Similarly, surely if every raven is black, then the chances of any raven you encounter being black is 100% (minus albinos, or whatever). It can only be otherwise if there is a chance that the ravens could be any other color, but what kind of meaningful "probability" could this be other than just your own disposition of confidence in the face of not knowing an important variable? — Wosret
If we're talking about the probability of him having this or that card, based on what we know, then we're really just saying that we know of no biases, or reasons to go one way or the other, so all things being equal, any guess is as good as any other. — Wosret
I have some questions about this. I don't see how H (hypothesis) logically implies E (evidence). — Arkady
Also, in order for the posterior probabilities not to matter here (because E cancels out), H and K must somehow imply the same "E". But, how can a hypothesis and its negation imply the same observational consequences? — Arkady
Also, the construction P(H&E|B) to describe confirmation seems odd. Wouldn't P(H|E&B) be more appropriate? — Arkady
My problem here is that I don't see how H logically implies E. Setting aside the propositional variables for a moment, I don't understand how this particular H ("all ravens are black") implies this particular E ("the sighting of another raven"). Again, I'm not sure what the E statement even means here.Well spotted!
A couple of things:
The truth value of H and HE are the same, because H logically implies E
p(HE) = p(H)*p(E|H)
The probability of E given that H is in fact true is 1, because H logically implies E.
So by Bayes theorem
p(HE|B) = p(B|HE)P(HE)/P(B)
= p(B|H)p(H)/p(B) = p(H|B) — tom
K is a particular hypothesis. To say that a particular hypothesis is "compatible with any evidence" means that there is nothing which can falsify K, even in principle. This is no bueno for a purportedly scientific/empirical hypothesis.K is compatible with any evidence. p(E|K) is still 1, and (KE) and (K) have the same truth value. I certainly seems weaker, but I can see no reason that K does not logically imply E, just as it implies not(E).
Yes, I saw that. I wasn't reading your post correctly; the ordering threw me, which is why I deleted that paragraph from my post (seemingly long before you replied to it; not sure if the forum software is getting glitchy here). But, I agreed with your ultimate presentation of the logic of confirmation, i.e. P(H|E&B), so we're good on this point.That's in all the equations!
Do you believe that an agent can have better or worse reasons for increasing or decreasing his confidence in a given hypothesis in the face of new evidence? That is, some types of evidence are "better" or "worse" than others? — Arkady
While for the frequentist a hypothesis is a proposition (which must be either true or false), so that the frequentist probability of a hypothesis is either 0 or 1, in Bayesian statistics the probability that can be assigned to a hypothesis can also be in a range from 0 to 1 if the truth value is uncertain.
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Broadly speaking, there are two interpretations on Bayesian probability. For objectivists, probability objectively measures the plausibility of propositions, i.e., probability corresponds to a reasonable belief everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.
In the frequentist interpretation, probabilities are discussed only when dealing with well-defined random experiments (or random samples) ... For any given event, only one of two possibilities may hold: it occurs or it does not. The relative frequency of occurrence of an event, observed in a number of repetitions of the experiment, is a measure of the probability of that event ... A claim of the frequentist approach is that in the "long run," as the number of trials approaches infinity, the relative frequency will converge exactly to the true probability ...
The frequentist interpretation is a philosophical approach to the definition and use of probabilities; it is one of several such approaches. It does not claim to capture all connotations of the concept 'probable' in colloquial speech of natural languages. As an interpretation, it is not in conflict with the mathematical axiomatization of probability theory; rather, it provides guidance for how to apply mathematical probability theory to real-world situations.
Personally, I think that even the "objective" Bayesian approach is fundamentally subjective, because it claims to measure confidence, degree of belief, or plausibility, all of which will vary from person to person. — aletheist
Then what about my example of the eggs? It certainly seems to make use of objective values and so won't vary from person to person, even though it's about confidence/plausibility. — Michael
I raised no objections to what you said in the post that you linked about the probability of all ten eggs being white before they are produced; or even the probability of all ten eggs being white after eight of them have already come out white, but before the other two eggs are produced. My objection was to your claim that the same reasoning still applies after all ten eggs have been produced. At that point, either all ten eggs are white (p=1) or at least one of them is non-white (p=0). — aletheist
If so then I think that the last issue of contention is my claim that evidence is anything that increases the objective Bayesian probability that a hypothesis is true. But then what else does it mean to count as evidence, I wonder? — Michael
Lots of different people have lots of different ways of counting something as evidence, and then weighing it with other evidence. Very few people have any idea what "objective Bayesian probability" is, let alone how to use it as a tool for mathematically gauging their confidence or degree of belief. — aletheist
I'm use Bayesian probability, as just mentioned. What I'm saying is that, given it's using objective values, it isn't something that will vary from person to person as you suggest. Hence it being objective Bayesian probability (as I understand it). — Michael
My point is that someone's actual confidence or degree of belief is not objectively measurable. — aletheist
Again, most people will never do this calculation, and only those who agree with its underlying assumptions will attribute any validity to it at all.
