Is all of probability fundamentally subjective and unneeded as a term outright?

Question

What is the real probability that a person will be murdered tomorrow somewhere in the world? It seems like there should be a right answer to this. In fact, most of us would bet tens of millions of dollars that atleast one person will be murdered tomorrow.

But what is the correct probability then? Is it 1? As in it’s guaranteed to happen? Why would one think it’s guaranteed to happen? It would likely be because someone was murdered every day for many years. But this is nothing more than an inductive inference. Where does probability come here or even needed as a concept to talk about whether there will be a murder somewhere in the world?

Let’s look at a seemingly harder example. What is the probability that there will be a murder tomorrow in your city? As you can imagine, it would depend on how you interpret probability and what class of past events you use. Do you look at the murder rates of your city or the country? What about the time period? What should that be? What makes one interpretation more valid than the other? If there is no clear answer to this, does this mean that the concept of probability is inherently subjective?

I can imagine a retort being that it is subjective but one can make models based on certain classes of past data and see which models predict the future better. But that doesn’t tell you what the “correct” probability is. It just tells you what model works the best. Where is the concept of probability actually needed here?

Answer 1

In fact there is no way to define unique objective probabilities for anything that happens (or not) in the real world. Every definition of probability requires model assumptions that cannot be verified (one can try to falsify some of them, which occasionally works, but even this usually incurs error probabilities, so any falsification won't be 100% reliable). So every definition requires some judgement that will ultimately involve subjective components.

What is usually referred to as "objective" probability is called like this either because it refers to data generating processes in the real world even if there is no objective knowledge how exactly they play out, or ("objective Bayes") because it refers to absence of personal judgement, which in practice is not possible, but rather seen as an ideal to aspire to.

We have argued that the subjective/objective-distinction is often unhelpful in statistics, and also often misunderstood, as the term "objectivity" means different things to different people or in different fields in this paper:

Andrew Gelman, Christian Hennig, Beyond Subjective and Objective in Statistics, Journal of the Royal Statistical Society Series A: Statistics in Society, Volume 180, Issue 4, October 2017, Pages 967–1033, https://doi.org/10.1111/rssa.12276

"Where is the concept of probability actually needed here?" You can find quantifications of uncertainty helpful for orientation even if they are not grounded in "objective truth". Such quantifications can also to some extent be empirically tested.

Your discussion of the murder rate example is known as Reference class problem by the way, and it is clear that compromises need to be struck in practice, i.e., you want a large enough reference class so that there is enough data, but a small enough one so that information is used as precisely as possible. This requires some subjective judgement, however that doesn't make it worthless.

PS: One thing that I think is special about probability (as opposed to other scientific concepts for which mathematical models exist) is that probability is always at least partly about something counterfactual, about what could happen that actually doesn't happen (and how likely that would've been). This makes it much harder to think about probabilities as "objective".

Answer 2

What is the real probability that a person will be murdered tomorrow somewhere in the world? It seems like there should be a right answer to this.

I would accept that there is a true, correct, mathematical probability for that, which is very close to (but less than) 1. This is an hypothesis that I cannot prove, but I believe it anyway.

Why would one think it’s guaranteed to happen?

One might think so because one has a flawed understanding of probability.

Or one might deploy the word "guaranteed" in a probabilistic sense, meaning that they believe the probability to be so close to 1 that the alternative is not pragmatically worth considering.

It would likely be because someone was murdered every day for many years. But this is nothing more than an inductive inference.

I think that's a straw man. Yes, if indeed one has that statistic to draw on (do we?) then by inference one could predict that there will also be at least one murder tomorrow. But that's no basis for assigning probability 1 to such an event, nor any other specific probability. Anyone coming to such a conclusion on that basis would fall into the "flawed understanding of probability" group.

Where does probability come here or even needed as a concept to talk about whether there will be a murder somewhere in the world?

Quantitative probability is not particularly useful in this area, because for most practical purposes, the exact mathematical probability of real-world events is not computable. On the other hand, qualitative probability is foundational. We can often estimate fairly well how likely or unlikely one event is either on its own or relative to an alternative. We make such estimates all the time, and rely on them in making decisions.

What is the probability that there will be a murder tomorrow in your city? As you can imagine, it would depend on how you interpret probability

Why?

and what class of past events you use.

