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As an amateur who has interest in logic and mathematics I've been reading about the concept of different probability perceptions. I'd like to have your opinions over the subject below.

When it comes to probability assessment/comparison of two unlikely events does logic give us a single definitive and universal answer? Please take this question into account over the examples below.

The events in my question concern one astronomically unlikely and measurable hypotethical event (let's call this Event 1) and another unlikely hypotethical event for which its probability cannot be easily measured yet its nature is recognizable due to being exposed to similar ones. (lets call this Event 2, more detaled explanation follows below)

To illustrate what I mean over two examples:

Event 1 is the likelihood of having an uniform picture let's say a cat photo on a random pixel generator.

Suppose we have a random pixel generator which has 1920 x 1080 screen resolution with 24 bit colors. For each pixel on the screen we have a 1 in 10^14981179 chance of being set at the correct position to generate any image we can think of hence a cat picture. (2^24^1920 x 1080) We end up with an unfathomably low probability.

I am taking the liberty of calling Event 2 a likelihood extremely surprising for some but just usual news to many people. Take crimes for instance, unfortunately every day on the news we come across several crimes therefore we are exposed to a sample unlike Event 1. Let's take hypotethically person A is committing a serious crime i.e robbery (You can name it to increase the degree of surprisingness) and it is extremely unexpected due to its not so easily explainable nature (no obvious motivation and reason for such action, completely opposite character of the person, serious consequences etc. but note that there is nothing supernatural about the action)

For me almost anything that can occur in this world would have much higher probability than the event 1 which is absurdly improbable. Let alone the lifetime of our universe, mathematically millions of universe wouldn't be enough to see a uniform real cat picture on a random pixel generator even it shuffles the pixels every second.

However can a person find Event 2 less likely than Event 1 just because his experience and belief over the person who hypotetically commits the crime?

Do logic and maths tell us that likelihood comparison of these events are subjective therefore we cannot argue about the odds?

Is Event 2 a case which is impossible to measure its probability? Or regardless of the peculiarity of the person who hypotethically commits the crime is taking into account other similar cases (i.e same type of crime ratios over a certain period) sufficient to conclude that Event 2 has higher probability without any doubt ?

Thanks for reading so far. Lastly is there any specific branch or body of work that focuses on such probability and logic topics?