But their actual confidence isn't relevant, as we're considering objective Bayesian probability ... each observation of a white egg is evidence that every egg is white. — Michael
And feel free to replace "probability" with "plausibility" if it really matters that much to you. — Michael
Your premiss here is that anything that increases the Bayesian plausibility of a proposition should count as evidence for its truth. Only someone who agrees with this - i.e., attributes validity to it - will actually be more confident about the truth of a proposition by virtue of such calculations. Your original example, where the observation a green apple somehow counts as evidence that all ravens are black, demonstrates the implausibility of this whole approach. — aletheist
A thing can be evidence even if it isn't taken to be. That's why "ignoring evidence" is a thing. — Michael
Again, there's the example of the eggs I gave earlier. It's an application of Bayesian probability (as I understand it), but it has nothing to do with how confident any particular person is. — Michael
Is there some objective standard that dictates which of them is right? — aletheist
Bayesian plausibility somehow counts the observation of a green apple as evidence that all ravens are black, but I do not. I have other criteria for something to count as evidence, including relevance.
Huh? Once the entire batch of ten eggs is produced, your calculations are only about how confident you are that all of them are white, as you observe them one by one. Again, objectively, either all of them white (p=1) or at least one of them is non-white (p=0).
If green apples are relevant to the proposition "if something isn't black then it isn't a raven" then it's relevant to the proposition "if something is a raven then it is black". — Michael
Again, the maths I used in the example shows that the Bayesian probability of every egg being white just is 0.510, and this is true even if I believe otherwise (hence the objectivity). — Michael
So, you are in essence saying that the same evidence would confirm "all ravens are black" as would confirm "not all ravens are black") (i.e. H and K, respectively, which constitute the hypothesis and its negation). I don't see how that could possibly be the case. — Arkady
I do not consider the observation of a green apple to be relevant to either of these propositions. Bayesian plausibility somehow counts it as evidence that all non-black things are non-ravens, but I do not - in this case, mainly because of the sheer number of non-black things. Even if we limit it to actual non-black things and assume that the quantity is finite, it will still be so large that the change in your calculated value is vanishingly small - certainly not sufficient to count as genuine evidence in my book. — aletheist
Again, this is not yet Bayesian plausibility, it is frequentist probability - the proportion of infinitely many ten-egg batches that would consist entirely of white eggs. It only becomes Bayesian plausibility when you claim that the increase to 0.59 when you observe that the first egg is white increases your confidence that all of the eggs in this particular batch are white.
If the principle holds when there are just two things to consider then it holds when there are a trillion things to consider. — Michael
And if you need the change to be sufficiently large enough then you just need to check a sufficiently large proportion of non-black things. — Michael
With the example of eggs, having checked 9 of the 10 eggs should count as evidence. — Michael
You said it isn't frequentist probability being that the probability (or plausibility, if you prefer) is 0.510, not either 0 or 1. — Michael
But, that's part of the paradox. Green apples are not unrelated to the universal statement "all ravens are black." It confirms the (logically equivalent) contrapositive, i.e. that all non-black things are non-ravens.We have a situation where the observation of a green apple purportedly supports an enormous number of unrelated universal statements, including the statement "all ravens are black". — tom
Perhaps such evidence cannot exist, but your purported proof to that effect seems flawed. Unless you can address my specific concerns, I can't accept it.The solution to this problem is to recognise that there is no such thing as epistemologically valuable corroborating evidence. It simply cannot exist.
Right...hence the paradox! It's counterintuitive (to put it mildly) that the observation of green apples confirms the hypothesis that all ravens are black.The corroborating evidence E points everywhere and thus nowhere. For some psychological reason we see this in the case of green apples, but not in black ravens.
Upon further reflection, it occurred to me that my thought experiment (whether or not it presents a valid point) has limited applicability to the raven paradox. The universal statement under consideration is "all ravens are black." The contrapositive is "all non-black things are non-ravens."This seemed to be Popper's view (as someone else pointed out). Let us consider for a moment the proposition that singular instances provide no confirmation of a universally-quantified hypothesis or statement (e.g. occurrences of white swans do not even marginally raise the probability of the hypothesis "all swans are white") by means of a thought experiment.
At the very least, this claim seems unintuitive under certain conditions. For instance, imagine that the world consists entirely of a carton of eggs, with a dozen egg cups, each containing exactly one egg. A "God's eye view" observer of the world formulates the hypothesis that "all eggs are white," and sets about inspecting each cup.
After the observer inspects, say, three of the eggs and finds that they're white, can he reasonably be more confident in the truth of his hypothesis to any degree whatsoever? After all, each cup which is found to contain a white egg is one less cup which can possibly hold a non-white egg (and we've stipulated that the world consists solely of this egg carton, so there is nowhere else for a non-white egg to hide). Does each observation of a white egg therefore confirm the hypothesis (even if only incrementally)? My intuition seems to say "yes," but of course, my intuition does not constitute any sort of rigorous proof. — Arkady
Assuming that we've already observed at least one white egg, observation of a white non-egg would confirm the hypothesis that "all eggs are white," because it would further diminish the probability of the carton containing at least one non-white egg (because the white non-egg, whatever it might be, is occupying space that might otherwise be occupied by a non-white egg). — Arkady
But, that's part of the paradox. Green apples are not unrelated to the universal statement "all ravens are black." It confirms the (logically equivalent) contrapositive, i.e. that all non-black things are non-ravens. — Arkady
Now that I think of it, I may have found another problem: you claim that H entails E, and so that p(he|b) = p(h|b). But, I'm not sure that this follows. Even if H entails E, unless they are necessary truths, the probability of their conjunction must be equal to or less than either of the conjuncts. P(H|E) or P(E|H) are not equal to P(H&E), even when H entails E (that is, P(E|H) = 1). — Arkady
Right...hence the paradox! It's counterintuitive (to put it mildly) that the observation of green apples confirms the hypothesis that all ravens are black. — Arkady
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