## Bayes Theorem is Best Theorem

Here’s a hypothetical scenario that I adapted from Luke Muehlhauser who, in turn, adapted it from Neil Manson:

Imagine you are on the battlefield and you are under attack by an enemy army containing 100 men. You know from intel that this army consists of 99 regular soldiers and 1 elite sniper, randomly distributed. One of these army members opens fire at you from 400 meters away and nearly hits you. If the elite sniper fired at you, there’s an 100% chance this would happen. However, the regulars are still good shots without sniper rifles, and could make the same shot about 0.1% of the time. What is the chance you were shot at by the sniper?

Thus enters Bayes Theorem, a way to assess probability.

## Applying Bayes Theorem

Well we know that sniper is randomly distributed and there is only one of him. But we also know that he definitely could have made that shot, whereas it is unlikely that a regular made the shot. So we have a problem of which probability to use: 1%, or 100%, or something in between? We’ll use Bayes Theorem to decide.

Bayes Theorem looks like this:
$P(H|E) = \frac{P(E|H)\, P(H)}{P(E)}$

But what does all that math-y stuff mean? Well, we have three things: We have H, which is our hypothesis (“A sniper has fired upon you”) and we have E our evidence (“The shot was made from 400 meters away”).

Lastly, we have the probability equation P(). P(H|E) is the probability of H given E, or what we want to solve for. This is the probability of our hypothesis being true given the evidence we have. In this scenario, it is the probability that a sniper fired at you given what we know about the shot made.

P(E|H) is the other way around, the probability of E given H, or the probability of the evidence assuming that our hypothesis is true. In this scenario, it is the probability that such a shot was made assuming you were fired upon by a sniper.

The last two P(H) and P(E) are the individual probabilities. P(H) is the probability of E assuming nothing else, or the probability that a sniper is firing upon you, assuming nothing else. P(E) is the probability that such a shot could be made at you, assuming nothing else.

#### Calculating P(H), P(E), and P(E|H)

So what do we need to know for Bayes Theorem? We want to solve for P(H|E), and to do we need values for P(E|H), P(E), and P(H).

First lets get P(H) and P(E|H) since they’re easy. What is P(H)? What is the probability that the sniper is firing upon you? Well since the sniper is randomly distributed among an army of 100, the probability the sniper is attacking you assuming nothing else is 1%. Therefore P(H) = 1% = 0.01.

Now what is P(E|H)? The symbols look complicated but it is an easy question: What is the probability the shot was made assuming it came from the sniper? We know from the story that this is 100%. P(E|H) = 100% = 1.

Now for the hard one: what is P(E), or what is the probability that such a shot could be made? Well the single sniper could have made that shot, but regulars can make the same shot 2% of the time. The way we solve this is to add two quantities: the first is the percentage of the army the sniper are multiplied by the percentage of times the sniper could make the shot, and the second quantity is the percentage of the army the regulars are multiplied by the percentage of times the regular could make the shot. P(E) = [(1%)(100%) + (99%)(0.1%)] = [(0.01)(1) + (0.99)(0.001)] = [0.01 + 0.00099] = 0.01099

#### Answering the Hypothetical With The Formula

Now we can put it together:

$P(H|E) = \frac{P(E|H)\, P(H)}{P(E)}$ $P(H|E) = \frac{1*\, 0.01}{0.01099} = \frac{0.01}{0.01099} = 0.9099$

So via Bayes Theorem, we know that despite the rarity of snipers, there is a 91% chance you were just attacked by the sniper. You better call in reinforcements! Congratulations on using probability effectively!

## Another Example

So let’s sharpen our Bayes Theorem usage by another, more relevant, example. This one is the classic example for mammograms, that I have adapted from Eliezer S. Yudkowsky, along with pretty much everyone else who has written an explanation of Bayes Theorem. It goes like this:

• 1% of women have breast cancer.
• If a woman has breast cancer, a mammogram will detect the breast cancer 80% of the time and fail (false negative) 20% of the time.
• If a woman does not have breast cancer, a mammogram will return a false positive 9.6% of the time.

Now we ask the key question: If you get a positive result from the mammogram, do you have cancer?

