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Objective probability (of specific state)

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Objective probability (of specific state)

Post by ThalesOfAthens » March 2nd, 2017, 6:13 pm

Physical systems have state. We can ask- what is the probability of this state we exist in now?

Exploring this question will allow us a better understanding of how natural systems work, how evolution works, and even lend insight into how communication works. It provides a conceptual framework which is distinct from other understandings of probability which allows us to understand more clearly how things function.

Understanding the question of objective probability of specific state will allow us to understand:

- What is the information content of a physical system?
- What is the complexity of a physical system?

Taking these ideas to a larger level, we will eventually be able to ask:

- What is the information content of the universe as a whole?
- What is the complexity of the universe as a whole?

To begin, we need to define some terms.

The objective probability of specific state is the probability at a specific point in time that a physical system exists in its state, relative to an earlier specific state at an earlier specific point in time, considered independently from any specific observers knowledge or expectation. This will be called the "Oposs probability". (sorry about the redundancy)

For any system, there are many probabilities that can be understood. For example, we can ask of a deck of cards- what is the probability we will randomly draw a red card? Or, what is the probability we will draw a spade? These are probability questions that have different answers. Neither of the questions is "more correct" but they have different answers, and it is important when looking at questions of probability to see if two questions are the same or different before accidentally conflating them.

Similarly, for any system, there can be different observers with different knowledge or expectations. We can imagine a machine which accepts a deck of cards and observers are told it "randomly spits out a card". Then we can ask- what is the probability it will spit out the Ace of spades? To the first observer, it appears to be 1 in 52. A machinist may open the machine and see that all of the red cards are discarded before the shuffle, and to this person it appears to be 1 in 26. A second machinist may observe that after the first separation, another card separation takes place and only the spades are left before the shuffle. This again changes the probability. These are all subjective probabilities, and are valuable, but different than the objective probability, which exists independently of any specific observers knowledge or belief.

When there is talk about the probability of physical systems, it normally is in reference to statistical mechanics. The Oposs probability is a distinct probability from statistical mechanics.

In statistical mechanics, there is a macrostate which contains many microstates, and the system moves probabilistically towards the macrostate which contains the majority of micro states. This is a different question of probability than Oposs, as statistical mechanics is a question of "what is the probability of the current state falling into a macro state", not a question of "what is the probability of a specific micro state".

For example, if one had a box full of deterministically moving molecules on one side, with a separation preventing them from moving to the other side, we can ask the question: when we remove the partition, what is the probability at a future point in time that we have reached a macro state where the molecules are evenly distributed? We can compute the probability, without knowledge of where the particles are specifically but only a gross knowledge of the energy, at any point in time in the future. However, this question of probability is not Oposs probability. We can see for this example, given the specific starting state and deterministic movement of the molecules, at every later time there is a 100% probability the molecules are at a given location. Because the molecules are moving deterministically, they do not change the probability of the Oposs. There is a 100% probability for each moment in time moving forwards that it exists at a specific state.

The Oposs is 100% at all times for this system.

The revelation here is that when one reads about "the probability of the system" or "the probability of the universe" the discussion is often about the statistical mechanic probability. This is a useful, but different, question from the Oposs probability. We shouldn't conflate them.

So now that we are clear there is a totally different probability from the statistical mechanical probability, we can ask- what is it? What does it mean?

If there are specific questions about how to interpret this notion of probability with regards to physical systems I'd like to hear them.

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Re: Objective probability (of specific state)

Post by Consul » March 2nd, 2017, 8:54 pm

You seem to be talking about the objective, physical probabilities called chances.

"Chance is objective single-case probability: for instance, the 50% probability that a certain particular tritium atom will decay sometime in the next 12.26 years. Chance is not the same as degree of belief, or credence as I'll call it; chance is neither anyone's actual credence nor the credence warranted by our total available evidence. If there were no believers, or if our total evidence came from misleadingly unrepresentative samples, that wouldn't affect chance in any way."