You seem to be assuming an inferrential approach. This is certainly the statistician's method for estimating the probability of an event, but it gives you only an estimate, often with an associated confidence estimate, not a true, exact probability.

Do you look at the murder rates of your city or the country? What about the time period? What should that be? What makes one interpretation more valid than the other? If there is no clear answer to this, does this mean that the concept of probability is inherently subjective?

No, it means that statistical inference is qualified by a set of subjective assumptions. The methodologies of statistical inference are related to probability, but they do not compute true probabilites in the sense you seem to mean.

I can imagine a retort being that it is subjective but one can make models based on certain classes of past data and see which models predict the future better.

Yes. That's more or less a description of science.

But that doesn’t tell you what the “correct” probability is.

True. So what?

Where is the concept of probability actually needed here?

There in particular, mathematical probability is the framework by which we understand statistical inference. However, exact numeric probabilities of real-world events indeed are not needed for most purposes.

No doubt one can go a very long way in philosophy without involving the concept of probability. Eventually, however, one has to come to grips with the fact that in the real world, we often^* need to make decisions without sure, advance knowledge of all the outcomes of each choice. A rational decision under these circumstances must be based on some evaluation of the likelihood of some of the outcomes of each alternative.

It does not matter whether the exact mathematical probability of various outcomes under various conditions is known or even hypothetically computable. I use probability if I judge, for example, that because I am plastered, the risk of being in an automobile accident or being arrested for DUI is is too great, and the consequences too severe, for me to accept driving myself home as a rational decision. (And I hope that I retain so much reason in that diminished state.)

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^*maybe always

Answer 3

All probability models are subjective, but some models are more useful than others

There are various ways you can come up with a probability for an event. Bayesian vs frequentists are the 2 main approaches to statistical inference, and within each of those, one can look at different variables or use different formulas. We can also map a continuous variable onto one of many probability distributions to predict the future value of that variable.

For each of these methods, we can see how reliable it is historically, and the more reliable it has been in the past, the more reliable it's likely to be in the future, and the more sense it would make to base any decisions we make on that.

So the method that's proven to be the most reliable is the most useful and is the one we should use (there may be some cases where we'd pick one model above another may be more useful, depending on the factors at play, but "most reliable = most useful" is a good rule of thumb).

This is inductive inference, yes, but not "just" inductive inference. Induction is universally considered to be a fact (there is some philosophical discussion questioning our reasons for believing this, but we still consider it to be a fact).

It is, however, not always easy to come up with a reliable measure, and there are plenty of ways you can go wrong. This is why statistics degrees and statistician occupations exist, and why there is additional knowledge sharing and why results are often checked by others (especially if it's used in science).

What is the "correct" probability?

That's arguably either 1 or 0 for any instance of an event, depending on whether or not the event happens.

But we don't know the future, so the best we can do is stick to our models.

Will someone die tomorrow?

For this problem, I might consider the number of people who die per day, model this as a probability distribution and then determine the probability of getting 0 on that distribution.

This shouldn't be too hard, although we'll need to take into account the hard cutoff at 0, which could cause problems if not careful. Also, the death rate changes over time, but we could limit the results to somewhat recently.

But this isn't a maths forum, so I wouldn't try to actually model this here.

There are most definitely other ways to solve the problem too, but you should end up with something very close to 1 (I'd say the probability shouldn't be equal to 1, but some reasonable methods may end up with that result).

Answer 4

Is all of probability fundamentally subjective and unneeded as a term outright?

Of a murder somewhere in the world tomorrow, "…what is the correct probability then? Is it 1?"

No, it is less than 1, because the question asks about the real world. To give another example, the probability of the sun rising tomorrow is less than one. Granted, the prediction of tomorrow’s sunrise is a safe bet, but the probability of the event is still some sliver less than one.

In a hypothetical world, governed only by assumptions run through formulas, the solution might in fact be exactly 1. It just depends on the assumptions.

When you mention inductive inference, you are on the right track. In an inductive analysis, probability is all that one ever gets, never certainty.