#### Using Bayes

Well again, we determine our hypothesis, our evidence, and find values for P(H), P(E), and P(E|H). In this example, our hypothesis is I have cancer and our evidence is I got a positive result from a mammogram.

Since only 1% of the population has cancer, P(H) = 1% = 0.01.

Since the mammogram will return a positive result 80% of the time if I have cancer, P(E|H) = 80% = 0.8

Since a mammogram returns a positive result 80% of the time if there is cancer and 9.6% of the time if there isn’t cancer, P(E) = [(0.01)(0.8) + (0.99)(0.096)] = [0.008 + 0.09504] = 0.10304. This is the percentage of all potential mammograms that return positive, regardless of whether or not the person has cancer.

$P(H|E) = \frac{P(E|H)\, P(H)}{P(E)} = \frac{0.8*\, 0.01}{0.10304} = \frac{0.008}{0.10304} = 0.0776$

Therefore, if you get a positive mammogram result, there is only a 7.76% chance you have cancer!

#### The Counterintuitive Lesson

This result at first seems rather weird. After all, isn’t this supposed to be an accurate test? Well then, why would it not really do much to tell me if I have cancer or not? Why is it that if I test positive for cancer there is still a 92.24% chance I don’t have cancer.

That’s because the rate of people who actually have cancer is so low that the chance of a false positive over a very large population greatly skews the test.

Therefore Bayes Theorem is teaching us some important lessons:

• Background evidence is important: as seen in the sniper example, the fact of the shot greatly outweighs the rarity of the sniper
• Tests can be wrong: tests can return false positives and false negatives.
• There is a difference between the event and a test: having cancer is different than testing positive for cancer.
• False results greatly skew our data: If you’re searching for something really rare and find it, it becomes more likely you got a false positive or false negative. (This is exactly why tests need to be replicable.)
• Everything we know is still a test: Since it is impossible to have all of the information, we can only make our best guess using Bayes Theorem.

## Bayes Theorem, Ghosts, Gullibility, and Replication

Now that we have an understanding of Bayes Theorem and the important philosophy behind it, we can apply it to an important point. Consider the number of people who have gone hunting for ghosts. This represents people preforming a test for ghosts. While we will never be able to figure out P(H), the probability that ghosts actually exist, and therefore cannot apply Bayes Theorem to get an actual probability of ghosts, we can apply the general philosophy here.

Since ghost hunting is a test, it also can return false positives at some rate. Someone can think they see a ghost when there really is no ghost at all. (Ghost hunting can also give false negatives at some rate as well.) Now, hypothetically imagine that we have gone testing for ghosts and have gotten a positive result. Should we assume ghosts are real?

Well, no. Perhaps the tests for ghosts are more like the mammogram example. Given that thousands of other tests for ghosts have failed for whatever reason, we have to reason to assume that we just got a false positive until more testing can be done. We would have to repeat our test a few more times, as well as have other, unrelated, unbiased people replicate our test before we have conclusive results. We just simply won’t know until we get more background information and evidence. Our knowledge is always incomplete.

The strong reality of false positives means that we need to adopt some skepticism. We shouldn’t believe in ghosts or ESP just because one test shows they exist, especially if hundreds of other tests show they don’t exist. Eliezer Yudkowsky’s essays “I Defy the Data” and “Your Strength as a Rationalist” elaborate on this.

Lastly, consider another example: We have a hypothetical friend named Bob who tells the truth 99% of the time. We both write down a number at random between 1 and 1000. I hand him my number and, without ever looking at either number, ask him if they match. He says “Yes, they match”. Should we assume Bob is telling the truth here, since he always tells the truth 99% of the time? Bayes Theorem says that we shouldn’t be so recklessly naïve. We should ask to see both slips of paper for ourselves, and get very suspicious when he refuses.

And also, we can block spam with it.

This is not meant to be an exhaustive explanation of Bayes Theorem, but a simple intuitive guide to get you going and to make a few points. If you want, here are a lot more references to fully get a grasp of Bayes Theorem, in order from simple to complicated:

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I now blog at EverydayUtilitarian.com. I hope you'll join me at my new blog! This page has been left as an archive.

On 31 Jan 2011 in All, Knowledge, Mathematics. 32 Comments.

