Is it possible for a probability to be unknown?
tjb
1. You cannot see inside it.
2. It contains at least one marble, and this marble is red.
3. It may or may not contain more marbles.
4. If it does contain more marbles, they may or may not be red.
Is it valid to say that the probability of picking a red marble is unknown
here? Or must we commit to 50%?
José Carlos Santos
Yes.
> Or must we commit to 50%?
Of course not! Why should we?
Best regards,
Jose Carlos Santos
tjb
>> I present you with a bag, with the following properties:
>>
>> 1. You cannot see inside it.
>> 2. It contains at least one marble, and this marble is red.
>> 3. It may or may not contain more marbles.
>> 4. If it does contain more marbles, they may or may not be red.
>>
>> Is it valid to say that the probability of picking a red marble is unknown
>> here?
>
> Yes.
>
>> Or must we commit to 50%?
>
> Of course not! Why should we?
Perhaps I'm wrong, but I'm under the impression that this is within the
definition of probability -- that probability is simply an estimate based
on what we know and what we don't know.
For example, if Jack tosses a coin and asks Jill the probability that it'll
land on heads, I expect that Jill will say "1/2". But what if Jack has
tested flipping the coin beforehand (without telling Jill), and found that
it actually lands on tails 99 times out of 100?
Here, it seems that Jill has assumed that the probability is 1/2 simply on
the basis that there are two possibile outcomes, and it's apparently far
from accurate (considering Jack's additional knowledge). But isn't this
what probability is all about -- simply assuming a value based on
everything we already know about the phenomenon, however little we might
know?
José Carlos Santos
Yes (well, sort of). And what we already know about the phenomenon of
tossing coins allows us to say that the probability of landing on heads
is 1/2.
On the other hand, if someone is about to shoot me at close range, I
might think: "there are two possibilities: either he will shoot and
I'll die or he will be struck by lightning and he'll die". But I
surely would *not* think that each event has a 50% probability!
So, no, when we ignore how likely each event is, we do *not* assign
probabilities to it. They are just unknown.
I think that Martin Gardner wrote a funny text about the error of
assigning a 50% probability to unknown events, but right now I can't
remember where.
Jonathan Hoyle
We certainly don't have to commit to anything. I would say that the
data is insufficient to hypothesize the assumptions necessary to
generate a probability.
Certainly not 50%.
For example, suppose the question were: "What is the probability that
life exists in the Alpha Centauri solar system?" We might be tempted
to say, "Of the choice 'life exists there' and 'no life exists there',
we do not have a preference for one over the other. Therefore, with
two equiprobable possible outcomes, the probability must thus be 0.5."
But to say that there is a 50/50 chance for the existence of life in
Alpha Centauri, based exclusively on the fact that we don't know, would
be ridiculous.
It's okay to simply say "We don't know."
Non-repeatable scenarios, or repeatable ones whose parameters we are
not privy to, make it difficult to construct a mathematical model so as
to estimate such probabilities.
Math1723
> not privy to, make it difficult to construct a mathematical model so as
> to estimate such probabilities.
How about a scenario like this:
Ask two people to each think of a number. What is the probability that
they are thinking of the same number?
(Note that I intentionally did not give a range to choose from.)
Jonathan Hoyle
That is at least a repeatable experiment. It does not matter whether
or not you gave them a range. You can continually run the experiment,
by selecting two new people for each trial, and calculate the hit
percentage.
Even without a range, most people will think of an integer, and the
vast majority will choose a single digit one at that. One might guess
that the hit rate would thus be around 10%, perhaps less for the few
who go outside that range. But certain numbers are more likely to be
favored over others: 3, 5 and 7 are (I would guess) more likely to be
picked than, say, 1, 2 or 9.
I would guess a hit rate closer to 20-25%, but that is pure speculation
on my part.
Jonathan Hoyle
Randy Poe
tjb wrote:
> José Carlos Santos <jcsa...@fc.up.pt> wrote:
> >> I present you with a bag, with the following properties:
> >>
> >> 1. You cannot see inside it.
> >> 2. It contains at least one marble, and this marble is red.
> >> 3. It may or may not contain more marbles.
> >> 4. If it does contain more marbles, they may or may not be red.
