distribution over them is similarly symmetric over translational shifts:

For all ∆, p(µ) = p(µ + ∆)
So p(µ) = c, for some constant c

Another common category of parameters are scale parameters. These are parameters that serve to rescale probability distributions without reshaping them. Formally:

p(x | σ) = 1/σ g(x / σ)

For this symmetry, the requirement for consistency is:

For all s, p(σ) = 1/s · p(σ / s)
So, p(σ) = 1/σ

In summation, by carefully analyzing the symmetries of the background information you have, you can extract out requirements for how to set your prior distribution that are mandated on threat of logical inconsistency!

These are the laws of probability, which we have proved to be necessarily true of any consistent set of degrees of belief. Any definite set of degrees of belief which broke them would be inconsistent in the sense that it violated the laws of preference between options … If anyone’s mental condition violated these laws, his choice would depend on the precise form in which the options were offered him, which would be absurd. He could have a book made against him by a cunning better and would then stand to lose in any event.

We find, therefore, that a precise account of the nature of partial belief reveals that the laws of probability are laws of consistency, an extension to partial beliefs of formal logic, the logic of consistency.

Frank Ramsey
The Foundations of Mathematics and Other Logical Essays, Volume 5

 

In this post, I’m going to describe one of the more famous arguments for Bayesianism.

These arguments are about how different types of epistemological frameworks will handle different series of wagers. Let me just lay out clearly what exactly we mean by a wager, so as to remove any ambiguity.

A wager on proposition A is a betting opportunity. It involves a payoff amount S and a buy-in quantity. In general, the amount that the buy-in costs will be some fraction f of the payoff amount, so we’ll write it as fS. If you bet on A and it turns out true, then you get the payout S, but still lost the initial buy-in. And if you bet on A and it turns out false, then you get no payout and lose the fS you already spent.

A	Net Payout
True	S – fS
False	-fS

From this payout table, you can calculate that an agent will find the wager to be favorable to them exactly in the case that P(A) is greater than f. That is, the agent will want to take the bet whenever the chance of a payout is greater than the proportion of the payout that is required to buy into the bet.

Now, imagine that somebody has a credence of 52% in a proposition A and a credence of 52% in the proposition ~A. How will they evaluate the following set of bets?

B₁: pays out $100 if A is true, buy-in of $51
B₂: pays out $100 if ~A is true, buy-in of $51
B₃: pays out a guaranteed $100, buy-in of $102

They will see both B₁ and B₂ as favorable bets, since the buy-in is a smaller fraction of the payout than the chance of payout. And they will see B₃ as an unfavorable bet, since clearly the buy-in is a larger proportion of the payout than the chance of a payout.

But B₃ is just the same as the combination of bets B₁ and B₂!

Why? Well, if you bet on both B₁ and B₂, then you are guaranteed to win exactly one of the two (since A and ~A cannot both be true, but one of the two must be). Then you will have paid in a net sum of $102, and gotten back only $100.

A similar argument can be made for any levels of credence C(A) and C(~A) that don’t sum up to 100%. And all of the usual axioms of probability theory can be argued for in the same way. Such arguments are called Dutch book arguments.

Dutch book arguments are standardly presented as revealing that if one does not form beliefs according to the laws of probability theory, then they will be able to be juiced for money by clever bookies.

This is true; somebody with beliefs like those described above can be endlessly exploited for profit. But it is much less impressive than the real conclusion of the Dutch book argument.

Recall that our agent above was found to believing a logical contradiction as a result of not having their beliefs align with probability theory (they had to believe that a bet was simultaneously favorable to them and not favorable to them)

Said another way, an agent not following the probability calculus may evaluate the same proposition differently if presented in a different form.

This is what Dutch book arguments really say: if you want your beliefs to be logically consistent, then you are required to reason according to probability theory!