1. #1 joseph says:

I am always at a loss when people self identify as Bayesian. I find in general life I am quite unaware of probabilities, and just use broad categories like “very low, low, average, high, very high”.

If I were to argue with a supernaturalist, I am sure they would argue that I was dismissing true positives as false positives because I was assigning a low probability to the existence of ghosts. They then could use Bayes the same way Young Earth Creationists insist on Carbon dating everything and everything and then claiming nothing is older than about 40,000 years.

Bayes also has one small fault in that in establishing the sensitivity and specificity of a medical test, say a feline coronavirus antibody titre in the detection of Feline Infectious Peritonitis, a Gold Standard test (histopath) is used, which of course has it’s own limitations.

2. #2 Peter Hurford (author) says:

I am always at a loss when people self identify as Bayesian. I find in general life I am quite unaware of probabilities, and just use broad categories like “very low, low, average, high, very high”.

I have the same general approach too — I don’t think anyone holds a precise percentage confidence value for their beliefs, but it definitely would be better if we somehow could.

I’m still unsure how Bayes would fit into a strategy to make sure our beliefs more accurately reflect reality, but I’m fairly certain that it does in some way.

~

If I were to argue with a supernaturalist, I am sure they would argue that I was dismissing true positives as false positives because I was assigning a low probability to the existence of ghosts. They then could use Bayes the same way Young Earth Creationists insist on Carbon dating everything and everything and then claiming nothing is older than about 40,000 years.

You’d have to then have an argument over the accuracy of the prior probabilities.

~

Bayes also has one small fault in that in establishing the sensitivity and specificity of a medical test, say a feline coronavirus antibody titre in the detection of Feline Infectious Peritonitis, a Gold Standard test (histopath) is used, which of course has it’s own limitations.

What is the fault?

3. #3 joseph says:

1/ Yep! Need more data is a bit of a default statment of mine.

2/ Indeed, often do.

3/ The fault is even the gold standard test has a sensitivity of lower than 100% and a sensitivity of less than 100%, but I don’t think there is anyway to correct for this. This is especially problematic in somewhat poorly defined diseases with highly variable symptoms (like FIP). You end up with more of a “within reasonable doubt” diagnosis than a “innocent until otherwise proven guilty”.

4. #4 Peter Hurford (author) says:

The fault is even the gold standard test has a sensitivity of lower than 100% and a sensitivity of less than 100%, but I don’t think there is anyway to correct for this.

What specifically makes this a problem for Bayes Theorem? It seems to just be a problem for the specific testing.

5. #5 joseph says:

Bayes Theorem then produces slightly incorrect results, due to limits on what is knowable. Because the “Gold Standard test” is difficult to examine with Bayes (there’s nothing better to compare it to). I can see how you can say that Bayes theorem is not wrong itself, I’d say that there are practical limitations in it’s use. The answers it produces depend on the quality of data being plugged in.

For example is I attempt to establish an “actual” number of cats with FIP in a community, say I set the standard at a positive pcr for a certain prevalent string of dna among FIP viruses, I high FIP antibody titre and indicative histopath, my “actual” which is required by Bayes, will still not be the real actual. My map will not be the territory. To use Bayes ideally, as I understand it, my false “actual” is not up to muster. The answers I’ll get are still better than not using it, but still not perfect.

It would be interesting to write some kind of meta-bayes formula that shows how bayes performs with low quality “gold standard tests” or, basically, ignorance.

6. #6 Peter Hurford (author) says:

Bayes Theorem then produces slightly incorrect results, due to limits on what is knowable.

I don’t think this is true, because Bayes Theorem isn’t telling you the actual answer, but telling you how confident you can be given the evidence you have.

Consider the example of cancer and mammograms again. It’s true that in the territory, you either have cancer or you do not — probabilities are only expressions of our personal uncertainty over the territory.

So we can initially know a prior that people usually have cancer 1% of the time. Then we can have a positive mammogram and count it as evidence. When we do, Bayes says that we now have enough evidence to move our estimate of cancer up to 7.76%.

