No resolution of the paradox is universally accepted, but the most widely accepted is what I will call the standard Bayesian response. In this article, I\u2019ll present this response, explain why I think it is incomplete, and propose an extension that might resolve the paradox.<\/p>\n\n\n\n
Click here to run this notebook on Colab<\/a>.<\/p>\n\n\n\n The paradox starts with the hypothesis<\/p>\n\n\n\n A: All ravens are black<\/p>\n<\/blockquote>\n\n\n\n And the contrapositive hypothesis<\/p>\n\n\n\n B: All non-black things are non-ravens<\/p>\n<\/blockquote>\n\n\n\n Logically, these hypotheses are identical \u2013 if A is true, B must be true, and vice versa. So if we have a certain level of confidence in A, we should have exactly the same confidence in B. And if we observe evidence in favor of A, we should also accept it as evidence in favor of B, to the same degree.<\/p>\n\n\n\n Also, if we accept that a black raven is evidence in favor of A, we should also accept that a non-black non-raven is evidence in favor of B.<\/p>\n\n\n\n Finally, if a non-black non-raven is evidence in favor of B, we should also accept that it is evidence in favor of A.<\/p>\n\n\n\n Therefore, a red apple (which is a non-black non-raven) is evidence that all ravens are black.<\/p>\n\n\n\n If you accept this conclusion, it seems like every time you see a red apple (or a blue car, or a green leaf, etc.) you should think, \u201cNow I am slightly more confident that all ravens are black\u201d.<\/p>\n\n\n\n But that seems absurd, so we have two options:<\/p>\n\n\n\n As you might expect, many versions of (1) and (2) have been proposed.<\/p>\n\n\n\n The standard Bayesian response is to accept the conclusion but, quoth Wikipedia<\/a> \u201cargue that the amount of confirmation provided is very small, due to the large discrepancy between the number of ravens and the number of non-black objects. According to this resolution, the conclusion appears paradoxical because we intuitively estimate the amount of evidence provided by the observation of a green apple to be zero, when it is in fact non-zero but extremely small.\u201d<\/p>\n\n\n\n It is true that when the number of non-ravens is large, the amount of evidence we get from each non-black non-raven is so small it is negligible. But I don\u2019t think that\u2019s why<\/em> the conclusion is so acutely counterintuitive.<\/p>\n\n\n\n To clarify my objection, let me present a smaller example I\u2019ll call the Roulette paradox. <\/p>\n\n\n\n An American roulette wheel has 36 pockets with the numbers Suppose we work in quality control at the roulette factory and our job is to check that all zero pockets are green. If we observe a green zero, that\u2019s evidence that all zeros are green. But what if we observe a red 19?<\/p>\n\n\n\n In this example, the standard Bayesian response fails:<\/p>\n\n\n\n As I will demonstrate,<\/p>\n\n\n\n In both cases we observe a non-green non-zero, but \u201cobserve\u201d is ambiguous. Whether the observation is evidence or not depends on the sampling process<\/strong> that generated the observation. And I think confusion between these two scenarios is the foundation of the paradox.<\/p>\n\n\n\n Let\u2019s get into the details. Switching from roulette back to ravens, we will consider four scenarios:<\/p>\n\n\n\n The key to the raven paradox is the difference between scenarios 2 and 4.<\/p>\n\n\n\n The reason for the paradox is that we imagine Scenario 2 and we are given the conclusion from Scenario 4.<\/p>\n\n\n\n It might not be obvious why the red apple is evidence in Scenario 4, but not Scenario 2. I think it will be clearer if we do the math.<\/p>\n\n\n\n We\u2019ll start with a small world where there are only I\u2019ll use We\u2019ll use a joint distribution to represent beliefs about Let\u2019s start with a uniform prior over all possible combinations of Now let\u2019s consider the first scenario: we choose a thing at random from the universe of things, and we find that it is a black raven.