A very original and thoroughgoing development of the theory of probability, which does not depend on the concept of frequency in an ensemble, has been given by Keynes. In his view, the theory of probability is an extended logic, the logic of probable inference. Probability is a relation between a hypothesis and a conclusion, corresponding to the degree of rational belief and limited by the extreme relations of certainty and impossibility. Classical deductive logic, which deals with these limiting relations only, is a special case in this more general development.

R. T. Cox
Probability, Frequency, and Reasonable Expectation
(Yep, that Keynes! He was an influential early Bayesian thinker as well as a famous economist)

Cox’s famous theorem says that if your way of reasoning is not in some sense isomorphic to Bayesianism, then you are violating one of the following ten basic principles. I’ll derive the theorem in this post.

Logic

1. ~~a = a
2. ab = ba
3. (ab)c = a(bc)
4. aa = a
5. ~(ab) = ~a or ~b
6. a(a or b) = a

Reasoning under uncertainty

7. Degrees of plausibility are represented by real numbers: {b | a}
8. {bc | a} = F({b | a}, {c | ab})
9. {~b | a} = G({b | a})
10. F and G are monotonic.

The first six, I think, need no introduction. I write “a and b” as “ab” for aesthetics.

The next four extend logic beyond the realm of the perfectly certain, and relate to how we should reason in the presence of uncertainty – how we should reason about degrees of plausibility. For this step, we need a new notation: a way to represent not the truth of a proposition a, but its plausibility. We do this with the {} notation: {a | b} = the plausibility of a given that b is known to be true.

I’ll make some brief notes about each of 7 through 10.

Number 7 perhaps sounds strange – plausibilities are states of belief, not numbers. But you can make sense of this in two ways: first, we can consider this a simple model of plausibilities, in which we are merely mirroring the properties of plausibilities in the structure of the system we set up around how to manipulate numbers. And second, if one is to design a robot that reasons about the world, it isn’t crazy to think about programming it to represent beliefs about the plausibilities of propositions as numbers.

#7 also contains an assumption of continuity – that it’s not the case that there are discontinuous jumps in the plausibility of propositions. Said another way, if two different real numbers represent two different degrees of plausibility, then every possible value between them should also represent a degree of plausibility.

Number 8 is about relevance. It says that the plausibility that b and c are both true given some background information a, is only dependent on (1) the plausibility that b is true given a and (2) the plausibility that c is true given a and b.

If you want to know how likely it is that b and c are both true given a, you can break the process down into two steps. First you determine how likely it is that b is true, given your background information. And second, you determine how likely it is that a is true, given both your background information and the truth of b.

In other words, all that you need to know if you want to know {bc | a} are {b | a} and {c | ab}. For example, say that you want the plausibility of Usain Bolt winning the 100 meter dash and also running an extra lap around the track at the end of the dash. The only pieces of information you need for this are (1) the plausibility of Usain Bolt winning the 100 meter dash, and (2) the plausibility of him running an extra lap at the end of the race, given that he won the race. And of course, each of these plausibilities are also conditional on all of the background information you have about the situation.

We give the name F to the function that details precisely how we determine {bc | a}.

Number 9 says that all we need to determine the plausibility of a proposition being false is the plausibility of the proposition being true. The function that details how we determine one from the other is named G.

And finally, Number 10 says that the plausibility relationships described in 8 and 9 are in some sense simple. For instance, if b becomes less plausible and ~b becomes more plausible, then if b becomes even less plausible, ~b should not suddenly become less plausible. If a change in the plausibility of a proposition a makes the plausibility of another proposition b change in a certain direction, then a greater change in the plausibility of a in the same direction should not result in a reversal of the direction of change of the plausibility of b.

There is an additional implicit assumption that is almost too obvious to be stated: If two propositions are just different phrasings of the same state of knowledge, then you should have the same plausibility in both of them. This is the basic requirement of consistency. It says, for instance, that “a and b” is exactly as plausible as “b and a”.

***

Now we can derive Bayesianism!

