/
What Physical Probabilities there Are and what Physical Pro What Physical Probabilities there Are and what Physical Pro

What Physical Probabilities there Are and what Physical Pro - PowerPoint Presentation

trish-goza
trish-goza . @trish-goza
Follow
388 views
Uploaded On 2017-06-22

What Physical Probabilities there Are and what Physical Pro - PPT Presentation

Rutgers September 262016 Two Faces of Probability subjectiveobjective Credences and Physical Probabilities T here are two kinds of probabilities 1 Probability as a subjective measure of degree of belief or credences constrained by principles of rationality the axioms of probability a ID: 562249

probability probabilities laws physical probabilities probability physical laws pps theory credences fundamental world dynamical conditional events objective belief deterministic

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "What Physical Probabilities there Are an..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

What Physical Probabilities there Are and what Physical Probabilities Are.

Rutgers

September 26,2016Slide2

Two Faces of Probability

subjective/objectiveSlide3

Credences and Physical Probabilities

T

here are two kinds of probabilities: 1. Probability as a subjective measure of degree of belief or credences constrained by principles of rationality (the axioms of probability and sometimes other constraints e.g. indifference).

2. Probability as an objective physical feature of reality; physical probabilities (PPs)

Some statisticians have no use for credences (think they are too subjective) and some (e.g. deFinetti) don’t believe in the existence of PPs. But I think both are important in science and that they are related to each other.Slide4

Physical Probabilities (PPs) are objective probabilities specified by scientific laws.

Physical probabilities are not credences and not subjective. They are objective mind independent features of reality that play a role in laws, causation, explanation and so on. They are empirical objects of scientific investigation.

Questions to be addressed in this talk:

1. What physical probabilities (PPS) are there?

2

.

What are PPs? (What facts about the world make it the case that, for example, the physical probability of an x-spin measurement on a y spin up electron yields down is .5 or the probability of rain tonight is .8) That they are specified by scientific laws doesn’t answer this question but helps pose it!

3

. How are PPs normatively related to credences, acceptance, betting, degrees of belief, etc.?

4. How is it possible for PPs (on various accounts of what they are) to provide normative constraints on belief etc.?Slide5

Philosophy of Science Assumptions

1. Realism: i) The aim of scientific theorizing (especially physics) is to discover true theories and ii) Contemporary science has been successful in discovering theories that we have reason to believe are true or approximately true.

2. Fundamentalism: i) the aim of fundamental physics is to discover theories that describe fundamental ontology and the laws that characterize it.

i

i) Fundamental laws are complete in that they cover all fundamental physical events and so all events and facts that supervene on them. Slide6

What PPs there are

1. Quantum mechanical probabilities as specified by Born’s rule. There are, roughly, three types of realist versions of quantum mechanics.

A) Spontaneous collapse theories (GRW) that include an indeterministic dynamical law that characterizes the evolution of the quantum state.B)Supplementary variable theories (Bohmian Mechanics) that include deterministic dynamical laws and a probability distribution over the initial values of supplementary variables (e.g. particle positions).

C) Everett Many worlds that include deterministic dynamical law (as in B) but no supplementary variables. There are difficulties understanding QM probabilities in Everett.

2.

S

tatistical mechanical probabilities.(these may be grounded in QM dynamical laws)

3. Special science laws specifying non fundamental dynamical and/or initial condition probabilities e.g. genetics, medical statistical laws, learning theory, evolutionary theory, meteorology…

4. Probabilities associated with repeatable processes e.g. coin tossing

5. Objective probabilities of individual events e.g. that Clinton (Trump) will win the electionSlide7

Probabilities and laws

Claims:

1. All PPs derive from laws. 2. Since quantum mechanics and statistical mechanics cover all micro and macro physical events all PPs ultimately derive from these probabilities. 3. There may be objective credences that are grounded in principles of rationality or logic (as e.g. Principles of Indifference or maximum entropy as Carnap, Jaynes, Williamson hold) but these are not PPs and cannot conflict with PPs.Slide8

Probability in Statistical Mechanics

D

eterministic dynamical laws (as in classical mechanics ) are not sufficient to explain the second law of thermodynamics and the probabilities that appear in statistical mechanics and other special science laws. There is a proposal for a framework that includes statistical mechanical laws that

arguably

also grounds special science probabilities and in addition explains temporal arrows of knowledge, causation and influence and via them our sense that “time flows” from past to future. (Boltzmann, Feynman, Albert, Carroll, Loewer)Slide9

The Mentaculus*

The proposal is that the correct theory of the world includes

1. physically complete dynamical laws (quantum mechanical deterministic or probabilistic)

2.

