Is the Bayesian idea of a continuous degree of belief incorrect?

Question

I'm having a hard time understanding how this concept makes sense or is rational in any shape or form. I would wager that this works with things that have prior probabilities and data behind them. But consider the case of where we have no prior data.

In Bayesian confirmation theory, you have a prior probability in a hypothesis H. An observation "confirms" your theory if it increases your probability of H. The only way this can happen is if an observation O occurs that is more likely under H than ~H (i.e. not H). In other words, after observing O, if P(O|H) > P(O|~H), Bayesian confirmation theory tells you that you must increase your P(H).

Now let's consider an example which to me seems to highlight the ridiculousness of this gesture. For example, suppose I am evaluating the hypothesis that an invisible monster is controlling my coin and causing it to land on heads every single time. Before tossing anything, my P(H) is obviously infinitesimally low. Suppose I now toss the coin one time. The P(O|H) in this case is 1. P (O|~H) which in this case is chance is 1/2. Bayesian confirmation theory now tells you to increase your credence in this monster hypothesis.

Sure, it might still remain extremely infinitesimal. But is this really rational? Why should a ridiculous observation like this update your credence in belief even in an infinitesimal way? Lastly, what does it even mean to have a credence of belief? Say you assign a probability of 0.000000000001% to the hypothesis that monsters were controlling your coin. What does this mean? I'm aware of the dutch book arguments for subjective probabilities but I fail to see how they are meaningful in any sense. It's not as if we have repeated trials where 0.000000000001% of the time, monsters would explain this.

Answer 1

You seem to be asking several questions here. First, does Bayesian conditioning work?

Bayesian conditioning, considered as an idealised model of rational updating of hypotheses based on evidence, has many useful applications. It is an important feature of a common approach to decision theory.

Second, is Bayesian conditioning what humans actually do when reasoning?

How human beings reason, draw inferences and update their beliefs has been studied extensively by cognitive psychologists for many decades. After all that work, there is no consensus among researchers on the subject. Some hold that people use mental rules akin to natural deduction rules. Others that people use mental models. Others that people make comparisons with paradigm cases. There are also many who hold that people reason by performing something that approximates Bayesian updating. A book defending this thesis is Oaksford and Chater, "Bayesian Rationality: The probabilistic approach to human reasoning" Oxford University Press (2007).

The fact that there is no consensus suggests that different people reason in different ways, so it may be futile to expect a one size fits all account of human reasoning. In any event, how people reason will only approximate some account, because logical and probabilistic calculation has a high degree of computational complexity. In most cases, the need to find a satisfactory solution to a practical problem as quickly as possible outweighs the desire to find an optimal solution that requires considerable time and resources. This is why many psychologists make use of the concept of bounded rationality.

Third, why does something inconsequential like a coin toss make a difference to an absurd hypothesis?

The simplest response is to observe that a single one-bit data point makes so little difference that it can be ignored as negligible. A further response would be to say that the absurd hypothesis should be compared with other competing hypotheses. If your coin keeps coming up heads, the coin may be biased, or the mechanism for tossing it may be biased. These are more plausible alternatives than the invisible monster.

Fourth, what does a tiny degree of credence signify?

Some of our beliefs are stronger than others. Some may be very weak indeed. In decision theory, if an agent is weighing up how to decide between a number of different options, account should ideally be taken of all the beliefs that are relevant to the decision, even the very weak ones. In some situations the weak ones may make a difference.

Answer 2

It depends on the premises which we are willing to allow.

If you want to say that a person either believes that something is true, or believes that it is false, and there is no middle ground permitted, then any non-bivalent system of logic or epistemology is going to be unsatisfactory, and there's nothing special about Bayesian reasoning in particular. This problem is answered in greater detail under your previous question, so I won't reiterate the arguments here.

For the sake of argument, let's assume you're willing to allow degrees of belief. The next question is, do your degrees of belief obey the Kolmogorov axioms? The axioms are as follows:

0. You have some sample space, which can be measured by some probability operator P(X). The word "measure" captures a whole lot of other axioms, which should be properly accounted for as well, but in short, a "measure" is analogous to the area or volume of a geometric set (you can take measures of individual subsets of the sample space, you can add non-overlapping measures together, etc.).
1. No portion of the sample space has a negative probability.
2. The probability of the space as a whole is one. Together with the prior axiom, this implies that every event has a probability of no less than zero and no greater than one.
3. You can add together countably (infinitely) many disjoint probabilities, to get the probability of their union. This is not actually required for simple examples, but you probably need it if you want to talk of continuous probability distributions.

