This seems to be just another “reading comprehension” issue. Obviously one of the ways for A to affect B is for it to affect the existence of B. If you look at the previous post from the one that I linked, you’ll see that the initial concern that lead me to the calculation was the “Back to the Future” movie series, where Marty McFly going back in time affects (that is, affects the existence of) himself.
Not that I’m surprised that something like this came up. This is what comes of getting ahead of yourself. In a previous post I advised you that you first needed to understand the “almost surely” issue of P(H) and P(E|H), in order to understand the “spiders on mars” calculation.
In your last post on that “almost surely” issue, you seemed to recognize the mistake that you’ve made, but it was surrounded by so much nonsense that it was hard to know for sure. So let’s start there. Do you acknowledge that P(H) is almost sure not equal to P(H|E), for any given H and E?
I promise you that, starting from here, you’ll come to a much more accurate understanding of what “evidence” is. You’ll actually even come to understand why “A affecting B” in general must mean that A is evidence on whether B exists.
So, again, here’s the starting point question: Do you acknowledge that P(H) is almost sure not equal to P(H|E)?
When you’re replying to someone, click the “reply” button directly below that post, not the one at the bottom of the page.
No. I deny that your geometric “proof” of this is relevant to the subject. You can use the same system to prove that any supposed ratio between P(H) and P(H|E) is also almost certainly wrong. In fact any particular value of P(H) is almost certaingly wrong too (has a probability of 0). You need to stop with the condescension.
Nor do you seem to realize that “affect the existence of B” is different from “affect the probability that B exists”. The first is about actual presence in the world, while the second is about our knowledge.
Thanks for the tip about the “reply” button. I will try to use one or the other as is appropriate.
I didn’t ask whether you thought my statement was relevant. I asked whether you acknowledged it was true.
I assure you that it’s extremely relevant, to the point of it being nearly the entire mathematical content of my statement about evidence. If you scroll up you’ll see that it’s nearly the first thing I said about it. But let’s take this one step at a time. For now, we don’t need to discuss anything about its “relevance”. A simple acknowledgement of a mathematical truth will suffice.
Yes. All of those statements are true, because the “system” I’m using is true. Your statement is confusing here, because you’re stating these like you expect these to be false, like you’re offering these as a proof by contradiction or a Reductio ad absurdum. But they’re all true statements.
Not that it matters - these are only miscellaneous truths. The important one is the one that I asked you to acknowledge, that P(H) is almost sure not equal to P(H|E).
This is merely a statement about what a Bayesian probability even means, and how numbers can be used to represent truth statements about the world. I’ll be glad to go over all of that with you, once we get the foundational math established (but you can read more about it here). So for now, once again, all you need to do is acknowledge a simple mathematical truth: that P(H) is almost sure not equal to P(H|E).
It’s true that if you pick a random point in the plane you define, it has a probability of zero of being on the line y=x. But of course you can’t pick a random point on a plane, and that’s not a good model of probabilities. So in fact the important question isn’t about truth but about relevance.
What does it mean for a system to be true?
And yet, when you have values for P(H) and P(H|E), they will indeed fall on one of those lines despite the probability being zero. And it may be that probabilities are not drawn from a uniform distribution by throwing imaginary darts. True as your model may be, whatever that means, I deny that it’s a valid model of the reality you claim to represent. And that’s my point.
But thanks for being somewhat less condescending this time. One hopes it was intentional.
Why, of course you’ll get an irrelevant statement if you add irrelevant things to it yourself. Did you think that the idea of “almost surely” only applies to a game of darts played on a square board?
“Pick a random point”? This is irrelevant. Of course, for any specific H and E, P(H) and P(H|E) will not be “random”. That doesn’t change the conclusion that these will almost surely not be equal.
“In the plane you define”? This, too, is irrelevant. You can choose to shape the space parameterized by P(H) and P(H|E) however you’d like. Of course, it’s convenient to choose to do so in the shape of a square, but that’s for your own ease of visualization. It doesn’t change the conclusion that these will almost surely not be equal.
“Uniform distribution”? This, too, is irrelevant. It was provided only for your own ease of understanding in the darts illustration, but is not required. Any distribution that spans the space will yield the same conclusion, that P(H) and P(H|E) will almost surely not be equal.
