A probabilistic theory can be considered as a function that maps random numbers to outcomes. It tells us to model the universe as a random number generator piped through that function. A deterministic theory is a special case of a probabilistic theory that ignores its random number inputs, and yields the same output every time.
Here’s an example: We can use the probabilistic theory of quantum mechanics to predict the outcome of a double slit experiment. If we feed a random number to the theory it will predict a photon to hit in a particular location on the screen. If we feed in another random number, it will predict another hit somewhere else on the screen. Feed in lots of random numbers, and we’ll get a probability distribution of photon hits. Believing in the probabilistic theory of quantum mechanics means we expect to see the same distribution of photon hits in real life.
Suppose we have a probabilistic theory, and observe an outcome that is unlikely according to our theory. There are two explanations: The first is that the random generator happened to generate an unlikely number which produced that outcome. The second is that our theory is wrong. We’d expect some number of unlikely events by chance. If we see too many outcomes that our theory predicts should be unlikely, then we should start to suspect that the theory is wrong. And if someone comes along with a deterministic theory that can actually predict the random numbers, then we should start using that theory instead. Yudkowsky’s essay “A Technical Explanation of Technical Explanation” covers this pretty well, I’d recommend giving it a read.
The takeaway is that quantum mechanics isn’t a decision theory of how humans should act. It’s a particular (very difficult to compute) function that maps random numbers to outcomes. We believe with very high probability that quantum mechanics is correct, so if quantum mechanics tells us a certain event has probability 0.5, we should believe it has probability 0.50001 or 0.49999 or something.
Also, in real life, we can never make real-number measurements, so we don’t have to worry about the issue of observing events of probability 0 when sampling from a continuous space. All real measurements in physics have error bars. A typical sample from the interval [0,1] would be an irrational, transcendental, uncomputable number. Which means it would have infinitely many digits, and no compressed description of those digits. The only way to properly observe the number would be to read the entire number, digit by digit. Which is a task no finite being could ever complete.
On the point about real-life measurements: we can observe events of probability 0, such as 77.3±0.1 when the distribution was uniform on [0,1]. What we can’t observe are events that are non-open sets. I actually think that “finitely observable event” is a great intuitive semantics for the topological concept of “open set”; see Escardó′s Synthetic Topology.
My proposal (that a probabilistic theory can be falsified when an observed event is disjoint from its support) is equivalent to saying that a theory can be falsified by an observation which is a null set, provided we assume that any event we could possibly observe is necessarily an open set (and I think we should indeed set up our topologies so that this is the case).
Believing in the probabilistic theory of quantum mechanics means we expect to see the same distribution of photon hits in real life.
No it doesn’t! That’s the whole point of my question. “Believing the probabilistic theory of quantum mechanics” means you expect to see the same distribution of photon hits with a very high probability (say 1−ε), but if you have not justified what the connection of probabilities to real world outcomes is to begin with, that doesn’t help us. Probabilistic claims just form a closed graph of reference in which they only refer to each other but never to claims of the form “X happens” or “X does not happen”.
I’ve already received multiple comments and answers which don’t actually understand my question and tells me things I already know, such as “a probabilistic theory is just some outcome space + a probability measure on it” or some analog of that. I know that already, my question is about the epistemic status of the probability measure in such a theory.
Okay, thanks for clarifying the question. If I gave you the following answer, would you say that it counts as a connection to real-world outcomes?
The real world outcome is that I run a double slit experiment with a billion photons, and plot the hit locations in a histogram. The heights of the bars of the graph closely match the probability distribution I previously calculated.
What about 1-time events, each corresponding to a totally unique physical situation? Simple. For each 1 time event, I bet a small amount of money on the result, at odds at least as good as the odds my theory gives for that result. The real world outcome is that after betting on many such events, I’ve ended up making a profit.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty. I explained above why the “continuous distribution” objection to this doesn’t hold.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty.
I agree with this in practice, but the question is philosophical in nature and this move doesn’t really help you get past the “firewall” between probabilistic and non-probabilistic claims at all. If you don’t already have a prior reason to care about probabilities, results like the law of large numbers or the central limit theorem can’t convince you to care about it because they are also probabilistic in nature.
For example, all LLN can give you is “almost sure convergence”, i.e. convergence with probability 1, and if I don’t have a prior reason to disregard events of probability 0 there’s no reason for me to care about this result.
I think davidad gave the best answer out of everyone so far, so you can also read his answers along with my conversation with him in the comment threads if you want to better understand where I’m coming from.