Now, for our purposes, this "probability" operator is taking on the role of the degree of belief, so now we just have to translate these axioms into more familiar terms:

0. You have a set of all possible beliefs (or perhaps "all possible beliefs that pertain to [some specific context]"). For any given subset, we can ask about your degree of belief, and you will answer with some real number. Your beliefs are arithmetically consistent with each other, so that if we ask you about different subsets, we can add or subtract your degrees of belief, and calculate your belief in the appropriate set union or difference.
1. You never claim to have a negative degree of belief in something. But you could say that you have zero belief. Zero is special because, using the addition property, we can see that any belief with zero probability can be included or excluded in any set union without changing the probability of the sum. Therefore, such a belief makes no difference, and may as well not exist at all. Informally, a belief of zero is a belief that something is strictly impossible.
2. The overall degree of belief you have in the set as a whole is one. Of course, one is ultimately just an arbitrary number; the point is that there is some upper bound on your overall belief, and it turns out that fixing this upper bound at one makes the arithmetic easier.
3. To the extent that you have (countably) infinitely many possible beliefs in the first place, addition over those beliefs still "works" - it converges to the appropriate value as we would expect. If your set of all possible beliefs is finite, then this is vacuously true.

If you agree with the above axioms, then Bayes' theorem logically follows from them. If you do not agree with one of those axioms, you should probably try to figure out which one it is that you disagree with, and why it is so unappealing to you.

Answer 3

There are 2 problems in your understanding here I suspect.

1. How improbable some things are is difficult to understand.

2. Alternative explanations for your experiment change how things work in unexpected ways.

Addressing the first:

One problem with Bayesian credence is humans are bad at understanding what 0.00000000001% means.

Imagine you had a hypothesis that an invisible dragon was making your coins always land heads. This hypothesis is very specific and hard to work out what the credence you have is. So let's do a slightly easier one.

The hypothesis that your coin, the one you just pulled out of your pocket with no special history, will almost always land heads. The invisible dragon implies this one, so this is a weaker hypothesis.

What is the credence you have for that hypothesis? Well, we can run Bayesian analysis backwards. How many heads in a row would it take for you to say "well, that actually seems likely"? 10? 100? 1000?

Suppose you are pretty sure this coin is utterly normal. So it would take 1000 heads in a row for you to be 50-50 that this coin will almost always land heads.

P(A|B)=P(B|A)P(A)/P(B)

Here P(A) is "coin will almost always land heads", P(B) is "1000 heads in a row".

0.5 = P(Coin will almost always land heads|1000 heads)=P(1000 heads|coin will almost always land heads)P(coin will almost always land heads)/P(1000 heads)

P(1000 heads) is 2^-1000, or 10^-300. P(1000 heads | coin will almost always land heads) is ~1.

0.5 = P(Coin cheats|1000 heads) =~ 1 P(coin will almost always land heads)/10^-300.

So P(coin will almost always land heads) =~ 0.5 in 10^300.

Or, 0.(place 301 0s here)5

Now we can chain it again. Given that you have a coin that (for no reason you yet know; there is no special history of this coin) almost always lands heads, what would be the probability it is an invisible dragon that is doing it telekinetically?

This requires it be an invisible creature, that this creature be using telekinesis, and that invisible creature is a dragon. Each of these is going to be less likely than the above.

So the "proper" initial credence value is going to be extremely hard to express, even exponentially, because you have to chain together a bunch of requirements each of which is exceedingly unlikely.

To work out how unlikely, you can again use inverse Bayes. You can work out the kind of evidence that would be required to make it reasonably credible, and how unlikely that evidence would happen if it wasn't true.

I suspect you'll need fancy notation in the end. Like maybe conway up arrow notation.

A bonus effect is that Bayes theorem lets you take even extremely unlikely things -- things with a 1 in 10^300 chance of being true -- and in a relatively short experiment (1000 coin flips) make them "ok, that is a reasonable explanation".

The real weakness is that knowing P(thing you don't believe in) is not easy.

When dealing with such extremely unlikely things, it might help to think logarithmicly.

lg(P(A|B)) = lg(P(B|A)) + lg(P(A)) - lg(P(B))

lg(P(unlikely|evidence)) = lg(P(evidence|unlikely)) + lg(P(unlikely)) - lg(P(evidence))

lg on probabilities are all negative. So lets define a new term -- E. E(X) = -lg(P(X)). E is positive.