Yes! That is in fact the very point of the “almost surely” concept! Any specific value combination of P(H) and P(H|E) has the probability of zero, despite the fact you can specify it. If you didn’t understand that from the Wikipedia article then you’ve missed its entire point. I suggest you carefully read it again.
Look, the mathematical truth I’m trying to communicate is so incredibly simple. Here it is, without your irrelevant addendums:
P(H) is a real number between 0 and 1.
P(E|H) is a real number between 0 and 1.
Then, given the properties of real numbers, P(H) is almost sure not equal to P(H|E), for any given H and E.
Genesis 2:7 says Adam was created from “dust” - ie, inanimate matter. Genesis 3:19 rules out the possibility that “dust” can be interpreted as a living organism:
"By the sweat of your face you shall eat bread
Till you return to the ground,
For out of it you were taken;
For dust you are,
And to dust you shall return.”
Now there I disagree entirely. Your dart analogy assumed a uniform distribution, and you never specified anything different. Anyway, there could be a probability distribution in which a majority of the distribution is smack on the line. It seems to me that such a distribution wouldn’t be continuous, but why must we assume a continuous distribution?
Sure, it’s a mathematical truth within the bounds of the particular model you specified. But does it have anything to do with the real world? You haven’t shown that at all. I flip a coin; that’s. What’s the probability that it’s heads? That’s P(H). If I tell you that there’s a wombat in Australia named Gus, that’s E. So now what’s P(H|E)? Show your work.
All animals are made from dust (that is, inorganic ingredients.) We know this from science and Bible readers know this from Eccl. 3:19-20.
Science and Genesis 2 agree: humans ultimately come from dust. And when we eat food, we are eating what came from dust/soil (whether directly from plants or by animals which ate plants.)
@Edgar , I agree with your line of thought, and favor the interpretation that Adam and Eve were created de novo. But it’s hard to completely rule out the possibility of other interpretations, like the one that @AllenWitmerMiller explains in his post.
In the original article, we try to cover as much space as possible, even including the hypothesis that Adam and Eve did not exist at all. But yeah, I think there’s much work to be done in narrowing things down more, without having to make absolute statements. I certainly have my specific interpretations, as I mentioned earlier in this thread.
Ah - so you did think the idea of “almost surely” only applied to a game of darts played on a square board.
I sincerely hope this was yet another issue with reading comprehension. The other possibility - that this mistake was caused by your inability or unwillingness to abstract from the specific to the general - is much harder to remedy.
But, assuming that this was a reading comprehension issue, let’s go over the “almost surely” article again. You see that the “Throwing a dart” section is under “Illustrative examples”? That means that that this is only one specific example, constructed specifically to help you understand a much broader principle.
The broader principle applies to our discussion of P(H) and P(H|E), because it maps directly to the dart example simply by the fact that P(H) and P(H|E) are real numbers between 0 and 1. This does NOT mean that probabilities are determined by throwing darts on a board, or that the application itself must involve darts or squares. This requires abstraction: since you’ve accepted the darts case, you should also accept any case that maps to it with an uncontroversial mapping (namely, that probabilities are real numbers between 0 and 1).
This is why I said near the beginning that “this conclusion relies only on the infinite cardinality of the real numbers, which all probability values must be.” So, once again, the truth that I want you to acknowledge is as follows:
P(H) is a real number between 0 and 1.
P(E|H) is a real number between 0 and 1.
Then, given the properties of real numbers, with no addition of any “particular model”,
P(H) is almost sure not equal to P(H|E), for any given H and E.
I’d be happy to discuss with you how probabilities apply to the real world afterwards, but let us first establish the mathematical foundations. I’d even be happy to discuss the “gus” problem - although you may abstract my answer from the “water molecule on Andromeda galaxy” article I linked earlier.
If you’re not going to read what I write, why bother responding?
That just doesn’t follow. Given the properties of real numbers we can say nothing about this. You need to add assumptions, perhaps that P(H) and P(H|E) are independently chosen from a particular distribution. But this might be clearer if you could address a real-world example. Why not try Gus? The water molecule in Andromeda is not relevant, as it isn’t about probabilities but about physical effects.
But do you not see that you’re just repeating your error at a higher level?
Look, you can overcomplicate the problem by insisting that I need a particular distribution. You rightly point out that it’s possible for this distribution to have all of its probability mass along the y=x diagonal. Fine, let’s say that I go with that.