The PH (the low

entropy

macro

condition

M(0) 17.82

billion years from today)

3.

a uniform probability distribution over physically possible

states compatible with the PH)

*“Mentaculus” comes from the Coen Brothers’ film “A Serious Man” where it is used by the nebishy brother of the film’s main character as a name of what he calls “a probability map of the world.”Slide10

Completeness of the fundamental probabilities

The uniform probability distribution over the states compatible with M(0) and the dynamical laws (whether deterministic or indeterministic) determines a probability distribution over

all micro histories of the universe compatible with the universe and thereby a conditional probability distribution over all pairs of physically expressible macro propositions. e.g. P(The U.S. wins the next world cup/everything we know) is well defined. Another consequence is that true special science probabilities most agree with fundamental probabilities if the fundamental probabilities are correct.Slide11

The Universe

According to

the Mentaculus:microphysical determinism and macro indeterminism withbranching toward the futureSlide12

The Mentaculus and Time’s Arrows

The Mentaculus entails (by the usual Boltzmann reasoning) that the temporally asymmetric second law of thermodynamics holds for the universe a whole and for its approximately isolated subsystems. It also grounds the temporal asymmetries of knowledge (we have records of the past), influence (we have some influence over the future but not the past), counterfactuals (small differences at a time can lead to large differences in the future but typically not the past) and causation (cause precede their effects). The reason to think that the Mentaculus is true is that it accomplishes the above. (For details see Albert 2000, 2015 and Loewer 2006, 2016)Slide13

The Mentaculus and special

s

cience lawsSince the Mentaculus specifies conditional probabilities for all pairs of macro propositions any special science law that takes the form of a conditional probability (e.g. Gresham’s law: the introduction of bad money into an economy under conditions C makes it likely that good money will be horded) are grounded in the Mentaculus. Of course this doesn’t mean that we can discover special science laws by deriving them from Mentaculus probabilities. But the Mentaculus does explain why frequencies typically provide good evidence for special science laws. Slide14

Fundamental chances

Fundamental chances (F,Cs) are objective dynamical conditional objective probabilities specified by fundamental indeterministic dynamical laws. For example, in GRW the fundamental dynamical law specifies the chances that the quantum state of a system evolves to one among a range of subsequent states. The laws may be local in that the probability of a system whose state is Q evolving in various ways depends only on Q and not on any other facts about states at the time of Q, or the backward light cone of Q For example, in QM the chance that a radium atom decays in the next hour is local and independent. (QM involves complications that we won’t go into here.) of the rest of the world and the past. FC’s are incompatible with determinism and are directed in one temporal direction (past to future) so not defined for past events.

However, even if the dynamical laws are deterministic there are conditional probabilities compatible with the Mentaculus that are very much like fundamental indeterministic chances. Call them “deterministic chances” DCsSlide15

Chance set-ups

A DC is a conditional probability that is robust and repeatable. For example, the probability that a well tossed ordinary coin lands heads.

Robustness: P(A/F&K) = P(A/F&K&Q) for a wide variety of Qs including perhaps the complete macro description at times prior to K . A= outcome, K= description of chance set up, F=trigger (e.g. flipping).D

Cs are maximally robust objective conditional probabilities.

If the laws are deterministic there still may be chances that are not FCsSlide16

Examples

The PP of this coin landing heads on the next flip is the

Mentaculus probability of its landing heads given its physical make up, how it is tossed and other macro information. This probability is robust since the outcome is independent of further macro information.

The physical probability of Clinton winning the election today is the Mentaculus conditional probability of Clinton winning given the current macro state. Polling provides information that supports and estimate of this probability.Slide17

Determinism and Physical Probability

The Mentaculus assigns physical probabilities to events (and conditional probabilities to pairs of events) even though its dynamical laws may be deterministic. If the fundamental dynamical laws are indeterministic as in GRW then the GRW dynamical probabilities entail statistical mechanical probabilities (Albert 2000). In this case thee is a probability distribution over physically possible all micro and macro histories of the universe.Slide18

Many think that physical probabilities are incompatible with determinism

“Today I can see why so many determinists, and even ex-determinists who believe in the deterministic character of classical physics, seriously believe in a subjectivist interpretation of probability: it is in a way the only reasonable possibility which they can accept: for objective physical probabilities are incompatible with determinism; and if classical physics is deterministic, it must be incompatible with an objective interpretation of classical statistical mechanics” Karl Popper, Quantum Theory and the Schism in Physics.