Answer 5

There are two parts to an analysis using probabilities:

1. Create an experiment to determine the probabilities of an event (coin flip). These experiments determine the probability that a coin will land heads or tails. The experiment can take the form of data gathering (The number of murders per day in the US). This part should be as objective as possible and in most cases is objective. The only subjective part in the post would be the definition of "murder"

2. Using the results of part one to make a prediction about a future event. This is the part that is nearly always subjective. If the coin in the experiment is 50/50, what makes one person choose heads and another choose tails? That decision is subjective.

Suppose that experiments and data collection can't be done. For example, determining the probability of intelligent life somewhere other than Earth. This is where subjective probability is useful in making predictions.

So not all of probability is subjective. It's only subjective in its use to predict future events and the future is notoriously unpredictable.

Answer 6

Joseph lives in New York. What is the probability of Joseph being murdered tomorrow? It is the number of murders in New York divided by the total population. This is the definition.

We cannot know this number in advance. We can only estimate it based on past data.

It doesn't mean anything to talk about probability of once-off events. What is the probability the Sun will rise tomorrow? It is the number of times the Sun rose in the morning divided by the number of mornings. That's just 1. What's the probability of the Earth blowing up tomorrow? Same problem.

Answer 7

I concur with the answer by user @Christian Hennig:, there is no "objective" way of ascertaining probabilities in the real world, so in practice the so called objective and subjective viewpoints are intertwined. We use the most objective random situations we can think about, such as coin-tossing or card-shuffling (often called classical probability) as measuring sticks, so less clear situations are assigned probabilities by comparison to these.

I have written more about this at our sister site: https://stats.stackexchange.com/questions/332026/can-we-think-of-a-probability-in-both-the-classical-and-subjective-sense-simulta/332218#332218

Answer 8

There is no such thing as subjective probability. All probabilities are based on objective data.

You cannot assign, calculate or estimate a probability for a single human action. A larger data set is required, either about earlier behaviour of the same person or a larger group of people in a similar situation.

Answer 9

You are just asking for things that are part of a basic college probability course.

The only philosophical aspect of this is that yes, future probabilities cannot be definitively stated without a complete mechanistic understanding of the process.

Invariably, it is not possible to have full mechanistic understanding of any real world process, because there may always be hidden variables. Coin tosses may be influenced by little gust of air or fluctuations in the Earth's magnetic sphere or invisible little trickster demons... Therefore, probabilities are typically calculated from an idealized model (imaginary coin in a room with perfectly still air, perfectly stable magnetic field and no demons) which is then assumed to be "close enough" to the real-world situation.

What is the real probability that a person will be murdered tomorrow somewhere in the world?

Impossible to know for sure, but can be estimated with various straightforward techniques. The simplest one is to simply calculate the murder rate for each past day, average them, and declare that tomorrow will be no different.

One thing you can say is that it will be almost 1. Because each person has probability p of being murdered tomorrow, and the probability of someone being murdered is equal to 1-q where q is the probability nobody will be murdered. Simple math shows (I won't) that q = (1-p)^n where n is the number of people in the world. Because n is so large and p for most people is not that small, q is going to be tiny, therefore 1-q will be almost 1.

But what is the correct probability then? Is it 1?

Of course not. Under any reasonable model there is a nonzero chance nobody will be murdered. Even a pre-planned murder has some small chance to unexpectedly fail.

When calculating the probability of a future event which is not fully understood (99.9% of real life cases) the two common approaches are frequentist and Bayesian.

To greatly simplify, a frequentist will look at how often someone was murdered in the past under the same conditions of note. Which conditions are of note is up to the analyst to decide and convince his audience of. If he chooses them poorly, his probability predictions will be off by more.

A Bayesian will select some probabilities for relevant independent events, which he thinks influence the event of interest (eg. murder rate). He will then use well established mathematical theorems to combine these probabilities and calculate a conditional probability for the event given those probabilities of other relevant events. The decision of which events are relevant and what their probabilities are is up to the analyst to decide and convince his audience of. If he chooses poorly, his probability predictions will be off by more, but it is often believed that Bayesian approaches are not very sensitive to slight inaccuracies in the priors. Sometimes frequentist approaches may be used, so that Bayesian probability becomes a sort of meta-technique for combining less reliable frequentist estimates into a more accurate Bayesian one.

This is of course very superficial, and there is a wealth of specialized techniques to either approach. I cannot possibly cover them here and they would be irrelevant to the question. Luckily, just about any college textbook on probability should cover them in detail.