The fact that Bayes might label someone who has cancer as only having a 7.76% chance given the evidence is not a problem, precisely because Bayes is about our personal confidence given our limited information in our map, not about what is actually happening in the territory.

Thus I think, unless I’m missing something, Bayes already takes into account your objection.

~

my “actual” which is required by Bayes, will still not be the real actual. My map will not be the territory. To use Bayes ideally, as I understand it, my false “actual” is not up to muster. The answers I’ll get are still better than not using it, but still not perfect.

Right. But that in itself is not a problem with Bayes, but rather a problem with us not knowing enough information to apply Bayes. Bayes is only as good as the information we can feed into it.

~

It would be interesting to write some kind of meta-bayes formula that shows how bayes performs with low quality “gold standard tests” or, basically, ignorance.

Right. Also, to be honest, I’m really not sure how to apply Bayes in my actual thinking or my theory of knowledge yet. Bayes for me is just this floating cool thing that I think about and don’t acutally use. Though, I do think I should apply it somehow.

7. #7 joseph says:

How do we know:

“prior that people usually have cancer 1% of the time”

I’d say the prior is realistically a little off. Does that change much? Doubt it, though that is not to say small mistakes are never important.

In agreement on the second and third points.

8. #8 joseph says:

Though think you meant “correct” rather than “incorrect”

9. #9 Peter Hurford (author) says:

I’d say the prior is realistically a little off. Does that change much? Doubt it, though that is not to say small mistakes are never important.

Right. Bayes is fairly “garbage in -> garbage out” as far as I can tell. It’s only as good as your priors and your evaluation of the evidence.

But it does seem to be a useful tool in many situations, such as answering questions like “If you get a positive result from the mammogram, do you have cancer?” or “If you observe a door slam with no one to close it, is there a ghost?”

~

Though think you meant “correct” rather than “incorrect”

Fixed.

10. #10 joseph says:

Yep, it’s a vital part of medicine.
Sadly when you tell people one test is not by itself enough they then believe:
1)Modern Medicine is not trustworthy (usually accompanied by a story of a grandfather that smoked cigarettes and drank whiskey his whole life and lived til 90).
2)That you are trying to get more money out of them, after all, who discusses the bill on House?

11. #11 Peter Hurford (author) says:

Since I stumbled upon this conversation, I’ll revive it with a bit more information about priors: apparently the way it can be done is to start out using calculations with a range of priors — the smallest acceptable prior and the largest acceptable prior — and calculate from there to come up with a range of probabilities based on the evidence.

Additionally, you can still use Bayes to argue that a certain thing is improbable given the evidence by using an absurdly high prior. Such as an argument that there is insufficient evidence for ghosts by starting with a 90% prior for ghosts.

12. #12 joseph says:

Thankyou for the further input,
So, without using, or even mentioning Bayes, to argue with my Grandmother, who often tells stories of witnessing ghostly presences, hearing voices, feeling cold draughts of air etc…if I start my argument with “Ghosts are ridiculous” she will typically reply “You don’t want to believe in Ghosts, you won’t even listen to the evidence”.

To put that is Bayespeakian, my prior for Ghosts is very high, and she argues that I am biased, seeking information that confirms my beliefs, disregarding that which does not.

So whilst I find Bayes great in situations where we can begin to number crunch, in fields that we can’t (and some of those aren’t necessarily even truth-apt, or falsifiable), I am left with the distinct impression that any prior I come up with is subjective. If it is subjective, then the addition of statiscal Bayesian terminology changes little, other than perhaps leaving my opponent feeling a little unbalanced. WLC seems to have perfected this technique, using Bayes to argue for the likelihood of the Ressurection of Jesus. The basis of this seems to be a much lower prior for resurrection than most agnostics will accept. I’d personally love to see the calculation behind that prior, I assume it’s God=Possible.

13. #13 Peter Hurford (author) says:

Thankyou for the further input,
So, without using, or even mentioning Bayes, to argue with my Grandmother, who often tells stories of witnessing ghostly presences, hearing voices, feeling cold draughts of air etc…if I start my argument with “Ghosts are ridiculous” she will typically reply “You don’t want to believe in Ghosts, you won’t even listen to the evidence”.

