Transcript Slides

What Physical Probabilities there
Are and what Physical Probabilities
Are.
Rutgers
September 26,2016
Two Faces of Probability
subjective/objective
Credences and Physical Probabilities
• There 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.
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.
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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.?
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.
• ii) Fundamental laws are complete in that they cover all
fundamental physical events and so all events and facts
that supervene on them.
What PPs there are
•
•
•
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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. Statistical 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…
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4. Probabilities associated with repeatable processes e.g. coin tossing
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5. Objective probabilities of individual events e.g. that Clinton (Trump) will win the election
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.
Probability in Statistical Mechanics
Deterministic 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)
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.”
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.
The Universe According to the Mentaculus:
microphysical determinism and macro indeterminism with
branching toward the future
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)
The Mentaculus and special science
laws
• Since 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.
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” DCs
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).
• DCs are maximally robust objective conditional
probabilities.
• If the laws are deterministic there still may be chances
that are not FCs
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.
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.
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.”
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.
Physical Probabilities are Normative
Physical 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?
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’?
• 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)
• Return to this question but first we need to look
at more specific formulations of physical
probability: degree of belief norms.
Lewis’ Principal Principle
1. 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.)
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.
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.
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).
Back to the Mystery
All the principles claim that the world or our
beliefs about the world- about what
fundamental probabilities there arenormatively 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 tnormatively constrain our belief about what will
happen after t?
What are Physical Probabilities
• An account of PPs should satisfy the following:
• 1. Connect PPs with laws
• 2. Allow for PPs both with deterministic and
indeterministic dynamical laws
• 3. 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
What are physical probabilities?
1.
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.
2.
Elimitivism/Instrumentalism (deFinneti):
3.
Frequentism: (von Mises, Reichenbach) a) actual frequencies b)
counterfactual frequencies
4.
Best System 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.
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.
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)
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 frequencies
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:
1. Applies to types only not token events
2. The counterfactual specifying the long run
frequency is not well defined.
3. Evaluating the counterfactual presupposes laws
that mention probabilities so the account is
circular.
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)
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.
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).
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.
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.
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).
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)
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.)
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)
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.
• 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.
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.
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.
•
The End