> >>
> >> Is it valid to say that the probability of picking a red marble is unknown
> >> here?
> >
> > Yes.
> >
> >> Or must we commit to 50%?
> >
> > Of course not! Why should we?
>
> Perhaps I'm wrong, but I'm under the impression that this is within the
> definition of probability -- that probability is simply an estimate based
> on what we know and what we don't know.
Bayesian probability is an estimate of our degree of belief in
different outcomes. A Bayesian would say you can assign
probabilities to events about which you have no knowledge
whatsoever, such as the existence of life on Mars.
Frequentist probability (and most of the time, I'm more
comfortable with this view) is more rigid. There is an actual
population of marbles and an actual probability that a given
marble is red, even if we don't know it. Experiments serve
to give us estimates of the fixed population values.
> For example, if Jack tosses a coin and asks Jill the probability that it'll
> land on heads, I expect that Jill will say "1/2". But what if Jack has
> tested flipping the coin beforehand (without telling Jill), and found that
> it actually lands on tails 99 times out of 100?
Well, it actually landed on tails 99 times out of 100 in Jack's
trials.
Then the Bayesian would tell you it's an unfair coin, and
the frequentist would tell you that there is a very low
probability it is a fair coin.
But in neither case would you say that you know for
certain it will always fall on tails exactly 99 times out
of 100 trials.
- Randy
tjb
> Bayesian probability is an estimate of our degree of belief in
> different outcomes. A Bayesian would say you can assign
> probabilities to events about which you have no knowledge
> whatsoever, such as the existence of life on Mars.
Ah. I take it then that Bayesian probability would put the probability at
50% in the example in the original post in this thread.
This may well be what I'm thinking of. Well, maybe I should give some
context by stating why I made this thread in the first place.
Basically, a book I'm reading quotes a definition of agnosticism (lack of
belief in God, but also lack of disbelief; i.e., fence-sitting), and the
book suggests that this quoted definition implies that God's existence and
non-existence are equiprobable. However, it's not actually stated within
the quotation that they are equiprobable, and I couldn't quite determine
how it was *implied* within it either.
My theory was that by implying that there is no evidence either way, the
quoted definition also implied that the probability was 50-50 (with the
author using the logic that if there is no evidence, it is 50-50). I
wonder if the author was using Bayesian probability then. Hmm.
Mike Kelly
Bob enters a competition. What is the probability that he wins? 50%
--
mike
Randy Poe
tjb wrote:
> Randy Poe <poespa...@yahoo.com> wrote:
> > Bayesian probability is an estimate of our degree of belief in
> > different outcomes. A Bayesian would say you can assign
> > probabilities to events about which you have no knowledge
> > whatsoever, such as the existence of life on Mars.
>
> Ah. I take it then that Bayesian probability would put the probability at
> 50% in the example in the original post in this thread.
In Bayesian language, the initial belief is called the "a priori"
or "prior" distribution. There isn't one "Bayesian probability".
I think you could make a case for distributions with p(red) = 50%,
p(red) = 100%, or p(red) = any nonzero value.
> This may well be what I'm thinking of. Well, maybe I should give some
> context by stating why I made this thread in the first place.
>
> Basically, a book I'm reading quotes a definition of agnosticism (lack of
> belief in God, but also lack of disbelief; i.e., fence-sitting), and the
> book suggests that this quoted definition implies that God's existence and
> non-existence are equiprobable. However, it's not actually stated within
> the quotation that they are equiprobable, and I couldn't quite determine
> how it was *implied* within it either.
You are correct that the author has made a mistake, thinking
that two possibilities implies two equally-likely possibilities.
I think a Bayesian agnostic could again put the a priori p(God)
as any value between 0 and 100%, exclusive. (If he/she chose
0 or 100%, that would indicate firm atheism or firm theism).
> My theory was that by implying that there is no evidence either way, the
> quoted definition also implied that the probability was 50-50 (with the
> author using the logic that if there is no evidence, it is 50-50). I
> wonder if the author was using Bayesian probability then. Hmm.
I think the author is following a common misconception about
probability.
