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Sep 27 2013

# Want to Literally Wear Bayes’ Theorem?

Surly Amy has kindly met my request to create a SurlyRamic of Bayes’ Theorem. I designed the graphic for her, and she has made the product. You can check it out here, and buy one if you are keen. In the interests of art (to make it look elegant and not a busy mess), I took two liberties: I didn’t put the two expressions in the denominator inside brackets, but just stacked them on either side of a plus sign to indicate that (obviously) the multiplications have to be completed before the addition. I also left out the variable b for background knowledge, though that is commonly done even by mathematicians. You should understand that it’s present in every single term (see my Bayesian Calculator for an explanation of this and the rest of the equation). For example, P(h|e) represents P(h|e & b) and P(h) represents P(h|b), and so on.

Now we can totally geek out the Bayesians.

1. 1
##### SallyStrange

Yes. Yes I would like to wear Bayes’ theorem. There’ve gotta be t-shirts, right?

1. 1.1
##### Richard Carrier

(I’ll get some more options up there soon. The art is mine but Amy has non-exclusive rights to it so she can use it freely on her jewelry, which is what I was blogging about. But why not a shirt? Great idea! So I re-opened my old Cafe Press store and put one in. I have had plans to put more stuff in there anyway. And as a bonus, of these t-shirt sales I get a cut.)

2. 2
##### Jason Goertzen

This is awesome!

3. 3
##### Gabriel

This is super cool. I feel like I would get beaten up for wearing it, but that it would be OK, because I would be wearing Bayes’ Theorem.

4. 4
##### Gary Slabaugh

I apologize up front for being a novice in the blogosphere and not knowing any better on where to start a thread. I admire your work. You seem to be onto something quite important, viz the integration of objective reality (an assumption of the scientific method) with historicity using Bates theorem. I have confidence (faith?) that the hypothesis of historical probability can be equally valid to scientific probability will be elevated to sound theory. Time will tell. My concern is in another (inter-related?) area, namely how to simply and concisely compare and contrast linguistically and epistemologically the concepts of “reality” and “probability.” How can there be any real common ground between two world views who define such basic assumptions so differently? Case in point: What constitutes evidence? Naturalism defines logos, rhema, reality, evidence, perhaps even probability and faith so differently than the definitions made by biblical literalist. From the New Oxford Annotated Bible, NRSV 4th ed. – Heb 11:1 “Now faith is the assurance of things hoped for, the conviction of things not seen.” followed by this commentary – ” The Gk word translated assurance has an objective quality and might be translated “reality.” Likewise conviction has the connotation of “evidence.” The people of faith who follow all possess insight into the reality of the invisible divine world.” I cannot fathom any common ground epistemologically between this “faith” and rationality. Can you enlighten me? Perhaps this is an old problem that is beyond the scope of logical argumentation.

1. 4.1
##### Richard Carrier

…the hypothesis of historical probability can be equally valid to scientific probability will be elevated to sound theory.

Note that this is very much not what I argue (in Proving History I am explicit that history cannot achieve scientific levels of certainty in most things and that science in fact differs from history in only that one single respect: the certainty with which its conclusions can be held).

How can there be any real common ground between two world views who define such basic assumptions so differently?

This question is confusing, because I don’t know which “worldviews” you are comparing here. History and science are not separate worldviews. They are parts of the same worldview (whichever one you mean to be talking about). I think from your subsequent question you mean to compare naturalism to biblical literalism, although it is unclear why the latter (what about non-literalist Christianity?).

What constitutes evidence?

Quite simply: anything that increases the probability of a hypothesis h over all competing hypotheses. Which all comes down to raw uninterpreted experiential data, from which we build and test hypotheses about what is causing that data and how it behaves (see my discussion in Epistemological End Game and, I think more importantly for you, Defending Naturalism as a Worldview).

Naturalism defines logos, rhema, reality, evidence, perhaps even probability and faith so differently than the definitions made by biblical literalist.

I’m not sure they actually do.

Your example, for instance, is not a disagreement on definitions, but on what causes certain evidence. To believe faith is evidence of something is simply to believe (falsely) that having faith in a hypothesis makes that hypothesis more likely to be true. It is fairly easy to demonstrate that that’s false, using evidence and reasoning even the literalist accepts. The question is whether they will accept the consequences of this, or go the other way to solve their cognitive dissonance by simply denying the evidence or even the applicability of logic. And that’s a delusion, not a difference of definitions.