That’s an ironic retort for a Ghost advocate, but they still are using Bayesian reasoning — start with a certain guess for how likely ghosts are, and then change your mind when encountering “evidence”, such as bumps in the night.

The problem here, I suspect, is that the notion of ghosts are incoherent and thus do not explain bumps in the night, but make the question go away. Consider we have no idea how ghosts actually cause bumps and such, they just do.

This is the weird application of Bayes to entities that always cause their circumstances to obtain by definition — it’s like saying the best explanation for why my carpet is green is that God wanted the universe that way.

It is thus only the dramatic ghost stories that can really verify if ghosts exist — the kind of stories of poltergeists flinging furniture around. Only these stories have no comparable non-ghost explanation. This is exactly why Bayes not only asks for the chance of bumps in the wall given ghosts, but the chance of bumps in the wall given no ghosts.

And when we investigate poltergeist stories, we only find fakery and fabrication, either intentional or unintentional.

~

To put that is Bayespeakian, my prior for Ghosts is very high, and she argues that I am biased, seeking information that confirms my beliefs, disregarding that which does not.

In this case, your prior for ghosts must be agreed upon before you can formalize the evidence for or against ghosts. There actually are mathematically rigorous ways of doing this (Kolmogorov complexity), but they usually cannot be applied in real life. Thus we have to ball park.

Perhaps we agree that the chance ghosts exist could be no better than 25% given what we know, because the past evidence for ghosts has been shoddy and we have philosophical misgivings about the possibility of spirits, yet we’re generous.

Now that we have a prior, we can begin to calculate the evidence. This will involve more ball-parking. Once I get a better grasp on Bayes myself, I’ll explain this better.

~

So whilst I find Bayes great in situations where we can begin to number crunch, in fields that we can’t (and some of those aren’t necessarily even truth-apt, or falsifiable), I am left with the distinct impression that any prior I come up with is subjective. If it is subjective, then the addition of statistical Bayesian terminology changes little, other than perhaps leaving my opponent feeling a little unbalanced.

This isn’t true for what I said above — even if you can’t calculate a specific prior, you can at least do so within margins of error. Perhaps you say that the highest prior that could possibly be assigned to ghosts existing is 50% and the lowest is maybe 10^-10%. Then running these through Bayes, you can get high and low bars for the chance of ghosts given all the evidence considered.

~

WLC seems to have perfected this technique, using Bayes to argue for the likelihood of the Ressurection of Jesus. The basis of this seems to be a much lower prior for resurrection than most agnostics will accept. I’d personally love to see the calculation behind that prior, I assume it’s God=Possible.

Bayes Theorem at least gives a reference frame for this debate to exist, and Richard Carrier definitely will be taking it on in his upcoming two books (“Proving History: Bayes’s Theorem and the Quest for the Historical Jesus” and “On the Historicity of Jesus Christ”).

14. #14 joseph says:

This got me thinking about the differences in assigning probabilities.

Let’s call a true positive experiencing a ghost i start with a prior of 1% or less because i think it’s an illogical concept and at odds with the hard earned knowledge humanity has clawed from the gods thus far. Granny Joseph thinks it’s at least 50% because….something something….dualism magic!

False Positives; self delusions, hallucinations, etc I’d put these at something like 99% of positives, Granny Joseph at most 50% because, she can’t be suffering with a medical condition, the Ghost told her something only the dead person would know, something that turned out to be true with more knowledge, multiple attestation etc.

Have to continue this later….

15. #15 joseph says:

As for the false negatives, where a ghost was present and not detected I am going to apply a prior of at most 1%, here my Grandmother’s prior will be much higher as she would say I wrongly rationalise away both my own “evidence” of ghosts, and the accounts of others, i don’t know, maybe her prior is 15% for sake of argument.

Next we have true negatives I’d say at least 99%, but my Grandmother 85% of all negatives.

We can use Kolmogorov complexity, again dualism would enter. I’d say it’s very high, my ghost breieving Grandmother would say it’s lower than a human which represents the conplexity of consciousness plus the biological complexity of having a physical form. I can’t defend physicalism except by saying Ockham’s Razor favours it, she could always pull a CL and say that I had tried to reduce or eliminate a necessary part of the explanation.