Geerts
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  • Seems your detailed thought process above already solved most of your questions, you're just not sure if it's possible to measure E2's probability, and your penultimate section should satisfy such possibility... – Double Knot May 20 '22 at 03:19
  • @Double Knot, would you be so kind to answer my questions in bold. Do you think for example likelihood comparison of E1 and E2 subjective and we cannot argue about the odds? Can you explain please? Thanks! – Geerts May 20 '22 at 05:27
  • It’s hard to know other people’s true intention who is not a *true* fellow, and after the crime is done we’ll all be surprised. So conversely had you *really* understand that person, you'll not be so surprised by that crime result. And since this is a common phenomena so you can reasonably rely on the stats due to law of large numbers. Also apart from objective frequency interpretation of probability, there's subjective degree of belief interpretation or Solomonoff's universal a prior probability, so subjectivity here in E2 can be explained as posterior Bayesian update about that person... – Double Knot May 20 '22 at 05:46
  • @DoubleKnot Does this mean likelihood comparison of E1 and E2 subjective and we cannot argue about the odds? Or does this mean a person who finds Event 2 more likely than Event 1 is illogical and without doubt logic tells us having a picture of cat on the random pixel generator is less likely? This person might think that Person A could be the last person on earth to commit such crime due to his character, lack of any motivation and obvious reason, sheer consequences etc. You can alter my robbery example with more shocking ones to increase contrasting feeling. – Geerts May 20 '22 at 06:10
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    How can you be so sure as an objective fact that "... that Person A could be the last person on earth to commit such crime due to his character..."? Thus it's totally logical that Event 2 is more likely than Event 1 in terms of probability which can be interpreted both as objective frequency and subjective degree of belief, since the true posterior crime probability of E2 has to be within the confidence interval of the usual level of the stats and your shocking feeling is just a Bayesian update about that person... – Double Knot May 20 '22 at 06:20
  • @DoublKnot Thanks for the insights. I'm pretty clear it's totally logical that E2 is more likely. I'm trying to understand if it's also totally logical to conclude E2 is less likely than E1. Such shocking crime in E2 can be just news for outsiders because 'seemingly' the same incidents occur. Let's say a person from the family is almost sure that Person A could be the type of last individual on earth to commit such crime. Such person from the family is therefore someone who experiences first hand hard to explain nature of E2 though it may just look like a statistics for a foreigner. – Geerts May 20 '22 at 07:36
  • @DoubleKnot Considering each single event itself is unique regardless of how similar they look -as a philosophical approach perhaps- can someone conclude that E2 is less likely than E1? (Character of Person A and conditions of the E2 are unique in the end regardless of seemingly same incidents that happen) To sum up based on your answers what I conclude logic tells us undoubtedly E2 is more likely. Would it be perfectly normal if someone finds E2 less likely? Can there be people who find E2 less likely? – Geerts May 20 '22 at 07:37
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    Re your above "Such person from the family is therefore someone who experiences first hand hard to explain nature of E2", the difficulty lies in the limitation of current psychology knowledge, some philosophers/neuroscientists even reject existence of psychology. Thus your shocking unbelievable emotion may not be objectively well founded. Indeed, sometimes even family members don't *truly* understand each other psychologically which actually occurs quite common as described in many ancient classic books such as [filial piety sutra](http://www.cttbusa.org/filialpiety/filialpietysutra2.htm)... – Double Knot May 20 '22 at 19:59
  • @DoubleKnot I see your point. It doesn't have to be a family member, in this example we can simply change the subject from Person A to ourselves. About the likelihood we can then say ‘It won’t ever happen because I know’. Please see the details that I explained in the comments of the answer of 'tkruse'. As 'tkruse' explains the fact that there are seemingly similar cases(statistics) about the Event 2 doesn't mean it has to have higher likelihood than Event 1. So it's also logical to conclude that Event 2 can have lower probability than Event 1, which is opposite of your view. – Geerts May 24 '22 at 08:58
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    Even you change person A to self, I'd rather still depend on stats since *know thy self* is extremely hard per the famous Greek Delphic maxim perhaps due to the similar limitation of folk (pseudoscientific?) psychology theory and also consider there're not that many people as denominator of your E2 probability compared to E1. But I speculate if one *truly* knows oneself such as attaining Buddhahood, then in such rare case E2 can be certain to be less than E1 as it can then be claimed infinitesimally approaching zilch. As reminded by Hegel your *sense certainty* is likely not a universal truth. – Double Knot May 25 '22 at 02:18
  • The field of mathematics that deals with questions like this or ones that are similar is *statistics and probability*. – niels nielsen May 23 '22 at 05:38
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    @DoubleKnot Just to inform you as you've given interesting insights. I got the opportunity to ask this question to a couple more people who have thorough knowledge about probability calculations and statistics. They indicate that essence of my question concerns conditional probability, the fact different people can have different probability assessments based on their different knowledge levels. What they indicate, serious crimes happen all the time thus it isn't a low likelihood given time in a population for a person to commit the crime. – Geerts May 31 '22 at 10:29
  • However this doesn't mean likelihood of a specific person to commit a specific crime has to be the same likelihood of any other person in the population. They therefore indicate that probability of Event 2 can be any value between 0 and 1, including literally 0%. Please note that I'm referring an action that looks completely arbitrary thus shockingly unexpected in serious crime cases like Event 1 (bank robbery, murder etc.) yet not supernatural. Let's say it isn't a case in which a person's mental state changes due to a sudden stroke causing him/her to commit the crime either. – Geerts May 31 '22 at 10:32
  • On the contrary, there is no obvious motivation/reason for the action and that person caanot be clearly considered mentally unhealthy. Do you still think that probability of Event 2 has to be higher than even the most absurd cases like having a specific image on the screen of rpg (Event 1)? I'm using Event 1 just as a reference, as it's a calculable low likelihood, to be able to compare how low likelihood of Event 2 can be. I'm just a curious amateur thinker. When reading about subjectivity concept of probability I pondered limits of extremely rare cases and such question came to my mind. – Geerts May 31 '22 at 10:35
  • Kudos for you follow-up spirit and cross check effort on other stats focused exchange... Bayesian update faction mainly rely on a posteriori conditional probability otherwise how they can update from a priori one? Of course for different person this "subjective" E2 event is different, and I've already indicated before for some saint like person it could be zero... Bear in mind though this subjective probability epistemology is very hard as evidenced by professional thinker like Carnap who spent most later life to propose his *degree of confirmation* logic for probability and not successful... – Double Knot May 31 '22 at 19:17
  • @DoubleKnot Your responses are appreciated. Allow me to comment lastly as I don't want to disturb you any more. I think your 0% approach which is extremely rare saint degree has confused me. Because based on other answers this literally 0% likelihood doesn't have to be so rare in a population. It could be 0% likelihood to commit the serious crimes mentioned (Event 2) for an average person who pursues a good life as long as some criteria are met by that person – Geerts May 31 '22 at 20:17
  • as such not having serious mental issues, finding such actions immoral, no other external factors that causes/provokes occurrence of the action, sheer/disruptive consequences of the action etc. So combination of these factors and knowledge level about that person makes the action shockingly arbitrary, if it is performed. Therefore possible occurrence of the action looks so arbitrary that its occurrence is evaluated with 0% likelihood even lower than most absurd cases like finding a cat picture on the screen of random pixel generator. – Geerts May 31 '22 at 20:17
  • As a thinker amateur or professional it’s never a disturb to search for truth and I believe you’re along this path… My early speculation is unconditional, if adding your “as long as some criteria are met” condition, of course more such condition as listed in your last comment above then smaller the conditional probability of E2 will be, nothing mysterious here and I agree… – Double Knot May 31 '22 at 22:04