But for every such degenerate pdf you specify, I can name infinitely more pdfs that do not have any probability mass along the diagonal, just by, say, simply smearing the pdf along the x direction. In other words, such degenerate pdfs are ALMOST SURELY not the “right” pdfs in any particular application (remember, “any given H and E”), just as ALMOST SURELY as P(H|E) and P(H) are not equal, and for the same reason: there are infinitely many alternatives. In fact, the cardinality of such pdfs is actually larger than the cardinality of the real numbers.
So, your pdf being the “right” one for any given H and E has a probability of zero, even if you can specify it. Just as P(H|E) and P(H) being equal has a probability of zero, even if you can specify it. This is, again, the exact idea behind “almost surely”.
A similar construction exists for the “gus” problem, although there the analogy is inexact because the problem is underspecified.
Or, instead of overcomplicating the problem, you can just acknowledge the simple truth:
P(H) is a real number between 0 and 1.
P(E|H) is a real number between 0 and 1.
Then, given the properties of real numbers,
P(H) is almost sure not equal to P(H|E), for any given H and E.
Technically, you’re incapable of naming infinitely many anything. But what does it matter? The distribution has to be chosen to match the phenomenon, and it doesn’t matter how many or few matching distributions there are. You’re mirroring your central error, claiming that the distribution, like the probabilities themselves, is chosen from a distribution that you have just specified, more or less.
Only if we’re picking a distribution at random from a distribution of distributions. But that’s a crap model of how we should do things.
Why not try it? I would claim that P(H|E), the probability of getting heads on a coin flip given that there’s a wombat in Australia named Gus, is exactly the same as P(H), 1/2. And I would further claim that P(E|H) is the same as P(E) in this case. Of course, by Bayes’ theorem, either both those claims are true or neither is. But go ahead and show why I’m wrong.
Your syllogism is missing a bit. “Given the properties of real numbers” doesn’t make the connection between the premises and the conclusion that you imagine it does. Perhaps if you specified the properties you mean? You are implicitly assuming that probabilities are randomly chosen from some continuous distribution; but they aren’t. Further, you are denying that any two events can be independent; if that’s the case, much of probability theory must be abandoned.
Well, I see that we’ve reached the “no U!” stage of the discussion. But it’s not hard to resolve this issue.
For P(H) != P(H|E), the probabilities themselves do NOT need to come from a specific distribution. That is why I did not specify any. It was you that specified the “probability along diagonal” distribution, and I’ve shown, through explicit construction, that there are infinitely many other alternative distributions which end with P(H) != P(H|E).
In other words, P(H) and P(H|E) can come from comes ANY distribution, and the distribution itself will almost surely not be one of the degenerate cases, so the probabilities will almost surely not be equal.
So then you say that I’m “picking a distribution at random from a distribution of distributions”. Again, not at all. It’s you that needs a specific distribution of distributions, to get P(H) = P(H|E). For me, I can have ANY distribution of distributions, and 100% of the time (that is, “almost surely”) we will end up with P(H) != P(H|E).
You can continue to carry this out to “distribution of distribution of distributions” and on to infinity, but it will end the same way. You’ll need something with infinite odds against you. I can choose anything, and be right 100% of the time. That’s why I agree with you that this is “a crap model of how we should do things”. What we should have done is simply acknowledged the simple mathematical truth I explained from the beginning, that P(H) is almost sure not equal to P(H|E), for any given H and E.
To recap:
You need an infinitely specific value of P(H|E) - one that is exactly equal to P(H).
I can have them be anything, and they will almost surely not equal one another.
You need an infinitely specific distribution of P(H|E) and P(H), one with its probability mass concentrated on the diagonal.
I can have any distribution, and they will almost surely result in P(H|E) != P(H).
You need an infinitely specific “distribution of distributions”, one which picks out the diagonal distributions.
I can have any distribution of distributions, and they will almost surely result in P(H|E) != P(H).
So on and so forth, up to infinity.
In fact, the cardinality of the infinities in the space you require increases each time. This is not surprising. It’s like when someone gets caught in a falsehood, and has to postulate some unlikely “what if” scenario to justify their position. Then when they’re caught on that, they repeat the process, but the second “what if” has to be even more fantastical than the first. That’s the scenario we’re in - except instead of “unlikely”, it’s actually just a probability value of zero. And you’re hanging your hope on a infinite string of such zero-probability events.