I think we could begin to number crunch on these kind of figures, but I am left thinking that my numbers will favour my conclusions and my Grandmother’s numbers her own…we could both claim to be Bayesian…

16. #16 joseph says:

Crunching the numbers further let us say we ask 100 people if they’ve seen a ghost.
25 say yes, 75 say no.
I think, because of my priors:

True positives 0.25
False positives 24.75
True Negative 74.25
False Negatives 0.75

My Grandmother thinks:

True positives 12.5
False positives 12.5
True Negatives 63.75
False Negatives 11.25

So I am left with 1 Ghost sighting, my Granny 23.75 .

17. #17 Peter Hurford (author) says:

I think I would have to get a better grasp of the subject myself and write multiple essays on these topics before I could adequately reply to what is going on here, so the best I can do right now is say that you have brought forward an intriguing point that merits future study.

Basically, your Grandma is, as we both probably agree, acting contrary to the intended use of Bayes, but the question would be proving that she is and quantifying by how much. I’ll put writing on this further on my list of things to do in the upcoming months.

But I can offer a few introductory comments:

Granny Joseph thinks it’s at least 50% because….something something….dualism magic!

This is obviously not the way to construct a usable prior as it’s basically no better than picking a prior at random. This is specifically why we have Kolmogorov Complexity.

I think the easiest way to resolve this is to adopt the strategy Carrier uses and talk about “typicalness” for priors — it isn’t the correct answer, but it’s usually pretty close. For instance, if you’re trying to prove that ghosts exist, we can talk about how often it is that other similar entities have been shown to exist, or how often it is that ghosts have been ruled out as the explanation of other seemingly bizarre claims.

We could also use another method of approximation: the explanatory virtues I talk about in “Making the Question Go Away”, where ghosts fail on nearly all of them.

All of these would argue for a best possible prior of 10% or so, with the true prior being highly likely to be much less than that.

~

False Positives; self delusions, hallucinations, etc I’d put these at something like 99% of positives, Granny Joseph at most 50% because, she can’t be suffering with a medical condition, the Ghost told her something only the dead person would know, something that turned out to be true with more knowledge, multiple attestation etc.

Here’s another good example where we should apply some more background knowledge — sure, Granny says she doesn’t have a medical condition, but is that accurate? Did she *really* receive information that only a dead person would know, or does she just think that she did? How do we know?

All of this has to be waited based on actual fact to the best that we can tell, not just based on what the reasoner believes — these are the ways in which the reasoner can be incorrect.

Also, a statement “Ghost told her something only the dead person would know” has to be treated with caution, because this, if true, would be 100% evidence for ghosts and make the ghost hypothesis absolutely true. Instead, it has to be “Ghost told her something only has a X% chance of being something only a dead person would know”, thus allowing us to sensibly use in Bayes Theorem.

~

We can use Kolmogorov complexity, again dualism would enter. I’d say it’s very high, my ghost breieving Grandmother would say it’s lower than a human which represents the conplexity of consciousness plus the biological complexity of having a physical form.

~

I can’t defend physicalism except by saying Ockham’s Razor favours it, she could always pull a CL and say that I had tried to reduce or eliminate a necessary part of the explanation.

Except, for Bayes, the fact that physicalism is justified by Ockham’s Razor is all the defense you need to say physicalism has much lower Kolmogorov complexity, because that’s what Kolmogorov complexity means. Non-physicalism will need more evidence to overcome the handicap due to higher Kolmogorov complexity.

However, I don’t think we’ll ever get beyond the Kolmogorov complexity step because non-physicalism is currently incoherent, so we have no idea how much Kolmogorov complexity it has. Thus we’ll always need less precise guesses and best-case-scenarios.

~

I think we could begin to number crunch on these kind of figures, but I am left thinking that my numbers will favour my conclusions and my Grandmother’s numbers her own…we could both claim to be Bayesian…

And this doesn’t reflect negatively upon Bayesian reasoning either, it just means you two have different epistemic states, and thus need more evidence. Perhaps you might also need more meta-evidence, or evidence about how reliable your current evidence is.