“To the question of how chance can be reconciled with determinism....my answer is it can’t be done....There is no chance without chance. If our world is deterministic there is no chance in save chances of zero and one. Likewise if our world somehow contains deterministic enclaves, there are no chances in those enclaves” . David Lewis in Postscript to “A Subjectivist’s Guide to Objective Chance.” Slide19

Deterministic Physical Probabilities

Many (like Popper and Lewis) philosophers think fundamental chances are all the objective physical probabilities there are, that all physical probabilities are temporally directed, that determinism is incompatible with non-trivial physical probabilities, that initial condition probabilities (e.g. Stat mech and Bohm) are not physical probabilities but must be understood epistemically. But I will describe an account of physical probabilities on which non trivial PPs are compatible with determinism.Slide20

Physical Probabilities are Normative

P

hysical probabilities are features of the world like causal relations, laws, perhaps like mass, charge etc. They are also supposed to normatively constrain our beliefs. As Al Hajek says

“Degree of belief is to objective probability as

Belief is to truth.”

Just as “truth” is a norm (what we ought aim at) by way of belief physical probability is a norm for degree of belief? Slide21

PP’s as norms of belief

That truth should be a norm for belief is uncontroversial. But that objective probability is a norm for degree of belief can appear to be deeply mysterious. Why should the fact that the physical probability that given its present condition a lump of radium at time t will emit an alpha particle at t’ or that the statistical mechanical probability of an ice cube melting in by t’ have any bearing on what our credences at t should be about what happens at t’?Slide22

Answer: Physical probability

is

the feature of the world that normatively constrains degrees of belief (Lewis at one point).But our question is what could this feature be? Does it really exist? (It seems “queer” in the way Mackie says objective norms are)R

eturn to this question but first we need to look at more specific formulations of physical probability: degree of belief norms.Slide23

Lewis’ Principal Principle

PP1: C(A/Ht,T)

= Pt(A/Ht,T)(where Pt(A/Ht) is the probability theory T assigns to A conditional on Ht, Ht is the history up to t, and C(A/Ht,T) is the credence one ought to have in A conditional only on Ht,T and nothing not implied by them.

2. PP2: C(A/(Pt(A)=x))=x and C(A/Pt(A)=x & Q)=x

(Q is “admissible” Lewis’ gloss..any proposition entirely about the history prior to t and the laws.)Slide24

Lewis on Chance I

Lewis was thinking of physical probabilities as fundamental dynamical probabilities; e.g. given the complete physical state of the world at t the physical probability of A obtaining at t’. He thought that determinism is incompatible with non-trivial physical probabilities and that whether or not determinism is true that physical probability of propositions concerning times prior to t are all trivial. He didn’t consider initial condition probabilities and theories that assign probabilities to all physically possible histories and to all conditional probabilities. Slide25

Connecting PPs and Credences

Externalist Principles:

3. Your UR conditional credences (your degrees of belief before you obtain any evidence) should match the physical conditional probabilities specified by the true complete theory.

4.Your credences at t should match the physical probabilities conditional on what you

know

at t; i.e. conditionalize UR credences on what you know (or what you ought to know given your situation at t)

Note 1: it will follow from reasonable accounts of knowledge on our proposals for complete theories that what you know about the future and past is limited to what you can infer (on the basis of conditionalizing the UR distribution on knowledge of the present). So you couldn’t know, for example, the outcome of a y-spin measurement on an x-spin electron.

Note 2: Admissibility is not needed in these formulations.Slide26

Internalist Principles

5

. Your credences at t should match the UR credences conditional on what you fully believe at t (if what your fully believe is compatible with the world’s probability distribution.)

6

. Your conditional credences

should match

those specified by

the complete theory you believe to be true.