E(unlikely|evidence) = E(evidence|unlikely) + E(unlikely) - E(evidence)

Using base 2 for logs, 0.5 probability corresponds to E of 1.

1 = E(fixed coin|1000 heads)

0 = E(1000 heads|fixed coin)

? = E(fixed coin)

1000 = E(1000 heads)

1 = 0 + 1000 - E(fixed coin)

E(fixed coin) = 999

0 in this scale is certain -- the bigger the number is, the more surprising it is.

To convert back to probability, just take 0.5^E.

This exponential scale might give you a better way to think about unlikely events. How many "otherwise even odds events" would have to behave in a way in sync with your unlikely hypothesis for you to say "well, that is now actually likely".

That is the "E" level of your unlikely hypothesis.

And the E level of something like "invisible dragon telekinetically making this coin land heads" might be a google.

Addressing the second:

The next problem is with the naive application of this theory.

The problem is that a pile of heads wouldn't ever cause you to actually think there is something as specific as a telekinetic invisible dragon. There are a pile of other reasons why the coin lands heads (it has heads on both sides, there is a magnet in it that lets it be remote controlled, whatever).

This impacts P(1000 heads). Because it isn't actually 1 in 2^1000. In fact, the more heads you have, the more likely you'll assume the coin is fixed, and the less unlikely new heads are!

When we are talking about a fixed coin, the alternative is the coin isn't fixed. So

P(fixed coin|1000 heads) = P(1000 heads|fixed coin) * P(1000 heads) / P(fixed coin)

but P(1000 heads) = P(1000 heads | fixed coin) * P(fixed coin) + P(1000 heads | fair coin) * P(fair coin).

Our assumption that P(1000 heads) = 1/2^1000 relied on the fact that P(1000 heads | fair coin) * P(fair coin) is 1/2^1000 and P(1000 heads | fixed coin) * P(fixed coin) is small.

However, when enough coin flips occur that P(fixed coin) hits 0.5 this assumption is no longer safe.

If there was a small chance of a coin being fixed (in any way) to always land heads, say 1 in 100, then P(1000 heads) is actually 1%! It doesn't generate boundless amounts of evidence.

Applied to the invisible dragon case, because there are many other ways to fix the coin than the invisible TK dragon, a boundless number of heads stops giving evidence of the dragon when it starts making the coin being fixed nearly certain. At that point to get more evidence of the dragon you need a test that distinguishes the invisible TK dragon from the other ways the coin can be fixed.

In log space, E(fixed coin) might be 100, and E(fixed by TK) might be 10^100, and E(dragon is doing it) might be 10^10^100, and E(dragon is invisible) might be 10^10^10^100.

And the probabilities might roughly multiply, which means the E (evidence required) roughly adds up.

P(unlikely event) = Sum P(unlikely event | S_i) * P(S_i)

You have to, in a sense, consider every situation when doing Bayesian analysis.

If you arrange your situations in a certain way, this is easy. Imagine your hypothesis was "the coin flips are fair" and "the coin are not fair". This covers the entire universe, so there are only 2.

But if you measure the chance of 1000 heads in a row as 1 in 2^1000, it implicitly does "there is an invisible dragon using TK to control the coin" and implicitly "the coin flips are fair" with the probability of the dragon being insanely small.

What really needs to be done is "fair coin", "invisible dragon", "alternative explanation for unfair coin".

With that model, large numbers heads in a adds evidence to the union of ("invisible dragon" and "alternative explanation"). And because the alternative explanations for an unfair coin are insanely more likely than the dragon, more and more heads doesn't actually move the dragon out of infinitesimal chances.

To use Bayes to pull the dragon out of infinitesmal chance of being true, you are going to have to provide more and more observations that split the invisible dragon from the alternatives. And the harder it is to detect a dragon the more extreme the evidence is going to have to be, mathematically.

Answer 4

You seem to roughly have these 2 possibilities:

• An invisible monster is controlling the coin and causing it to land on heads every single time.

• The coin is fair (or will always land on tails).

This is incomplete, because you're excluding other possible explanations for why the coin might land on heads.

The more general version of the first premise is simply:

• The coin will land on heads every single time.