A probabilistic theory can be considered as a function that maps random numbers to outcomes. It tells us to model the universe as a random number generator piped through that function. A deterministic theory is a special case of a probabilistic theory that ignores its random number inputs, and yields the same output every time.
Here’s an example: We can use the probabilistic theory of quantum mechanics to predict the outcome of a double slit experiment. If we feed a random number to the theory it will predict a photon to hit in a particular location on the screen. If we feed in another random number, it will predict another hit somewhere else on the screen. Feed in lots of random numbers, and we’ll get a probability distribution of photon hits. Believing in the probabilistic theory of quantum mechanics means we expect to see the same distribution of photon hits in real life.
Suppose we have a probabilistic theory, and observe an outcome that is unlikely according to our theory. There are two explanations: The first is that the random generator happened to generate an unlikely number which produced that outcome. The second is that our theory is wrong. We’d expect some number of unlikely events by chance. If we see too many outcomes that our theory predicts should be unlikely, then we should start to suspect that the theory is wrong. And if someone comes along with a deterministic theory that can actually predict the random numbers, then we should start using that theory instead. Yudkowsky’s essay “A Technical Explanation of Technical Explanation” covers this pretty well, I’d recommend giving it a read.
The takeaway is that quantum mechanics isn’t a decision theory of how humans should act. It’s a particular (very difficult to compute) function that maps random numbers to outcomes. We believe with very high probability that quantum mechanics is correct, so if quantum mechanics tells us a certain event has probability 0.5, we should believe it has probability 0.50001 or 0.49999 or something.
Also, in real life, we can never make real-number measurements, so we don’t have to worry about the issue of observing events of probability 0 when sampling from a continuous space. All real measurements in physics have error bars. A typical sample from the interval [0,1] would be an irrational, transcendental, uncomputable number. Which means it would have infinitely many digits, and no compressed description of those digits. The only way to properly observe the number would be to read the entire number, digit by digit. Which is a task no finite being could ever complete.
On the point about real-life measurements: we can observe events of probability 0, such as 77.3±0.1 when the distribution was uniform on [0,1]. What we can’t observe are events that are non-open sets. I actually think that “finitely observable event” is a great intuitive semantics for the topological concept of “open set”; see Escardó′s Synthetic Topology.
My proposal (that a probabilistic theory can be falsified when an observed event is disjoint from its support) is equivalent to saying that a theory can be falsified by an observation which is a null set, provided we assume that any event we could possibly observe is necessarily an open set (and I think we should indeed set up our topologies so that this is the case).
No it doesn’t! That’s the whole point of my question. “Believing the probabilistic theory of quantum mechanics” means you expect to see the same distribution of photon hits with a very high probability (say 1−ε), but if you have not justified what the connection of probabilities to real world outcomes is to begin with, that doesn’t help us. Probabilistic claims just form a closed graph of reference in which they only refer to each other but never to claims of the form “X happens” or “X does not happen”.
I’ve already received multiple comments and answers which don’t actually understand my question and tells me things I already know, such as “a probabilistic theory is just some outcome space + a probability measure on it” or some analog of that. I know that already, my question is about the epistemic status of the probability measure in such a theory.
Okay, thanks for clarifying the question. If I gave you the following answer, would you say that it counts as a connection to real-world outcomes?
The real world outcome is that I run a double slit experiment with a billion photons, and plot the hit locations in a histogram. The heights of the bars of the graph closely match the probability distribution I previously calculated.
What about 1-time events, each corresponding to a totally unique physical situation? Simple. For each 1 time event, I bet a small amount of money on the result, at odds at least as good as the odds my theory gives for that result. The real world outcome is that after betting on many such events, I’ve ended up making a profit.
It’s true that both of these outcomes have a small chance of not-happening. But with enough samples, the outcome can be treated for all intents and purposes as a certainty. I explained above why the “continuous distribution” objection to this doesn’t hold.
I agree with this in practice, but the question is philosophical in nature and this move doesn’t really help you get past the “firewall” between probabilistic and non-probabilistic claims at all. If you don’t already have a prior reason to care about probabilities, results like the law of large numbers or the central limit theorem can’t convince you to care about it because they are also probabilistic in nature.
For example, all LLN can give you is “almost sure convergence”, i.e. convergence with probability 1, and if I don’t have a prior reason to disregard events of probability 0 there’s no reason for me to care about this result.
I think davidad gave the best answer out of everyone so far, so you can also read his answers along with my conversation with him in the comment threads if you want to better understand where I’m coming from.