7. Your UR* credences should be

weighted (by your subjective

probabilities over all

complete

physical probability

theories that qualify as Best Theories (more soon). Slide27

Back to the Mystery

All the principles claim that the world or our beliefs about the world- about what fundamental probabilities there are- normatively constrain our beliefs in non probabilistic propositions. How can that be? How can the fact that at t the physical probability that a radium atom will decay in the minute after t- a fact that obtains at t- normatively constrain our belief about what will happen after t? Slide28

What are Physical Probabilities

An account of PPs should satisfy the following:

1. Connect PPs with laws2. Allow for PPs both with deterministic and indeterministic dynamical laws3. Be applicable to the PPs that occur in QM, SM special sciences and so on.

4. Explain and justify the Principles connecting PPs with credences Slide29

What are physical probabilities?

Primitivism/Propensities (Popper): Physical Probabilities (or truth makers of propositions specifying probabilities) are fundamental irreducible features of reality not supervenient on the totality of categorical facts/events that have the power of producing frequencies and normatively constraining credences.

Elimitivism/Instrumentalism (deFinneti):

Frequentism: (von Mises, Reichenbach) a) actual frequencies b) counterfactual frequencies

Best

S

ystem Account (Lewis, Loewer): Physical probabilities (true probability propositions) are supervenient on/ reducible to the patterns of fundamental categorical facts/events. They are specified the theory that best systematizes the totality of fundamental facts.Slide30

Troubles with Primitivism

What are these primitive features of reality that normatively constrain credence? Don’t think that Laws of large numbers are any help (Strevens)

One primitivist view is that PPs are degrees of propensity to produce. In GRW the probabilities are degrees of the propensity of the wave function at one time to produce various subsequent wave functions.

The notions of production and degree of production are utterly mysterious (not garden variety e.g. the acorn produced a tree).

Since propensity PPs don’t supervene they can be arbitrarily altered while the categorical facts remain the same. This makes it utterly mysterious how they can provide norms for belief about the categorical facts.

Propensity PPs don’t apply to initial conditions. Slide31

Trouble with Elimitivism/Instrumentalism

I agree with elimitivists that if PPs (and laws) must satisfy all the conditions placed on them by our folk concepts it is plausible that there are none. But we need laws and PPs for science.

It is incredible that the Heisenberg uncertainty principle, the Bohmian prohibition on superluminal signaling, the second law are all merely bits of advice and not made true and laws by facts.

Why do QM, Stat mech probabilities provide “good advice”? If the answer is that it is something about the categorical facts that makes certain advice good and certain not so good then this can be parlayed into a reductive/supervenient account (but one would still want to know why these facts makes the advice good)Slide32

Actual Frequentism

The probability of A occurring on trial E is the actual frequency of As occurring on E.

Problems: 1. Applies only to types of events not to token events

2. Not every frequency is a probability

3. There are probabilities that occur in laws that differ from actual frequenciesSlide33

Hypothetical Frequentism

The probability of A occurring on E is the limit of frequencies of A on E that would be obtained were E repeated infinitely many times.

Problems:Applies to types only not token events

The counterfactual specifying the long run frequency is not well defined.

Evaluating the counterfactual presupposes laws that mention probabilities so the account is circular.Slide34

Lewis’ BSA

Lewis’ own view is that probabilities supervene on the totality of fundamental categorical facts via the Best Theory account of laws and chances:

Laws:“Take all deductive systems whose theorems are true. Some are simpler better systematized than others. Some are stronger, more informative than others. These virtues compete: An uninformative system can be very simple; an unsystematized compendium of miscellaneous information can be very informative. The best system is the one that strikes as good a balance as truth will allow between simplicity and strength. How good a balance that is will depend on how kind nature is. A regularity is a law iff it is a theorem of the best system.” (1994a p.478) Slide35

Humean account

Lewis’ BSA is a Humean account of laws in that whether an equation or proposition is a law is determined by the totality of occurrant events. The Best System of a world is the best (or one of the best) summaries of the events of the world. On non-Humean accounts (e.g. Maudlin) laws are ontological items over and above the totality of occurrant events that in some sense “generate” events.Slide36

Extended to include PPs

“Consider deductive systems that pertain not only to what happens in history, but also to what the chances are of various outcomes in various situations - for instance the decay probabilities for atoms of various isotopes. Require these systems to be true in what they say about history....Require also that these systems aren't in the business of guessing the outcomes of what, by their own lights, are chance events; they never say that A without also saying that A never had any chance of not coming about

.”