First step: conditional probabilities. We use the associativity of “and”:

{bcd | a} = F( {b | a}, {cd | ab} ) = F( {b | a}, F( {c | ab}, {d | abc} ) )
{bcd | a} = F( {bc | a}, {d | abc} ) = F( F({b | a}, {c | ab}), {d | abc} ) )

So F(x, F(y, z)) = F(F(x, y), z)

Any monotonic function that satisfies this functional equation is isomorphic to ordinary multiplication on the interval [0, 1].

In other words, there exists a function W such that:

W(F(x, y)) = W(x) · W(y)

From which we can see:

W({bc | a}) = W({b | a}) · W({c | ab})

Which is exactly the form of conjunction in ordinary probability theory!

This means that any form of reasoning about the plausibility of conjunctions of propositions is either violating one of the 10 axioms, or is equivalent to probability theory up to isomorphism.

So if somebody walks up to you and presents to you an algorithm for computing the plausibility of the conjunction of two propositions, and you know that they are following the above ten rules, then you are guaranteed to be able to find some function that takes in their algorithm and translates into ordinary Bayesianism.

This translation function is W! Notice that although W looks a lot like ordinary probability, it is a different thing entirely. For instance, if somebody is already reasoning with ordinary probability theory, then {b | a} = P(b | a). To convert {b | a} into P(b | a), then, W need not do anything! So W({b | a}) = {b | a} = P(b | a), or W(x) = x, which is clearly different from the probability function.

Next, we reveal the nature of the function G.

The first step is easy:

{~~b | a} = G(G({b | a}))
So G(G(x)) = x

This means that G is an involution – a function that is its own inverse.

Next, we use the commutativity of “and”:

W({bc | a}) = W({b | a}) · W({c | ab})
= W({b | a}) · G( W({~c | ab}) )
= W({b | a}) · G( W({b~c | a}) / W({b | a}) )

So W({cb | a}) = W({c | a}) · G( W({~bc | a}) / W({c | a}) )

Now if we let c = ~(bd) for some new statement d, and rename W(b | a) = x and W(c | a) = y, we find:

x G( G(y) / x ) = y G( G(x) / y )

The only functions that satisfy this functional equation as well as the earlier involution equation are the following:

G(x) = (1 – x^m)^1/m, for any m

With some algebraic manipulation, we see that this is equivalent to:

W({a | b})^m + W({~a | b})^m = 1

This equation almost perfectly represents the normalization rule: that the total probability must sum to 1! It seems mildly inconvenient that this holds true for the function W^m rather than W. If we knew that m had to equal 1, then we’d have a perfect translation function both for conditional probabilities and for normalization… But if we look back at our conditional probability finding, we notice that it can be equivalently represented in terms of W^m instead of W!

W({bc | a}) = W({b | a}) · W({c | ab})
W({bc | a})^m = W({b | a})^m · W({c | ab})^m

Now we just define one last function Q = W^m, and we’re done!

Q takes in any system of reasoning that obeys the starting ten rules, and reveals it to be equivalent to ordinary probability theory.

The rest of probability theory can be shown to result from these basic axioms (we’ll drop the brackets now, and will call Q({a | b}) its proper name: P(a | b).

P(certainty) = 1
P(impossibility) = 0
P(bc | a) + P(b~c | a) = P(b | a)
P(a or b | c) = P(a | c) + P(b | c) – P(ab | c)
And so on.

This derivation of probability theory is great because it isn’t limited by the ordinary frequency interpretations of probability theory, in which a probability is defined to be a limit of a ratio of an infinite number of experimental results. This is not only ugly theoretically, but leaves us puzzled as to how to talk about the probabilities of one-shot events or of hypotheses that can’t be directly translated into empirical results.

The definition of probability invoked here is infinitely deeper. Instead of frequencies, probabilities are defined according to fundamental principles of normative epistemology. Any agent that reasons consistently according to a few basic maxims will be reasoning in a way that is functionally identical to probability theory.

And probabilities here can be assigned to any propositions – not just those that refer to empirically measurable events that are repeatable ad infinitum. They represent a normative rational degree of certainty that one must possess if one is to reason consistently!