3 Answers3

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For very simple events, like particle interactions or dice rolls, we can derive quite rigorous models. Our intuition is often misleading about these cases, because while we recognise the cat is unlikely, we struggle with the idea a specific image of static is equally unlikely. We perform a 'chunking' process, where we merge many distinct states. We should understand thermodynamics like thus, with there being many similar equilibrium states like the static, and few with special properties, like the cat.

For more complex events where the parts or interactions are beyond being precisely modelled, we need Bayesian inference, where we begin with priors, or expectations. These can come from almost anywhere, we are only required to think of the best likelihoods we can. Then we adjust them over time.

It's important to understand all probability is based on a fantasy: counterfactuals. That is, we imagine the event could happen again from the same initial conditions. For simple cases, we can mimic the starting conditions closely enough, and prove this by statistical analysis. Bayesian reasoning can be used to predict complex systems, like the intentions of other humans. We know from the Dunbar Number our neocortex evolved primarily for predicting other humans, and gives us a cognitive bias towards narrating subjectivities. See: Is the idea of a causal chain physical (or even scientific)? Hume's Problem Of Induction shows us that in truth relying on the past as a guide is not a logical choice, but one derived from experience. Popper who dismissed induction's role in science, essentially argued science is about conjecture and criticism including by experiment, which fits with the Bayesian perspective.

So. Can you model the situation unambiguously? First approach. Should ve able to agree. Is the situation to complex to model completely, and unambiguously? All the computing power on Earth is limited to the quantum states of a lump of matter about the size of a tennis ball, so anything like biology that uses a substantial fraction of the available complexity, is far beyond what we can currently model explicitly. So we need conceptual chunking, and learning from experience, which will depend on framing and our past.

CriglCragl
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  • Your insights are interesting. However with my limited mathematics I don't think I can model Event 2 unambiguously. Its likelihood indeed is hard to calculate, not straightforward like Event 1. That's why I've posted this question. Does the fact that there are seemingly similar cases for the Event 2 (robbery statistics) necessarily mean that it has to have higher likelihood than Event 1 which has infinitely small likelihood? Do you think it can be completely logical that Event 2 has less likelihood than Event 1? Please see the comments under the answer of 'tkruse' for the details. – Geerts May 23 '22 at 12:24
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Psychology at least tells us that most humans are extremely bad at judging probability from observation, even for frequently occuring events. There are also biases like selective perception, selective memory, confirmation bias...

It's also difficult to objectively apply maths to model real world events, as any two mathematicians or statisticans can reasonably disagree on how reality should be modeled, depending also on the purpose of such an effort.

So there are pragmatic, psychological and political limitations at least to answer questions about mathematical properties of reality.

Regarding probabilistic predictions of the future, it is not possible to define one probabilistic prediction of a single event as correct. Assume a 1% chance of rain tomorrow, this is "true" both if tomorrow it rains or does not rain.

For methods of prediction, we can measure their usefulness by comparing them to outcomes over many events, but that does not make any single prediction correct or better. A wise man might much more often predict well the future than a fool, but the fool might still be right about the next day when the wise man is wrong.

Events need not be rare for this, rarity just makes it harder to compare the methods. But single predictions remain incomparable even for frequent events.

Unpredictability is similar to rarity in a way. For predicting the final position of a ball at a roulette table, similar questions in the quality of individual predictions arise.

For your random pixel generator, it is practically impossible to ever show a specific cat picture. But if you take any random picture that your generator already has generated, the likelihood for generating that random image again is still the same as for generating your cat photo. And it was the same as for your cat picture even before. So every time your generator creates a random image, it creates something that was practically impossible to happen a moment before, according to statistics. So observing anything happening does not allow use to say it was likely to happen because it happened.