So much for the specific distributions you brought up. Next, let’s talk about another specific issue you want to consider, the problem of “Gus”. I can merely note that since this is only one specific problem, I can ignore it in the space of “any given H and E”, even if it turns out that P(H|E) turns out to be equal to P(H) - yet another application of “almost surely”. But let’s go ahead and tackle it anyway. So you think P(H) = 1/2?
The assumption there, upon which “almost surely” depends, is that the probability of a match is zero. But in order for that to be true the probabilities must be randomly sampled from a continuous distribution. You have to show that this is the way in which probabilities arise. I deny that they do. In order for the distribution of probabilities not to be of the sort I envisage, in which points on the diagonal are occur often, you have to argue that such distributions are an infinitesimal proportion of all distributions and that in addition the distribution is chosen randomly from among that infinite set. I claim that it isn’t.
You think it isn’t? It’s just the canonical probability of heads in a coin flip. I suppose you will claim that every wombat has some influence on every event in the universe. But does the fact that his name is Gus have any influence? Also, does the fact that his name is Gus depend in any way on whether my coin comes up heads? If not, Bayes’ theorem must be discarded.
P(A ∩ B) = P(A) · P(B), then A and B are independent?
Such that when A and B are independent:
P(A | B) = P(A)
If P(A | B) ≠ P(A), as @naclhv initially stated, then he is claiming A is conditional upon B–which can be tested.
If the conditional probability is going to be used to justify some proposition, the assumptions of dependence should be met first–which is what @John_Harshman is correctly arguing for.
@naclhv is essentially smuggling in the assumption of dependence for any H and E. It’s not a justified assumption in this case.
According to Genesis 1, God said “Let the earth bring forth” all the non-human creatures, an ambiguous use of words which could leave room for some kind of evolution. But It seems to me that the gist of Gen 2:7 is different; it conveys a definitive idea - that Adam was inanimate matter until God breathed life into him. There was no life … then there was life - which is vastly different to the evolutionist interpretation - there was life … then there was more life.
And there is no getting away from the implications of Gen 3:19 - “dust” is dead matter; it cannot possibly refer to a living organism.
Furthermore, evolution cannot account for the immense gap between humans and non-humans, so it makes scientific sense that humans were created separately from non-humans.
I think Scripture supports this idea - apart from Gen 2:7, we are told humans were made in the image of God - ie, not in the image of some sort of primate.
The Genealogical Adam and Eve concept addresses these issues. And there is no conflict between the evolution of Homo sapiens and all other species (a history which is clearly revealed in God’s creation) and a de nova Adam and Eve created from dust as described in Genesis. All humans today can be descendants of Adam and Eve as well as descendants of ancient primate ancestors.
Sounds like the Argument from Personal Incredulity fallacy. And a declaration of personal opinion is not evidence.
Then describe how one can use scientific methods to rule out the evolution of humans.
After some 2000 years of debates, theologians still are not in total agreement on the meaning of the Imago Dei—but virtually all concur that it is not focused on genetic factors and physical characteristics per se. Instead, they list “possessing a soul”, having a human will, being inherently self-consciousness of God, and having the ability to have a relationship with God.
I don’t know your definition of the image of a primate but humans ARE primates----as the Christian taxonomist Karl Linnaeus established long before Darwin. For centuries now Christians have recognized that humans can have primate morphology while still being described theologically as bearing the image of God. Or do you follow the traditions of some who assume that “the image of God” refers to humans having physical bodies just like they imagine God to have? (No. God is not a biological organism with a physical body.) So don’t assume a false dichotomy where an evolved human somehow can be created in the Image of God.
Yes, all animals and plants are made from dust. That is entirely compatible with the idea of God creating all living things. All biological life comes from the dust and the Bible makes the same claim.
… which unfortunately represents a blatant denial of the implications of Gen 3:19. This is what theistic evolutionists are forced to do - deny Scripture.
You don’t agree with my claim that evolution cannot account the immense gap between humans and non-humans? Sounds like the Argument from Personal Incredulity fallacy.
Interpreting the evidence to conclude that Adam evolved from a primate is ultimately a personal opinion.
How many genes are there in your “immense gap,” Ed? Can you identify 10 genes of the ~20000 that chimps have but humans don’t (i.e., no orthodox), and vice versa?
Can you do the same for mice and humans?
The real evidence might make you rethink some things.