<\/p>\n\n\n\n The likelihood for this observation is: In this scenario the posterior probability of A is 20%. The posterior probability is higher than the prior, so the black raven is evidence in favor of To quantify the strength of the evidence, we\u2019ll use the log odds ratio, which is 0.81. Later we\u2019ll see how the strength of the evidence depends on the prior distribution of Before we go on, let\u2019s also look at the marginal distribution of As expected, observing a black raven increases our confidence that all ravens are black. The posterior distribution shifts toward higher values of In Scenario 1, the likelihood depends only on Finally, let\u2019s visualize posterior joint distribution of Because we started with a uniform distribution and the data has no bearing on In summary, Scenario 1 is consistent with intuition: a black raven is evidence in favor of In this scenario, we choose a thing at random from the universe of The likelihood of this observation is: In this scenario, the posterior probability of So the red apple is not evidence in favor of However, the red apple is evidence about And here\u2019s the posterior joint distribution of Because the red apple has no bearing on In summary, Scenario 2 matches our intuition: a red apple (chosen at random) is not<\/em> evidence about whether all ravens are black.<\/p>\n\n\n\n In this scenario, we choose a raven first and then observe that it is black.<\/p>\n\n\n\n The likelihood for this observation is: In fact, the entire posterior distribution is the same in both scenarios. That’s because the likelihoods in Scenarios 1 and 3 differ only by a constant factor, which is removed when the posterior distributions are normalized.<\/p>\n\n\n\n In summary, Scenario 3 is consistent with intuition: if we choose a raven and find that it is black, that is evidence in favor of In the last scenario, we first choose a non-black thing (from all non-black things in the universe), and then observe that it is a non-raven.<\/p>\n\n\n\n The likelihood of this observation is: This likelihood depends on both<\/strong> The posterior probability of A is about 15%, which is greater than the prior, so the non-black non-raven is evidence in favor of Here is the marginal distribution of And here\u2019s the marginal distribution of Finally, here\u2019s the posterior joint distribution of In Scenario 4, the likelihood depends on both<\/strong> And in Scenario 4 a non-black non-raven (chosen from non-black things) is evidence in favor of In all four scenarios, the results are consistent with intuition. So as long as you are clear about which scenario you are in, there is no paradox. The paradox is only apparent if you think you are in Scenario 2 and you imagine the result from Scenario 4.<\/p>\n\n\n\n In the context of the original problem:<\/p>\n\n\n\n In these examples, we started with a uniform prior over all combinations of In general, different priors lead to different posterior distributions, and in this case they lead to different conclusions about the strength<\/em> of the evidence. But they lead to the same conclusion about the direction<\/em> of the evidence.<\/p>\n\n\n\n To demonstrate, let\u2019s see what happens if we observe a series of black ravens (in Scenario 1 or 3). For simplicity, assume that we sample with replacement.<\/p>\n\n\n\n The following function computes multiple updates, starting with the uniform prior and then using the posterior from each update as the prior for the next.<\/p>\n\n\n\n This table shows the results in Scenario 1 (which is the same as in Scenario 3). For each iteration, the table shows the prior and posterior probability of The Problem<\/h2>\n\n\n\n
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The Roulette Paradox<\/h2>\n\n\n\n
1<\/code> to 36<\/code>, and two pockets labeled 0<\/code> and 00<\/code>. The non-zero pockets are red or black, and the zero pockets are green.<\/p>\n\n\n\n\n
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The Setup<\/h2>\n\n\n\n
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The Math<\/h2>\n\n\n\n
N = 9<\/code> ravens and M = 19<\/code> non-ravens. Then we\u2019ll see what happens as we vary N<\/code> and M<\/code>.<\/p>\n\n\n\ni<\/code> to represent the unknown number of black ravens, which could be any value from 0<\/code> to N<\/code>, and j<\/code> to represent the unknown number of black non-ravens, from 0<\/code> to M<\/code>.<\/p>\n\n\n\ni<\/code> and j<\/code>; then we\u2019ll use Bayes\u2019s Theorem to update these beliefs when we see new data.<\/p>\n\n\n\n(i, j)<\/code>. For this prior, the probability of A<\/code> is 10%. We\u2019ll see later that the prior affects the strength of the evidence, but it doesn\u2019t affect whether an observation is in favor of A<\/code> or not.<\/p>\n\n\n\nScenario 1<\/h2>\n\n\n\n