(Lewis, David. "Humean Supervenience Debugged." 1994. Reprinted in Papers in Metaphysics and Epistemology, 224-247. Cambridge: Cambridge University, 1999. p. 227)

"Chances are real features of the world. They show up, [for example], in how often people get cancer and coin tosses land heads, and they are affected by whether people smoke, and by how coins are tossed. Chances are what they are whether or not we ever conceive of or know about them, and so they are neither relative to evidence nor mere matters of opinion, with no opinion any better than any other."

(Mellor, D. H. Probability: A Philosophical Introduction. London: Routledge, 2005. p. 8 )

Chance versus Randomness:

Philosophy of Statistical Mechanics:
"We may philosophize well or ill, but we must philosophize." – Wilfrid Sellars

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Re: Objective probability (of specific state)

Post by ThalesOfAthens » March 4th, 2017, 9:52 am

Chance, as defined above, includes the notion of objective probability but neglects to talk about the type of probability under consideration. Here we are interested in the objective probability of a specific state.

In the Stanford Encylopedia of Philosophy the entry on “Chance vs Randomness” [1] states:

Loewer (2001) draws on the best systems analysis of chance to argue that, in worlds like ours (with entropy-increasing processes, and apparently fair coin tosses, etc.)

Here the notion of chance covers both entropy increasing and that of fair coin tosses, and the purpose of objective probability of specific states is to disambiguate these notions of chance. Therefore, the notion of specific state is critical to an understanding of probability.

The usage of objective probability alone leads to confusion in a lot of literature about how things can be moving into more probable, and less probable, states at the same time. For example, if one googles “entropy vs evolution” a lot of discussions come up attempting to explain this situation. Here is a typical explanation from Cornell: [2]

Even if it is true that the processes of life on earth result in an entropy decrease of the earth, the second law of thermodynamics will not be violated unless that decrease is larger than the entropy increase of the two heat reservoirs.

This is an attempt to explain this apparent contradictory statement:

Physical systems move into more probable states (entropy) at the same time they move into less probable states (dice land with low probability).

However, if we clarify this statement, we can see that the two halves of the sentence are discussing completely different types of probability.

Entropy is a consideration of the probability of a microstate within a specific macrostate. A macrostate is defined as a state of the system where the distribution of particles over the energy levels is specified.[3]

In the objective probability of specific state, the probability is of a microstate alone, considered against all other possible microstates. The contradictory statement becomes clearer when restated:

Physical systems move into more probable macrostates with respect to a specified energy level while moving into a less probable microstate with respect to the probability of that microstate by itself.

This is similar to asking “How is it possible that I draw a red card with 50% odds while I draw a spade with 25% odds?” They are simply different questions, and do not need to be reconciled with each other.

The macrostates of statistical mechanics are aligned with specified energy levels. This is useful if one has a thermometer and wants to assess the probability with respect to this. However, we can consider arranging the set of all possible microstates into other collections of macrostates. For example, we could collect all the microstates together where it rains tomorrow at a specific location, and put them in a macrostate and we could collect all the microstates together where the reader wins the lottery and put them in a macrostate. These are not invalid organizations of macrostates, and lead to different answers with regards to the probability of a macrostate arriving. The probability of arriving in a macrostate including the microstate of the reader winning the lottery is not at odds with the odds with the probability of the macrostate being at a new energy distribution. They are simply different probability questions about the same final state.

We could consider arranging macrostates with regards to the objective probability of a specific microstate. For example, take all of the microstates with the same self probability, and pool them into the same macrostate, and take all the microstates with a slightly higher probability, and pool them together as well. This is a new macrostate arrangement which is also not at odds with the macrostates of statistical mechanics- it is simply a different arrangement, and leads to different answers about the probability of arriving at a specific state.

[1] Stanford Encyclopedia of Philosophy the entry on “Chance vs Randomness”
[2] Cornell website on astronomy
[3] Statistical Mechanics by Alfred Huan

I do not have permission to place links yet.

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