"Probability Fallacy"

[uDevil]

I hadn't checked out the "plog" feature at Amazon until today, when I found the following post by the author of a number of technical books.

Regarding the part in bold, I'm guessing this is an old argument, but my uneducated view is that there is no logical inconsistency and that multiple universes are neither implied nor precluded.

What say the probability forum regulars?


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Probability Fallacy
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by Stan Gibilisco at 6:03 PM, March 20, 2007
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Several months ago, someone took me to task about the following line of reasoning, saying it is overly simplistic and explaining why. I've lost that e-mail, but it made good sense. What am I missing?
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When scientists formulate theories, they sometimes say that an event "probably" occurred in the distant past, or that something "might" exist, as-yet undiscovered, right now. Have you ever heard that there is a "good chance" that extraterrestrial life exists? Such a statement is meaningless, I say, from the point of view of a logician. Either there are space aliens, or there aren't.
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If I say "I believe the universe began with an explosion," I am stating the fact that I think something is true (but I do not know for sure). If I say "The universe began with an explosion," the statement is logically sound, but it is a hypothesis, not a proven fact. If I say "The universe probably started with an explosion," then what? Does this imply a claim that there were multiple pasts and the universe had an explosive origin in more than half of them? How can probability be assigned to something that happens -- or fails to happen -- only once?


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[Siegmund]

I think most everyone would agree his first example - the chance that aliens exist - isn't meaningless. People come up with back-of-the-envelope estimates of what percentage of planets have the necessary ingredients for life, and the chance of life starting randomly, and astronomers can give us estimates of how many stars there are and how likely stars are to have planets. New stars and planets are forming (somewhere in the universe) all the time. This particular experiment really IS being performed billions of times.

Of course, some estimators calculate the expected number of planets with life is very near zero, and others calculate that number to be extremely large. Very few calculate that number to be close to one.

His second argument - the universe probably started with an explosion - might offend hardcore frequentists. To Bayesians it's a fairly normal question to ask: we exist; here's a list of ways that we might have come to exist; here's an estimate of which of these ways we think is most plausible (priors, and perhaps the results of some experiment to see which models gave the right predicitions); so we calculate a conditional probability that, given that the world exists as it does today, this is how it started.

In fact there is a whole sub-discipline of Bayesian statistics called "model selection", based on the idea that scientists often start with several competing hypotheses and then collect additional data that supports some and falsifies others.

Some frequentists would have your quoted objection. Other frequentists would accept that even if the universe only *did* have one past, there are various things that *could* have happened in the past.

[T50_Omaha8]

Say you walk up to a craps table right after everyone cheers and you ask what the person just rolled. They reply "he rolled a 4". From this information you deduce that there is a 66% chance that one of the two dice showed a 3 on the last roll. Even though the event happened in the past, there is uncertainty as the what the precise outcome of the event was, so you use information you know to assign probabilities to all the possible outcomes.

If the event in question is the begining of the universe rather than a roll of dice, you are merely assigning different probabilities for the same outcome, so asserting that the universe "probably" started with an explosion just means you attatch a >50% chance to that cause, just like you attach a 66% chance to one of the dice being a 3 even though it did have some specific outcome.

In other words, I see very little difference between the creation of the universe and a crapshoot.

[mt_david]

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In other words, I see very little difference between the creation of the universe and a crapshoot.

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You really don't think craps is +EV do you? [img]/images/graemlins/grin.gif[/img]

[DWarrior]

My guess would be the author is confused between probability and reality.

In T50's example, a 3 was *probably* rolled, but it either did or it didn't.

[f97tosc]

I think we have to start by asking what we mean by probability.

According to the frequentist definition, a probability is a limit of (citing Wikipedia here) "relative frequency in a large number of trials." The author of the post you cited seems to subscribe to this view. Then, if there is only one trial, then as the quoted article says, the probability is ill defined. I think by "trial" it is usually meant something that can be observed, and since we can't observe things in other universes (if they exist), we couldn't reasonably make a statement about the probability that our universe started in such a such way. This is not to say that other universes can't exist, it just means that (according to this definition) it is meaningless to make claims about their probability.

According to the alternative, Bayesian, view, a probability is fundamentally a statement about the observer, and to which extent he or she can rationally believe a statement to be true. According to this definition, one could well talk about a probability for the universe starting off from nothing. To make a somewhat contrived example, suppose we found that only two basic physical models of equal simplicity can describe our universe. Suppose that one of them suggested that the universe was caused by another universe, and the other that it came from nothing. Then, in the Bayesian view, we may well say that the probability that the universe came from nothing is 0.5. This is not a statement about the origins of the universe as much as it is a statement about us - it says that we have no particular reason to favor one model over the other.

[jason1990]

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If I say "The universe probably started with an explosion," then what? Does this imply a claim that there were multiple pasts and the universe had an explosive origin in more than half of them? How can probability be assigned to something that happens -- or fails to happen -- only once?


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There is a tendency to assume that probability statements can only be interpreted in one of two ways: using the frequency interpretation or the Bayesian interpretation. People tend to forget the classical interpretation, which states that the probability of an event is the ratio of outcomes in that event to total outcomes. The classical theory only applies in situations with outcomes that are equally likely. This would seem to severely limit its scope, and may be one reason it tends to be forgotten. But in practice, the classical theory is more applicable than it may seem. Any time we can partition the collection of outcomes into equally likely events, we can assign probabilities without reference to long runs of independent experiments or to subjective assignments of probabilities. In practice, this partitioning is typically done using symmetry.

One thing which might be meant by the bold statement is: "I have a model which describes the possible ways the universe could have begun, given what we know today. This model contains certain physical symmetries which allow these possible ways to be partitioned into collections which are equally likely. More than half of these collections consist of ways that involve an explosion."

As another poster mentioned, it is similar to the following. I roll a die inside a closed box. A camera photographs the result, and the die is destroyed. What is the probability that the die landed on 1? The roll happened in the past. It happened only once and it will never happen again. Yet we still say the probability is 1/6. When we say this, we are not necessarily claiming something about a long sequence of hypothetical die rolls. And we need not be claiming that 1/6 is our subjective opinion on the matter. The claim that the probability is 1/6 is simply the claim that the die is symmetric. As the quote says, this is a logically sound hypothesis, although not a proven fact.

If I postulate certain symmetries about the possible origins of the universe, then I can make similar probabilistic claims. Those claims would not refer to multiple universes, but only to this one. And they would not be subjective claims, since I would be asserting a hypothesis about these physical symmetries -- an assertion which is logically sound and which is either true or false.