18. #18 joseph says:

What I am really getting from all this is that Bayes is a middle step, if someone entirely disagrees with you about subjects like the acquisition of knowledge, logic, the need for falsifiability, whether dualism is likely or not, then you will not agree with them on either how to apply Bayes, or the results produced. Then you have the final step of applying what Bayes has told you. As such, while I will go on using Bayes in a clinical setting, I don’t think I can whole heartedly refer to myself as Bayesian…I am much more keen on approaching these kinds of questions in terms of whether they are logically consistent, what sorts of falsifiable hypotheses could we form based on such theories.

19. #19 joseph says:

I also have a quibble with kolmogorov complexity as sometimes simple laws can produce complexity, so while a ghost would have high complexity so do we, and yet we arise from simple rules of physics describing chemical reactions etc. A quasicrystal is more complex than a crystal, yet the laws that describe the formation of both are identical. Again, this is a question of both correct and fair application, rather than questioning the theory itself.

20. #20 joseph says:

Interestingly when I die I intend to argue against Ghosts the same way Harry Houdini did, as well as specify times, places and people for hauntings and blessings, and possibly a control group…

21. #21 Peter Hurford (author) says:

What I am really getting from all this is that Bayes is a middle step, if someone entirely disagrees with you about subjects like the acquisition of knowledge, logic, the need for falsifiability, whether dualism is likely or not, then you will not agree with them on either how to apply Bayes, or the results produced.

Sure. You have to agree to Probability Theory and such before Bayes makes sense — you probably couldn’t just give someone Bayes and expect someone to accept it. There are previous steps as you mentioned. However, this doesn’t make Bayes incorrect.

I suspect that you and I have different expectations for what Bayes is supposed to do — I expect it to be a mathematically rigorous definition of what correct reasoning looks like to the point where any deviations from Bayesian reasoning are simply wrong and provably so, whereas you seem to expect Bayes to produce magic statements that convince every onlooker to suddenly change their mind.

~

As such, while I will go on using Bayes in a clinical setting, I don’t think I can whole heartedly refer to myself as Bayesian…I am much more keen on approaching these kinds of questions in terms of whether they are logically consistent, what sorts of falsifiable hypotheses could we form based on such theories.

I don’t think Bayesian reasoning is yet ready to be used for philosophical arguments, though a few (notably Yudkowsky and Carrier) are trying to get it to be used as such — right now it is just too impractical to apply and not defined in such a way that can be understood by lay people.

I do think that Bayesian reasoning is a model of what correct deductive / inferential reasoning looks like, though. Bayesian reasoning can be correct, even if not practical.

~

I also have a quibble with kolmogorov complexity as sometimes simple laws can produce complexity, so while a ghost would have high complexity so do we, and yet we arise from simple rules of physics describing chemical reactions etc. A quasicrystal is more complex than a crystal, yet the laws that describe the formation of both are identical. Again, this is a question of both correct and fair application, rather than questioning the theory itself.

I don’t think there is actually an objection here, but rather a confusion over two different meanings of the word “complex”. For instance, the output of Conway’s Game of Life certainly looks intricate and complex, but given that it only involves four rules, it is really really simple from a Kolmogorovian standpoint.

~

Interestingly when I die I intend to argue against Ghosts the same way Harry Houdini did, as well as specify times, places and people for hauntings and blessings, and possibly a control group…

Make sure to publish your results in several academic journals, and don’t forget to collect your \$1 million from James Randi, though perhaps you might need a friend or family member to collect it on your behalf.

Being a Ghost might actually be more fun than Heaven, maybe.

~

Eventually I’ll begin a series of essays where I will expand on the claims I’ve made in this comments section.

22. #22 joseph says:

“Bayesian reasoning can be correct, even if not practical”
“…you seem to expect Bayes to produce magic statements that convince every onlooker to suddenly change their mind.”

Well I do use Bayes, so I think it is practical, at least in some situations, (and correct in all situations). So when I read Luke Muelhauser advocating Bayesian updating I am intrigued, but sceptically so, as to how this is done.

Here I’ve deliberately (unfairly) even sketched out the most troublesome kind of argument I can think of, to see how Bayes would be applied. Because there is no use using it badly. I would love it if Bayes could be used to answer philosophical questions (and magically settle disputes), and that is the impression I get from Less Wrong, but I seem to be missing something.