(Technically, the coin can land on heads, say, 99% of the time, or 95% of the time, or any other percentage of the time, and this could be hard to differentiate from 100% with only a few throws, but let's assume it's 100% and 50% are the two possibilities, for argument's sake.)

In this case, yes, the coin landing on heads will increase the probability of always landing on heads.

One could say that the probability of an invisible monster increases as well, but this is simply in relation to the coin being fair. It doesn't increase in relation to any of the other explanations for why it landed on heads, especially not the more reasonable explanation of the coin simply being biased, because those probabilities would be increasing at the same time.

So even if you grant that each one of infinitely many flips will land on heads, this still doesn't get you to an invisible monster being a likely explanation.

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The hypothesis that an invisible monster is controlling the coin is unfalsifiable and not demonstrable.

So I don't think you can reasonably assign a concrete numeric probability to it. Also, 0.000000000001% is not infinitesimally low. Numeric probabilities work when you have some basis for what those probabilities should be. If you flip a coin, you can determine how likely it is to be fair. Most beliefs are not quite so simple and measurable that one can easily assign a reasonable probability value to it.

Instead, beliefs should generally be determined through a solid epistemological framework that would typically be a bit more fuzzy than concrete mathematics.

Answer 5

I'm not sure how well this plays into Bayesianism directly, but suppose that beliefs are physical states with a probabilistic basis (metaphysically, as grounded in the limit in quantum fields; or epistemically, in that we can only probabilistically justify predictions about what we or others will believe as time goes on). Would it not be possible to characterize beliefs as having real-valued degrees to their names?

Consider, for example, fuzzy logic, for which there are 2^ℵ₀-many truth-values (albeit perhaps not quite so much as Fregean such objects, to be sure). Since there are acceptable mutations of set theory according to which we can force 2^ℵ₀ to equal an absolutely infinite number of different cardinals (or even to equal absolute infinity itself, depending on how much our theory has mutated), this suggests an absolutely infinite range of degrees of possible beliefs. Even if the Continuum Hypothesis is true, there are still enough degrees of truth to allow for enough degrees of belief that "degrees of belief" talk has some obscure metaphysics-of-mathematics basis.

Or then consider an elaboration of the above theme such as fuzzy multisets. If we have beliefs about such entities, would it not be possible for those beliefs to come in degrees? Or, for that matter, for generic belief in those things to end up being the same thing as there being degrees of specific belief.

Again, not sure if Bayesianism involves such background logic/mathematics, but so this is mostly a rejoinder to your question on a general level: "Is it rational to believe in degrees-of-beliefs, and if so, how?" is answered by, "The form of rationality itself involves degrees of the relevant form."

Answer 6

Now let's consider an example which to me seems to highlight the ridiculousness of this gesture. For example, suppose I am evaluating the hypothesis that an invisible monster is controlling my coin and causing it to land on heads every single time. Before tossing anything, my P(H) is obviously infinitesimally low. Suppose I now toss the coin one time. The P(O|H) in this case is 1. P (O|~H) which in this case is chance is 1/2. Bayesian confirmation theory now tells you to increase your credence in this monster hypothesis.

I think the problem here is conflating evidence with prior belief.

Say we had a different hypothesis, which is that the person is flipping a coin with a head on both sides (perhaps to decide who bats first in a game of cricket). If this person has a long history of winning coin tosses, then we may have some prior reason to believe that they do indeed have such a coin. Lets take H0 to be the usual null hypothesis - the coin is fair - p(head) = p(tail) = 0.5, and H1 be the hypothesis that the coin has two heads - p(head) = 1 and p(tail) = 0. In this case we estimate p(H1) = 0.6 and p(H0) = 0.4 (the two hypotheses are not necessarily exhaustive, but we'll assume they are to keep things simple).

The thing to do is to separate out the terms relating to the evidence from the terms belonging to the terms relating to the priors. The Bayes factor is the ratio of the [marginal] likelihoods of the observations under the two hypotheses:

K = P(D|H1)/P(D|H0)

Where P(D|M1) is the probability of observing a head if H1 is true. Using Bayes rule, we can write that as

K = P(H1|D)P(H0)/P(H0|D)P(H1)

where P(H1) represents our prior belief (before witnessing the head) that H1 is true. Equating and rearranging,

P(H1|D)/P(M0|D) = P(D|H1)/P(D|H0) x P(H1)/P(H0)

So the Bayes factor shows us by how much the evidence changes our prior relative belief in the two hypotheses to give us our posterior relative belief.