i \/ (N + M)<\/code>, because i<\/code> is the number of black ravens and N + M<\/code> is the total number of things. <\/p>\n\n\n\nA<\/code>.<\/p>\n\n\n\ni<\/code> and j<\/code>.<\/p>\n\n\n\ni<\/code> (number of black ravens) before and after.<\/p>\n\n\n\n![\"_images\/73acb5dc58b1e987037e9ef21aff0c9bc014e6a3f841f015b7365f9352d648c7.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/73acb5dc58b1e987037e9ef21aff0c9bc014e6a3f841f015b7365f9352d648c7.png\")
<\/figure>\n\n\n\ni<\/code>, and the probability that i = N<\/code> increases.<\/p>\n\n\n\ni<\/code>, not on j<\/code>, so the update doesn\u2019t change our beliefs about j<\/code> (the number of black non-ravens). <\/p>\n\n\n\ni<\/code> and j<\/code>.<\/p>\n\n\n\n![\"_images\/3f8f1dc012592d11ac19f20c5698984fc6134a93c7eeec35c1ba1aed5913a1a2.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/3f8f1dc012592d11ac19f20c5698984fc6134a93c7eeec35c1ba1aed5913a1a2.png\")
<\/figure>\n\n\n\nj<\/code>, the joint posterior probabilities don\u2019t depend on j<\/code>.<\/p>\n\n\n\nA<\/code>.<\/p>\n\n\n\nScenario 2<\/h2>\n\n\n\n
N + M<\/code> things, and it turns out to be a red apple \u2013 which we will treat generally as a non-black non-raven.<\/p>\n\n\n\n(M - j) \/ (N + M)<\/code>, because M - j<\/code> is the number of non-black non-ravens and N + M<\/code> is the total number of things.<\/p>\n\n\n\nA<\/code> is the same as the prior. In fact, the entire distribution of i<\/code> is unchanged.<\/p>\n\n\n\nA<\/code> or against it. This is consistent with the intuition that the red apple (or any non-black non-raven) is irrelevant.<\/p>\n\n\n\nj<\/code>, as we can confirm by comparing the marginal distribution of j<\/code> before and after.<\/p>\n\n\n\n![\"_images\/b99c195fe739e438b9f9d21fbbfc7e12e264b93bff2de276ae0dbe7effc395dc.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/b99c195fe739e438b9f9d21fbbfc7e12e264b93bff2de276ae0dbe7effc395dc.png\")
<\/figure>\n\n\n\ni<\/code> and j<\/code>.<\/p>\n\n\n\n![\"_images\/de8e2c353c1109b5324bd83ac12ec0262d1267021a2736b91d8f83ec641bc11d.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/de8e2c353c1109b5324bd83ac12ec0262d1267021a2736b91d8f83ec641bc11d.png\")
<\/figure>\n\n\n\ni<\/code>, the posterior probabilities in this scenario don\u2019t depend on i<\/code>.<\/p>\n\n\n\nScenario 3<\/h2>\n\n\n\n
i \/ N<\/code>, because i<\/code> is the number of black ravens and N<\/code> is the total number of ravens.
In this scenario, the posterior probability of A is 20%, the same as in Scenario 1.So we conclude that the black raven is evidence in favor of A<\/code>, with the same strength regardless of whether we are in:<\/p>\n\n\n\n\n
A<\/code>.<\/p>\n\n\n\nScenario 4<\/h2>\n\n\n\n
(M - j) \/ (N - i + M - j)<\/code> because M - j<\/code> is the number of non-black non-ravens and N - i + M - j<\/code> is the total number of non-black things.<\/p>\n\n\n\ni<\/code> and j<\/code>, unlike Scenario 2. This is the key difference that makes Scenario 4 informative about whether all ravens are black.<\/p>\n\n\n\nA<\/code>. The log odds ratio is about 0.47, which is smaller than in Scenarios 1 and 3, because there are more non-ravens than ravens. As we\u2019ll see, the strength of the evidence gets smaller as M<\/code> gets bigger.<\/p>\n\n\n\ni<\/code> (number of black ravens) before and after.<\/p>\n\n\n\n![\"_images\/90ae0b51191d2f3b068cc9a4d382e172f23d8ec19a5708c3e5f51677d0b966cd.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/90ae0b51191d2f3b068cc9a4d382e172f23d8ec19a5708c3e5f51677d0b966cd.png\")
<\/figure>\n\n\n\nj<\/code> (number of black non-ravens) before and after.<\/p>\n\n\n\n![\"_images\/f7b88c2307331de2664b17bd98bdd246a61d436d1f5fc263d05e80ea706ce875.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/f7b88c2307331de2664b17bd98bdd246a61d436d1f5fc263d05e80ea706ce875.png\")
<\/figure>\n\n\n\ni<\/code> and j<\/code>.<\/p>\n\n\n\n![\"_images\/83ec65bdd92130d86aad0ff90c7b99232096fffa2e14c5ec96647f66e8137d0c.png\"](\"https:\/\/www.allendowney.com\/blog\/wp-content\/uploads\/2025\/12\/83ec65bdd92130d86aad0ff90c7b99232096fffa2e14c5ec96647f66e8137d0c.png\")
<\/figure>\n\n\n\ni<\/code> and j<\/code>, so the update changes our beliefs about both parameters.<\/p>\n\n\n\nA<\/code>. This might still be surprising, but let me suggest a way to think about it: in this scenario we are checking non-black things to make sure they are not ravens. If we find a non-black raven, that contradicts A<\/code>. If we don\u2019t, that supports A<\/code>.<\/p>\n\n\n\n\n
Successive updates<\/h2>\n\n\n\n
i<\/code> and j<\/code>. Of course that\u2019s not a realistic representation of what we believe about the world. So let\u2019s consider the effect of other priors.<\/p>\n\n\n\nA<\/code>, and the log odds ratio.
<\/p>\n\n\n\n