- Randy
dl
> Bob enters a competition. What is the probability that he wins? 50%
>
I agree. In my opinion a probability can never be unknown. It reflects
our knowledge - or lack of. In complete lack of knowledge the
probability of exclusive and complete events should be equally
distributed. We could reformulate your example as:
An experiment can give only one of the two results A and B. What is the
probability of getting result A?
With no knowledge of what the experiment is, nor what the results are,
but just that they are mutually exclusive and that one of the two
_will_ take place, I think we must say 0.5 (the letters A and B could
be exchanged without modifying what we know).
To apply this to the original example, if we accept its non-realistic
formulation (no limit is specified to the number of marbles) then we
should say 0.5.
To keep it simple, if there was a limit of total 2 marbles then:
Probability that there is just one marble: 0.5 (and it _is_red, we
know).
Probability that there are two marbles: 0.5
Probability that the second marble, if present, is red: 0.5.
Probability of picking a red marble: 0.5 * 1 + 0.5 * (0.5 * 1 + 0.5 *
0.5) = 0.875
Increasing the limit, this approaches 0.5
imagin...@despammed.com
No! Never 50% - you should go for 65% if you're selling, 35% if you're
buying.
oo-._.-._.-._.-._!
Actually this "50%" is uncannily like the Orlovian "Big'un": Bill
enters a competition. What is the probability he will win? Why, 50%.
And the probability he will come third? Why, 50%. And the probability
he will come in the top three? Why, 50%. But once we formulaically
declare that one of these "50%"s is the Real "Who'nos", the others can
be calculated. Suppose the 50% for "top 3" is the real "Who'nos", then
the probability of coming top is "Who'nos/3". Well, given any specific
enough question, I'm sure I could make up a story to answer it. (Which
of course is TO's MO)
Brian Chandler
http://imaginatorium.org
tjb
> For example, suppose the question were: "What is the probability that
> life exists in the Alpha Centauri solar system?" We might be tempted
> to say, "Of the choice 'life exists there' and 'no life exists there',
> we do not have a preference for one over the other. Therefore, with
> two equiprobable possible outcomes, the probability must thus be 0.5."
> But to say that there is a 50/50 chance for the existence of life in
> Alpha Centauri, based exclusively on the fact that we don't know, would
> be ridiculous.
I can kind of see what you're saying, but I'm still not entirely sure about
this. Probability, to me, seems to be simply a measure of what we don't
know. Let's say that Jane just tossed a coin and observed it land on
tails. Bob hasn't observed this. If Jane then asks Bob the probability
that it landed on heads, Bob might say "50%". But to *Jane*, the
probability that it landed on heads is 0%. Does that mean that Bob was
somehow wrong to commit to 50%?
It seems to me that the "there are two choices; that's all we know" thing
is very similar to this coin example -- we have incomplete knowledge, and
we state a probability anyway.
Let's say that we want to determine whether or not X exists (where X is any
arbitrary phenomenon). We currently have no evidence either way. But
then, a few weeks later, we make a discovery that in effect slightly doubts
that X exists. Does the existence of X now have a probability? Would it
now be only (something like) 40% probable? If so, wasn't it 50% to begin
with?
Jonathan Hoyle
> is very similar to this coin example -- we have incomplete knowledge, and
> we state a probability anyway.
That there are two choices is fine. The assumption that those choices
are equi-probaable is the problem. With the coin, it's a reasonable
assumption. With the life on Jupiter problem, on what basis can you
make the assumption that those choices are equi-probable?
> Let's say that we want to determine whether or not X exists (where X is any
> arbitrary phenomenon). We currently have no evidence either way. But
> then, a few weeks later, we make a discovery that in effect slightly doubts
> that X exists. Does the existence of X now have a probability? Would it
> now be only (something like) 40% probable? If so, wasn't it 50% to begin
> with?
No. If you don't know either way, you can't assume 50%. You can
assume this only if you have reason to believe the choices are
equi-probable.
When you say "slightly doubts", do you mean "believed to be less
probable than before"? If so, that means that you had some idea before
what the probability was to think it's less now. With the life on
Jupiter scenario, "more doubtful evidence" would likely take the figure
down much more than 40%, because it would be the *only* evidence.