In this case P(H1)/P(H0) = 0.6/0.4 = 1.5 and P(D|H1)/P(D|H0) = 1/0.5 = 2, so

P(H1|D)/P(M0|D) = 2 x 1.5 = 3.

So our belief in the person having a double headed coin has increased, but only very slightly as a single coin flip is only very weak evidence.

For the invisible monster, the reasoning is the same, but the priors are different. So the question is, why would the priors beliefs being numerically different invalidate the analysis, when it works fine for pretty much all reasonable examples?

I think the problem here is that your true prior belief in the existence of the invisible monster is actually precisely zero. If that is the case, no amount of evidence will change that, and the Bayesian analysis accurately reflects that.

P(H1|D)/P(M0|D) = 2 x 0 = 0.

Unfortunately providing good numerical values for prior beliefs on propositions that are essentially infinitely unlikely is bound to be tricky. If you introduce infinities into mathematics, you are likely to run into problems. In reality only things that are logically certain (tautologies) or logically impossible have probabilities of exactly 1 or 0, but you can get infinitesimmally close to 0 and 1, and end up with numbers that are too large or too small to be meaningful to us.

I'd say that 0.000000000001% is an absurdly high prior probability for invisible monsters - for a computational analysis I'd probably use the smallest representable floating point value. Not impossible, but as near as you will get. Twice that is still basically nothing and likely to be lost as soon as you do any calculation with it.

One last thought - also the choice isn't that there is an invisible monster or the coin is fair, there could be other explanations, for instance they are using a double headed coin, or they have learned the skill of flipping it so that it always comes up the side they want it to, which are much more plausible than invisible monsters. The Bayesian analysis is not limited to just two competing hypotheses, and I suspect if you have a broader set of hypotheses, then the posterior belief in the invisible monster will rise less than if the only alternative is a fair coin.

Answer 7

Suppose we are social constructionists about mental illness. The mean GAF score for inpatients is, let's suppose, 41.2 The fact that this time no-one can actually score the mean will change how we categorise our inpatients and - by extension - how impaired the inpatients are (mental illness changes when we assess it).

If we are likewise social constructionists about the field we are using Bayesian reasoning about, then how finely grained our beliefs can be will have a bearing on what is actually happening.

Does that mean that for social constructions Bayesian reasoning is less reliable - slightly off - from what it otherwise would be? I think that's the motive for the question, I'm not sure (as pointed out, it is based on a mathematical triviality, so the question can't be trying to refute that, just how useful etc. it is).

Given the fact that no-one has a 41.2 GAF score changes actual inpatient functioning, then statistics has to track that additional change. I would be wary of saying this does not make the 'mean' less reliable, even-though in this instance it'd probably say more about the usefulness of GAF itself than statistics (i.e. that the shift from 'mean' is dwarfed by sitting GAF assessments).

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The simple take home message is that updating your belief and not updating your belief, as well as whether you can or cannot, will - in some cases - influence what is actually happening and so how reliable your Bayesian results are.

Answer 8

If you like things Bayesian and also like things cognition, I recommend you look into Karl Friston's research. He specializes in computational neuroscience. For me the theoretical physics behind his work has explained the totality of individual human behavior and human social organization.

It's all about minimizing free energy, avoiding entropy. It's why atoms self-assemble into molecules, it's why molecules self-assemble into non-equilibrium steady-state systems (life), and for our discussion it's why we hold beliefs and both constantly change them and act upon them. Our brains are sensory-expectation engines that hold a generative model of itself and the world. All action and perception is in service of minimizing free energy: reducing the gulf of prediction error between the model and the incoming sensations. We change "beliefs" within our model to match the incoming sensations. We act upon the environment to change the incoming sensations to match "beliefs" within our models.

All social organization can be explained as yet another stage of emergent self-organization that seeks to minimize free-energy, just at this level at a human-collective level. How much prediction error is reduced with the legal concept of a stop sign given tens of millions of vehicles? Without a stop sign one might feel the consciousness awareness of that prediction error: physiological arousal, like stress, where we frantically seek to maximize model evidence (reduce that error).

I'm writing a book that provides a more reductive mechanics to social psychology in areas like politics and economics using his work. I'm also working with a few research partners on a molecular mapping technology for this computational neural mesh. I think once completely actualized it will reduce a lot of complexity surrounding our day-to-day lives as humans, by ultimately reducing our humanity to equations.