5. 5
##### Michael Sommers

You still need the brackets. It looks like P(h|e) = (P(e) P(e|h)) / (P(h) P(e|h)) + P(~h) P(e|~h)

1. 5.1
##### Richard Carrier

It’s art. What is “needed” is not a meaningful statement in that context. Brackets would make the image too cluttered and ugly. The wearer will certainly understand the notation and can explain it.

Indeed, anyone sharp enough to use the equation need only ask themselves why not just write P(h|e) = 1 + P(~h) P(e|~h), since that is what the equation would be saying if you read it the way you suggest. Logic dictates that that can’t be what it is saying, and therefore the terms under the divider are meant to be added. This is even clearer when you realize what we are seeing is a single fraction with a single divider (everything above is the numerator, everything below is the denominator), not a fraction separated from another expression. That’s how art works.

6. 6
##### lpetrich

I think that I would have preferred the odds form:

P(h|e) / P(~h/e) = P(h) / P(~h) * P(e|h) / P(e|~h)

I find it a bit easier to use for qualitative applications.

1. 6.1
##### Richard Carrier

Indeed. I agree that form is useful (I use it in my next book), but in my experience teaching, people unfamiliar with Bayes’ Theorem have a hard time understanding it (you can teach them to use it, but they won’t know why it works; most also don’t know how to convert the odds result back into a probability), and it really just conceals the core equation (which, unlike the odds form, is more explicitly an expression in the law of total probability), so looking at the full equation is looking at its full glory. The odds form is a stripped down emanation.

I also had trouble creating a graphic of the Odds Form that worked. It’s actually longer, and thus I feared it wouldn’t fit on a SurlyRamic unless shrunk to the point of being hardly visible at anything more than an intimate distance. I couldn’t find a font skinny enough to make it work. Whereas I found the long form can be fit if I stack it unusually like I did. I just went back and did some comparisons, though, and the length difference is small enough that I don’t think this would matter.

7. 7
##### orestes6

Reply to 4.1 Sorry that my pasting of quotes is sad. You said in “Proving History” I am explicit that history cannot achieve scientific levels of certainty in most things and that science in fact differs from history in only that one single respect: the certainty with which its conclusions can be held

I apologize for the error that historical uncertainty could approach scientific uncertainty. (That the application of Bayes’ theorem to historical uncertainty could someday result in the genuine confirmation of historical truths is probably too idealistic.)

The two worldviews to which I meant to refer were not science and history (apology for the confusion) but the worldviews of metaphysical naturalism (which is thoroughly validated) with a faith based world view. (NB, I am not advocating for the second worldview- just questioning it’s assumptions and seeking a deeper understanding, the meaning and the validity of opposing assumptions.) I meant to articulate that, as a tautology, faith could be defined biblically as “the reality of things hoped for; the evidence of things not seen” – reinforcing the concept of “people of faith …possess insight into the reality of the invisible divine world.” I suppose it’s a value judgement to dismiss an alternative worldview altogether in favor of one’s own vs seeking deeper insights into the assumptions that underlie alternative worldviews from one’s own.

What are the assumptions that make up a faith based world view? Do the primary sources for alternative worldviews teach those assumptions? My example was attempting (poorly?) to describe faith as an ontology and as “raw uninterpreted experiential data” at least from the perspective of the author of Hebrews and those who agree with that perspective. (Again, I am not advocating for that perspective, just evaluating it – having already validated to a small degree of uncertainty the truth of metaphysical naturalism.) Is that what the author really meant in order to define faith? Is faith still defined that way? What do the words really mean? Is it valid? defensible? Ought it be summarily discarded as delusional because it claims evidence for invisible sky fairy ordered worlds prepared by the word of said sky fairy so that what is seen was not made out of visible things.

Dr Carrier, is it not a rather profound question to ask “What causes certain evidence?” It takes a courageous intelligence to ask insightful questions. For what it’s worth, I believe that you have both courage and high, functional intelligence.