Give me an example of a problem in which we have no idea which of two
choices is correct (nor do we have reason to believe the choices to be
equiprobable), and some discovery causes a "slight doubt" in our
otherwise complete ignorance.
marcus_b
Maybe the right question to ask is: if you were forced to make
a bet on whether a marble taken out of the bag at random would
be red, what odds would you offer? Your decision is primarily
subjective, since you have very little information, and if you are
a Bayesian, you don't hesitate to take into account everything
you know. E.g., someone has told you that the bag has at least
one red marble. Do you trust that person? Did they look you in
the eye when they talked to you, or did they look sneaky?
[Bayesians should make good poker players ... I am not sure that
in real life they actually do.] Or are they hoping you will bet on
red, knowing that there are 12 marbles in the bag, and only one is
red? Or are they just giving you a neutral fact? Or a hint? You
cannot see inside the bag, but you can see how large it is. If it
is so small that it could contain only one marble, you know that
marble is red. A person betting against you who does not make use
of this information, who just estimates the probability with no basis,
should be at a disadvantage. I.e., as some have argued, the
Bayesian approach is 'coherent' or rational. If you are right in your
subjective guesses, you should have an advantage in betting.
Marcus.
John Coleman
Randy Poe wrote:
> tjb wrote:
> > Randy Poe <poespa...@yahoo.com> wrote:
> > > Bayesian probability is an estimate of our degree of belief in
> > > different outcomes. A Bayesian would say you can assign
> > > probabilities to events about which you have no knowledge
> > > whatsoever, such as the existence of life on Mars.
> >
> > Ah. I take it then that Bayesian probability would put the probability at
> > 50% in the example in the original post in this thread.
>
> In Bayesian language, the initial belief is called the "a priori"
> or "prior" distribution. There isn't one "Bayesian probability".
> I think you could make a case for distributions with p(red) = 50%,
> p(red) = 100%, or p(red) = any nonzero value.
>
> > This may well be what I'm thinking of. Well, maybe I should give some
> > context by stating why I made this thread in the first place.
> >
> > Basically, a book I'm reading quotes a definition of agnosticism (lack of
> > belief in God, but also lack of disbelief; i.e., fence-sitting), and the
> > book suggests that this quoted definition implies that God's existence and
> > non-existence are equiprobable. However, it's not actually stated within
> > the quotation that they are equiprobable, and I couldn't quite determine
> > how it was *implied* within it either.
>
> You are correct that the author has made a mistake, thinking
> that two possibilities implies two equally-likely possibilities.
He doesn't mention what the book is, but a (wild) guess might be that
it is "The Probability of God" by Stephen Unwin. This book explicitly
brings a Bayesian approach to the question of religious belief. The
book (though not the cover - curse those marketing types) is fairly
modest in that its main point seems to be that Bayesian methods have
the potential to clarify a person's subjective beliefs and it doesn't
claim to have once and for all settled the existence (or nonexistence)
of God. His conclusion is that it is more likely than not that God
exists, though like a true Bayesian he acknowledges that other people
with other prior beliefs might come to different conclusions. At once
place he writes (since you need a prior probability to get off the
ground) "Bayesian analysts would generally agree that the choice of
prior probabilties must be considered and justified on a case by case
basis ... Here, I think that the expression of complete ignorance [of
whether or not God exists] is a good case for the 50-50 argument" (pg
57). In the next couple of pages he tries to give some argumentation to
back this up, so he is not simply unaware of the issue or naively
assuming that uncertainty implies probability 50%. Somewhat humorously,
he has an appendix for setting up an Excel spreadsheet to play around
with his calculations in which the initial prior probability is a
parameter that anyone could modify if they see fit. If this isn't the
book, I would be interested in hearing from the OP what it was.
C6...@shaw.ca
> Randy Poe <poespa...@yahoo.com> wrote:
> > Bayesian probability is an estimate of our degree of belief in
> > different outcomes. A Bayesian would say you can assign
> > probabilities to events about which you have no knowledge
> > whatsoever, such as the existence of life on Mars.
>
> Ah. I take it then that Bayesian probability would put the probability at
> 50% in the example in the original post in this thread.