We can also trivially find events that are even less likely than your cat photo generator, by using a generator with more pixels, as an example. We can find examples that are more likely, by using a generator with fewer pixels. If we observe some rare event in our lives, then without further knowledge it would seem rational to assume we might observe it again, but again, every time we observe your generator generating a random image, we can be sure we'll never see that same image again, our knowledge of the space of possibilities is just as important as observation of events.

tkruse
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  • What you mean as disagreement is a situation as follows? In Event 2, for robbery case let’s say Person A is waiting a bus in front of a bank office. He knows that likelihood of attempting a bank robbery for him is 0%, impossible. There is no reason at all for him to do such action as he isn't an immoral person and he isn't mad either psychopath. He's looking forward to arriving home, having dinner with his beautiful wife and lovely children etc. However there is also Person B who is passing by the bank. – Geerts May 22 '22 at 10:00
  • For Person B, Person A is a complete stranger. When asked Person B, I think he cannot say Person A’s likelihood of commit such crime is 0%. He doesn’t know him, he cannot rule out statistical bank robbery cases. Thus Person B will give a different likelihood, not 0%. I hope my example is clear. We can change the subject Person A with ourselves to stress that ‘It won’t happen because I know’. How do you explain such phenomenon? Which perspective of likelihood is correct? Apologies for giving such disturbing example, this is just mathematical/philosophical discussion in relation to the topic. – Geerts May 22 '22 at 10:03
  • Those are estimates under different knowledge. It would be usually much more surprising if in such an situation were exactly the same. However if the people involved exchanged a lot of knowledge and opinions, we can expect their estimates to approach each other. However, their combined estimate might still be far off a scientific estimate done by experts. Untrained humans are overall incredibly bad at statistical judgement. – tkruse May 22 '22 at 13:36
  • It's clear to me that untrained humans are overall bad at such judgements. I see from your reply that indeed there is one correct likelihood judgement. Your reply is appreciated but I'm still struggling with finding answers. Is it then possible that Person A's judgement which is 0% hence impossible can be the correct one? Relating to my original post can likelihood of Event 2 be lower than Event 1 which is so extremely small? – Geerts May 22 '22 at 14:56
  • So individual estimates cannot be correct or incorrect before the event. Regarding tomorrow's weather over New York at 1pm local time, one forecast might give a probability of rain of 10%, another at 20%. Neither can be defined usefully as more correct than the other. Only the underlying methods of prediction can be judged after observing them for many days, counting their success. But even if we find one method has better success over 100 days, that does not mean it provides necessarily a better prediction for the next day. It's impossible to define such quality criteria for single estimates. – tkruse May 22 '22 at 23:58
  • Sorry but I need to ask for the clarification. According to your answer I'm concluding that likelihood of Event 2 can be lower than the infinitely small likelihood of Event 1. Let me know please. As there is already statistics from similar sample of the phenomenons of Event 2 (robbery) there is another view that eventually likelihood of Event 2 should be higher than likelihood of Event 1. Please see the comments from 'Double Knot' under the original post. However I'm not treating Event 2 just like a statistical case. That's why I stressed its uniqueness as 'It won't happen because I know' – Geerts May 23 '22 at 03:37
  • Just because you have observed a rare event 2 does not mean the likelihood of event 2 is higher than that of event 1. Though typically in real life it would be, and it's often more rational to assume something you observed was not that unlikely. But that's no guarantee, just a heuristics. See my extended answer. – tkruse May 23 '22 at 05:03
  • The likelihood of getting 1 when throwing a 20 sided dice is 1/20. The likelihood with a 6 sided dice is 1/20. If you throw a 6 sided dice and a 20sided dice together, and only the 20sided dice shows 1, it does not mean the probability of the 20sided dice showing 1 is higher than 1/6 – tkruse May 23 '22 at 05:09
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The concept of probability in the second event is subjective. In the first case, it is not. The second event suffers from the reference class problem. Depending on how you characterize the situation, the probability will be different.

It is not that you don’t know the probability of the second event. It is that the very notion of a probability given the ill defined nature and context of the problem becomes vacuous.

One can probably still phrase the question a different way: If one of these events happened tomorrow, and only one, which one would you bet on if money is on the line? Most would say the second. However, this is just a hypothetical, since we don’t actually know if one of these events happened.

Arguably though, the first event seems of a kind that is physically impossible for all practical purposes. The second seems possible. For this and only this reason I would argue it makes more sense to bet on the second one over the first IF you knew beforehand that one and only one of them occurred. If you don’t know this, it is most honest to say we simply don’t know if the second event happened but that we can reasonably be certain that the first one did not.

thinkingman
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