“but rather a confusion over two different meanings of the word “complex””

Yes, always problematic, if I ask a theist the kolmorogov complexity of God do I mean “every specific detail” or metely the “principles said God would arise from”, I suspect though that a theist would have more problems with the idea of describing a consistent God arising from basic principles than I would.

Sorry mean to stop commenting, and wait for further essays.

23. #23 joseph says:

Though I note that from Eliezer’s writings that Karl Popper’s falsificationism can be couched in terms of Bayes, in that case I think I see what they mean, but still wonder if it means doing things any differently.

24. #24 joseph says:

I went back to the problem of priors and found a few methods mentioned:

http://en.wikipedia.org/wiki/Jeffreys_prior

Seemed promising…but lack the mathematical skills to know.

25. #25 Stephen R. Diamond says:

Joseph,

The problem with the Yudkowskyite application of Bayesian decision theory is Kolmogorov Complexity and Salomonoff induction.

They’re trying to turn subjective probability into objective probability, and it can’t be done. Bayesians methods are sometimes useful, sometimes not; but any claim that Occam’s Razor is an objective standard for assigning priors is bogus; “simplicity” is relative to a language. In my opinion, Salomonoff Induction is the root of all Yudkowskian nonsense: from the Many Worlds Interpretation of qm to the Singularity.

I think you may find this interchange interesting — http://tinyurl.com/8anofto
Set the comments to “Sort by rating,” and see the first 11.

26. #26 Stephen R. Diamond says:

Oh, you might need to read or skim the OP. (You might find “Pascal’s Mugging” interesting; I consider it a reductio of Salomonoff Induction (if it even needs one — Dmitryl’s argument worth noticing in the discussion).

27. #27 Stephen R. Diamond says:

@Joseph So I am left with 1 Ghost sighting, my Granny 23.75 .

How many ghosts sightings would you have admitted without the benefit of Bayes? (I hope it’s much less than 1 :))

The moral is that Bayes is useful to the extent you have a justification for the priors. I think I can more meaningfully say there are no ghosts than I can partition my belief into priors and subsequent evidence. Bayesian decision theory doesn’t demonstrate that we are generally better at estimating plausibility by using Bayesian procedures. Often, our intermediate probabilities are very obscure; we really don’t have the slightest idea of what they are. Why should I know how much I would believe in ghosts absent evidence?

28. #28 Peter Hurford (author) says:

Seemed promising…but lack the mathematical skills to know.

I lack the mathematical knowledge to even recognize if it is promising or not.

29. #29 joseph says:

Thanks Stephen, actually I thought of posting this on your blog, but a search for Bayes didn’t find much. I have to agree with you that the selection of Salomonoff Induction seems to be pandering to the audience, programmers largely. Thankyou for the link. I suppose another point raised by my ghostly example was even if your priors sound low, Bayes will tell you that given a suitably large sample size it probably happened. So the fact that we have thousands, if not millions, of ghost sightings according to Bayes would mean 1 or 2 were probably actual ghost sightings.

Nevermind Peter Hurford, thanks for having a look.

I also stumbled on another of Russell’s quandries, which was something like how to most randomly draw a line across a circle.

30. #30 Stephen R. Diamond says:

I suppose another point raised by my ghostly example was even if your priors sound low, Bayes will tell you that given a suitably large sample size it probably happened. So the fact that we have thousands, if not millions, of ghost sightings according to Bayes would mean 1 or 2 were probably actual ghost sightings.

A little bit of care is required here. You can only use the “multiplication rule” to determine expected frequency of sightings from reported sightings if each reported sighting is a statistically independent event, which isn’t the case with ghost sightings (whether due to fads or culture).

31. #31 joseph says:

if each reported sighting is a statistically independent event, which isn’t the case with ghost sightings (whether due to fads or culture)

Couldn’t a somewhat post-modernist comment be made here that things like the result of a mammogram are not entirely statiscally independent events (and are influenced by fads and culture)….?

32. #32 joseph says:

These were another two articles that interested me, the second is even semi-understandable:

http://en.wikipedia.org/wiki/Principle_of_maximum_entropy