No. There is no such thing as a single, uniquely-defined Bayesian
probability. A Bayesian named Jones might put p = 0.5, Smith might use
p = 0.4 and Schwartz might use p = 0.0001. Much depends on the extra
information (valid or not) they bring to the problem. For example, are
they allowed to look at the bag? They can judge from the bag's size
that it likely will not hold millions of balls, marbles, or whatever.
They might differ in their judgement of exactly how many colours exist
in the world (that could be alternatives to "red"), etc. Indeed, the
true Bayesian would assign a probability distribution to the value of p
(so that p is truly uncertain). They might then use a summary
statistic, such as the expectation of the distribution, as a "point"
estimate of p, but that would not mean that they regarded a single
value as "true"---just most likely, perhaps. If a Bayesian has a
non-informative prior (a uniform distribution) the expectation of p
would be 0.5, but still, any value from 0 to 1 would be equally likely.
I think you might benefit by looking at the book by E. Jaynes,
"Probability Theory (the Logic of Science). Jaynes rejects the standard
"Kolmogorov" formulation of probability, and opts instead for a
formulation based on "plausible reasoning". His treatment is
refreshing, but sometimes a bit too extreme and rebellious for my
taste. He rejects notions of "randomness" completely, so for him,
probability is all about degrees of knowledge. It sounds like his
approach is in synch with what you want. But you would still come away
realizing that your simple p = 0.5 estimate is baseless.
R.G. Vickson
Herman Rubin
Of course it is possible for a probability to be unknown.
This is the reason that statistical inference is needed;
USUALLY the probability distribution is unknown in practice.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Robert Maas, see http://tinyurl.com/uh3t
> if you were forced to make a bet on whether a marble taken out of
> the bag at random would be red, what odds would you offer?
Being forced to make a bet presumes you're in a contest with an
adversary. (Nobody except an adversary would force you to make a
bet when you'd rather not make any bet.) In that situation, you
need to know the payoff matrix of the contest, the complete set of
individual strategies available to your adversary and to yourself,
and the sequence of playing.
- If you move first, then your adversary can manipulate the
situation based on knowledge of what you did, then from your point
of view the game is totally hopeless:
-- If you guess high probability of red, opponent manipulates the
situation so it's something non-red instead.
-- If you guess low probability of red, opponent manipulates the
situation so it's red.
-- If you guess exactly 50%, and if payoff rules are that you win
or lose equal amounts depending on statistical estimate of winning
based on your probability, then opponent delierately biasses the
results long enough to trick you into betting differently, at which
point one of the earlier manipulations happens.
But you really must read the fine print as to how much you win or
lose in different circumstances.
Now if your opponent must establish the situation before you make
your move, but you can't see the situation until after you make
your move, then the minmax per game theory applies. Your best
strategy is usually a mixed strategy that maximizes the expected
value of the worst that can happen to you per all possilble mixed
opponent strategies.
So if you are playing against a supernatural being who can change
reality out from under, you can't win, so don't worry about it,
there's no point worrying about something unfixable. Likewise if
you're playing against a sleight-of-hand artist who can change
reality out from under you by distracting you from seeing the
change, you can't win. But if you are really sure your opponent
can't possibly cheat, and the rules say he must set up things
before you make your move, and he can't change anything after
you've made your move, then minmax game theory applies.
So whoever postulated this contest, please tell me what the payoff
matrix is for you, given the fact of what probability you guessed,
and the fact of what color ball was selected.
Herman Rubin
Robert Maas, see http://tinyurl.com/uh3t <rem...@Yahoo.Com> wrote:
>> From: "marcus_b" <marcus_bruck...@yahoo.com>
>> if you were forced to make a bet on whether a marble taken out of
>> the bag at random would be red, what odds would you offer?
>Being forced to make a bet presumes you're in a contest with an
>adversary. (Nobody except an adversary would force you to make a
>bet when you'd rather not make any bet.) In that situation, you
>need to know the payoff matrix of the contest, the complete set of
>individual strategies available to your adversary and to yourself,
>and the sequence of playing.
Not necessarily; nature is not an adversary. If there is
an adversary, game theory needs to be invoked, not merely
probability. Statistical inference is based on the idea
that the laws of nature are set, and are not going to
change either for or against you.