As for subjectivity, when using models of probability, just like any other model, you must choose some assumptions and be confident they correspond to reality closely enough. The choice of these assumptions and what counts as "close enough" is subjective. Everything else (the calculations from those) is objective. In this sense probability is subjective, but in this sense no study of the natural world can be objective.

Answer 10

This question basically boils down to the question of determinism. Do we live in a deterministic universe? If so, then with enough information, anything can be predicted. The way a die will land can be determined by calculating the forces acting on it. Probability just functions as a tool to guess what will probably happen when not all information is known or there is so much information that needs to be taken into account that making a prediction is unfeasible. And yes, the more information you have, the more accurate your probability estimations can be.

However, the physicists currently believe that we in fact do not live in a deterministic universe. The current consensus of physics is that on the most microscopic level of quantum mechanics, there are indeed things that happen truly random. And those effects do occasionally affect our macroscopic world to some extent.

And then there is the question if human behavior is perfectly predictable. Which boils down to the age-old question "Do humans actually have free will?". This is an open question philosophers constantly debate about since the antique and which we will never be able to answer conclusively. Probably.

Answer 11

The mathematical theory of probability is logically as sound as anything we have. The issue is, what is its significance to the real world?

A lot of philosophers here seem to be saying that we can't define the real-world probability of, say, a particle experiencing radioactive decay, which would come as a surprise to most physicists. If the real world has infinite branching universes, we can define probability objectively using physics: something with probability p happens in universes whose probability mass adds up to p.

Without that, the best we can do is: if we have a large number of events with probability x%, approximately x% of them will happen.

Answer 12

I think your use of the word subjective is somewhat loaded- as if you considered any assessment of probability to be purely a personal matter. Probabilities can be clear-cut in certain circumstances, but in others you must fall back on estimates. In some respects it is no different to making any other form of estimate.

If I am laying an irregular patch of concrete on uneven ground I cannot know exactly how much concrete I need, so I make an estimate. I get an idea of the average length, width and depth and I calculate a volume. I only know how close my estimate was when I lay the concrete. If you were to say that my estimate of the volume of concrete was 'subjective', you would be overlooking the fact that most other people would approach the question in the same sort of way and it could be relied upon to give broadly the right answer, notwithstanding the fact that some degree of personal judgement pays a role.

Now take the question of whether a murder is likely to happen in my town tomorrow. I can estimate the probability in a number of ways, some more sophisticated than others. I could look at the average number of murders in the UK over some period, assume the number is related to overall population, then scale it to the population of my town, and so on. I might see if the murder statistics vary by time of year, by socio-economic conditions, etc, and take them into account in my estimates. There are many approaches I might adopt, and whether I picked one over another might be a matter of personal preference (or ignorance, laziness etc), but I think you are going too far by writing off the whole exercise as purely subjective.

Answer 13

Probabilities in quantum mechanics don't face these issues.

The problem with classical probabilities is that they are born out of ignorance, and there is no limit to ignorance. There can be infinite number of things that you are not considering. When you try to predict the probability that a coin will land heads, you usually reject the possibility that the coin is rigged.

If you don't reject that possibility, you are faced with an infinite number of sub-possibilities : The coin could have probability p for heads and q for tails, where p and q are any real numbers between zero and one.

Even after you consider the above, you need to assume another probability distribution over the space of possible coins, indicating the probability density that you have any particular coin. You can be even further ignorant about this ignorance.

This process can keep on exploding forever. There is no limit to ignorance. In practice, we stop this process somewhere and then apply Bayes' theorem over our thus obtained sample space. Classical probability works in practice

The only way to escape this process without an artificial full stop is when you have zero ignorance : i.e. when you already know the definite state of the coin to be, say, Heads. Then the probability of Heads is 1, and Tails is zero.

Now we come to quantum mechanics

Quantum probabilities are not born out of ignorance. If you have collapsed a particle into a (non-degenerate) eigenstate, you already know the definite state of your system. So you no longer need to consider ignorance.

Does this mean that all the probabilities become 1 or 0 just like what happens classically when you have zero ignorance?

No. In quantum mechanics, a state can be definite with respect to one observable but probabilistic with respect to others. This probability is completely given by the Born rule, and you do not need to consider an endless explosion of ignorance when you calculate this probability.