What causes raw uninterpreted experiential data? Nature causes the stimuli and nature causes the physical organs receiving the raw stimuli. Sensation. Consciousness? I believe I am an observer, therefore I observe. How can I be an actively involved observer if I don’t believe my observations are real? That the only real things are Platonic forms or Kantian noumena. Observations cannot be trusted as not being illusions or delusions without being proven.Nothing can be one hundred percent proven. Nihilism. An alternative: Where is the proof or the high probability almost to certainty that observations are real? Consensus. What about other functions than sensation? Less consensus, higher probability of illusion or delusion. Like the “flash of genius” or the “one per cent inspiration.” Can that still be defined as raw uninterpreted experiential data? How about non logical intuitive heuristic – a discovery of previously undiscovered raw uninterpreted experiential data point? As long as it’s falsifiable and confirmed by repeated scientific inference, is not the initial discovery the first evidence? A disciplined feeling function able to “hear” the “sound of sheer silence”? Evidence? Insanity?

Dr Carrier, I hope you don’t perceive me to be a jerk, but as a seeker. I am simply asking if the robustness of Bayesianism, probability theory, metaphysical naturalism, logic, and the rules for evidence really allow for raw uninterpreted experiential data outside the realm of physical sensation?

1. 7.1
##### Richard Carrier

That the application of Bayes’ theorem to historical uncertainty could someday result in the genuine confirmation of historical truths is probably too idealistic.

I don’t know what this sentence means. What is “genuine confirmation”? Are you saying no historical knowledge exists? That no historical truth can be authentically confirmed…not even that the Holocaust happened?

I am assuming that’s not what you meant. But the opposite of genuine is inauthentic/fake, so if no historical truth can be genuinely confirmed, then there is no historical truth. So you must mean by “genuine” something more than what we already do now to confirm what’s true about history. And I cannot guess what that is.

I meant to articulate that, as a tautology, faith could be defined biblically as “the reality of things hoped for; the evidence of things not seen” – reinforcing the concept of “people of faith …possess insight into the reality of the invisible divine world.” I suppose it’s a value judgement to dismiss an alternative worldview altogether in favor of one’s own vs seeking deeper insights into the assumptions that underlie alternative worldviews from one’s own.

That would be a bad epistemology. It can easily be demonstrated to be self-defeating. So if anyone comes at you denying facts using that argument, attack the defects in their epistemology first.

Is that what the author really meant in order to define faith? Is faith still defined that way? What do the words really mean? Is it valid? defensible?

Now you seem to be asking what Paul (the author of that text two thousand years ago) meant, which is an entirely different question (I don’t imagine it’s likely we will ever be arguing with people who have been dead for two thousand years). The answer to that question I give in detail in Not the Impossible Faith, pp. 236-40 (and on the defects in such an epistemology and its effect on the success of Christianity, see ibid. chs. 7, 10, 13, and 17).

Ought it be summarily discarded as delusional because it claims evidence for invisible sky fairy ordered worlds prepared by the word of said sky fairy so that what is seen was not made out of visible things.

Yes.

Although we still have to deal with the deluded, and the damage they do to society. So though the nonsense itself is summarily discarded (i.e. we needn’t spend any more time considering whether to adopt it ourselves), the belief system still has to be dismantled and discredited, and we have to find practical ways to work with and live with the delusional people among us.

For how we should explore the ontological and epistemological questions you asked, see my book Sense and Goodness without God, which deals with these matters extensively.

I am simply asking if the robustness of Bayesianism, probability theory, metaphysical naturalism, logic, and the rules for evidence really allow for raw uninterpreted experiential data outside the realm of physical sensation?

I don’t understand the question. What do you mean by “raw uninterpreted experiential data outside the realm of physical sensation”? Raw uninterpreted experiential data just is. Whether it is inside or outside “the realm of physical sensation,” indeed whether there even is a “the realm of physical sensation,” is what one then attempts to probabilistically infer from the raw data, and as such can only ever be an inference from that data, not a part of it. This is equally true of a God, incidentally–who can never be certain he is actually omniscient rather than merely fooled into believing he is, and who can never know anything except through the mediation of raw uninterpreted experiential data, which he can never know for certain is coming from anywhere. This is simply an innate reality for all logically possible conscious agents.

Bayes’ Theorem describes the only logically correct way to draw inferences from this (or any) data. All other methods are just disguised forms of Bayesian reasoning, or logically invalid. I demonstrate this in Proving History, pp. 106-14 (that the conclusion there will extend even to mathematical truths one can deduce from the point made on pp. 23-26 and in the endnotes related there).

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