(1995 p.480). Slide37

Probability is a device for permitting informative and simple laws.

The idea is that probability is introduced into candidates for best systems as a device for enabling a law to express information simply.

e.g. the sequence hthhthtttthththhthtt……may be simply and informatively described by saying that it fits the probabilistic law that the members of the sequence are outcomes of independent trials each with a probability of .5.Slide38

Deterministic PPs

Although Lewis thought that PPs are all dynamical and incompatible with determinism his Best System account applies to initial condition PPs and applies even if the dynamical laws are deterministic. For example, the Mentaculus is a proposal for the Best System of a classical mechanical world similar to the actual world. Slide39

The Big Bad Bug

Before proceeding to see whether the Best System account can do better wrt normative role of PPs we need to confront the Bug. The Best Theory of a world @ (BT@) may assign a non-zero probability

ε to the proposition Q that the world is one whose Best Theory is incompatible with BT@. In that case your cred(Q/BT@) should be 0 not ε

.

There is a big literature on responding to this. I propose simply to conditionalize the UR distribution on the Best Theory account (equivalent to the NP).Slide40

Can the Best System’s account ground probability-credence norms?

Lewis famously says:

“Be my guest—posit all the primitive unHumean whatnots you like. … But play fair in naming your whatnots. Don't call any alleged feature of reality ‘chance’ unless you've already shown that you have something, knowledge of which could constrain rational credence. I think I see, dimly but well enough, how knowledge of frequencies and symmetries and best systems could constrain rational credence.

“(

Lewis 1994:484)Slide41

Proposal

I don’t know what Lewis saw but here is a proposal. Introducing an uninterpreted expression P(A/B) for a probability function into the language for comparing candidate best theories increases informativeness with not much cost in simplicity. Here is how. Evaluate informativeness of a theory in terms of what it says you ought to believe or what degrees of belief you ought to have about the categorical facts. We need some principle to make this connection. The PP is such a principle. We can then evaluate the theory in terms of its being the assigning a high probability to true propositions and low probability to false propositions. In this way we provide an interpretation for P(A/B); i.e. we associate with each candidate best theory a class of worlds for which it is the best theory. P(A/B)=x is true at a world w if it is entailed by the best theory for w. [lots of details need to be worked out since there are infinitely many propositions, we need scoring rules etc. but the idea is clear enough.)Slide42

OK

. But why should the fact that on this account P(A/B) is true guide my credences?

Answer: Because in seeking a best theory you committed to a principle connecting PPs to credences in order for that theory to inform you about the categorical facts. To ignore the principle is to reject the commitment. You could have chosen a different principle e.g. the anti-PP (in which case you would interpreted P(A/B) differently.) Or you could have chosen no principle (in which case you turned your back on the idea that you were seeking a Best Theory of the World)Slide43

Slightly different proposal

Think of a Best Theory with PPs as informing you about the world by telling you what credences to have via some principle e.g. the PP, the anti-PP. Then assume that there are a finite number of candidate (sufficiently simple etc.) Best theories and a uniform subjective distribution over them and say that a candidate T is obtains at a world w just in case your credence C(T/w) is much higher than for any alternative C(T*/w). *

*what if there is no winner? Similar question wrt laws.Slide44

As on the earlier proposal if you are committed to seeking a Best Theory and characterizing its informativeness (and simultaneously interpreting the physical probability function) via a particular normative principle you are committed to that principle. Slide45

Avoiding a misunderstanding

This is not a “proof” of the PP or any principle connecting physical probabilities to credences. It is a “vindication” or a “rationalization” provided within the context of a Best System account of laws and probabilities. If one is committed to seeking a Best systematization of the categorical facts and adds probability expression to the language to abet this then one would be committed to a principle to connect probabilities to credences in order to provide content to that expression and to extract information from the Best Theory.Slide46

Conclusion

What Physical Probabilities Are there?

The probabilities implied by quantum mechanics and the Mentaculus. These assign a probability to every physically possible history of the universe and thereby conditional probabilities over all pairs of physically specifiable propositions.What are Physical Probabilities? PPs are the probabilities entailed by the best systematization of the actual fundamental history of the universe. I conjecture that system is the Mentaculus. Slide47

The End