Basic Probability

This is a discussion on Basic Probability within the The Foyer forums, part of the Off the Grid category; Originally Posted by A Dashing Blade Assuming that a statistically significant number of people did this experiment (>1050 for a ...

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Old Apr 8, 2009, 9:23am   #76
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Originally Posted by A Dashing Blade View Post
Assuming that a statistically significant number of people did this experiment (>1050 for a 2SD level of confidence) + independence of outcomes (ie non-rigged coin) then the answer is yes, this would tally.

However, if you asked a question along the lines of . . . "if you tracked the log of daily changes in a market (any market, S&P, USD/JPY etc) would you expect these changes to statistically conform to a gaussian distribution?" . . . then my answer would be "no".
It's the latter that interests me more deeply and why we generally assume we can fit these type of dispersal from the mean into classical theory. Even acknowledging fatter tails and skewness, it doesn't address the glaring discrepancies.
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Old Apr 8, 2009, 9:28am   #77
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No? Any idea why they wouldn't, or why you think they wouldn't? "Classical probability theory" says that you would expect the ratio of heads to tails from single coin tosses to converge on 1:1 over the long term (with the usual assumptions: fair coin, etc.). Equivalently, you'd assign a probability of a coin landing heads on any single toss as 0.5. But you wouldn't be surprised if you got five heads in a row. I assume you'd accept that, so why not accept the 0.875 probability given for the four-coin question? I'm clearly not 'getting it'.
I didn’t either which is what prompted the initial query from me. Using even a basic MC on this simplified problem, using genuine random seeding, you get an increase in divergence away from not toward the mean. In classical theory you ‘should’ converge and we’ve come to get comfortable with that notion over the years, but ‘things’ don’t – otherwise they’d pretty much be where they started. While that statement might be a little off the wall, more pragmatically, in my trading I note that using classical probability does not give me the results that I experience on a daily basis in the markets. More importantly, these differences are driven by others setting prices so we’re all operating in a universe which although it acknowledges classical theory as ‘useful’ does not actually operate on that basis on a day-to-day level

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Please do, it's intriguing. We might get more illumination sticking to the coin toss problem first, though - if we can't clear that up we may not solve a more real-world, less-obvious problem.
Good point. I think we’ve already lost those that aren’t going to hang on regardless of the set chosen.
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Old Apr 8, 2009, 9:47am   #78
 
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I didn’t either which is what prompted the initial query from me. Using even a basic MC on this simplified problem, using genuine random seeding, you get an increase in divergence away from not toward the mean. In classical theory you ‘should’ converge and we’ve come to get comfortable with that notion over the years, but ‘things’ don’t – otherwise they’d pretty much be where they started. While that statement might be a little off the wall, more pragmatically, in my trading I note that using classical probability does not give me the results that I experience on a daily basis in the markets. More importantly, these differences are driven by others setting prices so we’re all operating in a universe which although it acknowledges classical theory as ‘useful’ does not actually operate on that basis on a day-to-day level

Good point. I think we’ve already lost those that aren’t going to hang on regardless of the set chosen.
why do you get a divergence from the mean?
I think I can see why a random distribution might diverge from the norm.
random coin-tossing isnt like the markets. are you saying they should be treated as such?
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Old Apr 8, 2009, 10:39am   #79
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why do you get a divergence from the mean?
Divergence from the mean. The constantly changing values in any series define the mean. Bit like price in relation to MAs and Bollies. It is within classical probability, under specific circumstances; you expect regression to the mean. And it’s valid for a subset of data and applications, but not all. I think the problem is that we tend to use a one-size-fits-all approach to probabilistic theory and while that’s going to be just fine for most things, if you’re getting deeply into the real life aspects on any event or series of events for which we need to use an estimation of likelihood (or unlikelihood), to work with an inappropriate model is worse than no model at all.

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random coin-tossing isnt like the markets. are you saying they should be treated as such?
I’m saying that the probabilities we use to asses events related to coin tossing are typically assumed to fall within classical theory, but even this application appears not to necessarily fit. We should, I’m suggesting, be looking for a model that supports what we perceive about chance in any specific endeavour at any given time, rather than rely on standard models.

I want to go into subjective probability to cover this, but it’ll be later on tonight.
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Old Apr 8, 2009, 2:34pm   #80
 
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It's the latter that interests me more deeply and why we generally assume we can fit these type of dispersal from the mean into classical theory. Even acknowledging fatter tails and skewness, it doesn't address the glaring discrepancies.

Mandelbrot : The (Mis) Behavior Of Markets
should deffo be your next port of call. Very accessible, non-technical, should give you a few ideas, it did for me.
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Old Apr 8, 2009, 2:38pm   #81
 
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Old Apr 8, 2009, 4:07pm   #82
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Mandelbrot : The (Mis) Behavior Of Markets
should deffo be your next port of call. Very accessible, non-technical, should give you a few ideas, it did for me.
Have read and re-read this particular volume. I suspect both of Taleb's more popular works were also instrumental in causing this havoc.

What's really taken me over the edge is a couple of books on Behavioural Finance, which do give a reasonably empirical basis for establishing a start point in reviewing what we think probability is and what it actually translates to in the financial markets.

While we probably don't want to go anywhere to wishy-washy (quantum mechanics, Intent fields, Power of Unconscious/Conscious Thought), there is something to be said for considering the very tangible role human psychology may have in skewing probabilities in the specific field of trading.
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Old Apr 8, 2009, 11:13pm   #83
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TheBramble started this thread Subjective probability is a measure of a state of knowledge rather than a probabilistic frequency which is what classical or frequentist probability is all about. It is where the probability of an event is proportional to the product of the likelihood and the prior probability. The rather unpleasant result of accepting subjective probability as a possibility (arf) is that there is absolutely no reason whatsoever why an infinite number of coin tosses will approximate toward a 50/50 split. We’ve just been educated to believe this so strongly as obvious and a given, that any challenge seems nonsense.

This isn’t just Bayes or Laplace - John Maynard Keynes proposed the idea that 'probability' should be interpreted as 'subjective degree of belief in a proposition'. And Keynes was one of us in almost every respect. Alan Turing also was a supporter of this view. These guys actively pursued the same question I am here precisely because of the limitations and inconsistencies in classical probability. It’s at odds with reality.

The problem with researching for some form of ‘acceptable’ support tonight for what I’m struggling with has turned up so many ‘names’ that did and are saying pretty much the same thing that I’m wondering why it’s taken me so long to stumble upon the ramifications of it.

If you set up a simple MC to drive say a 20K list of 1s or 0s and use the result of the average to determine the probable outcome of the next run, you’ll get a very solid approximation to the data distributions you get in the markets. The data has memory.

If we take a winning trade as a Head (or a 1) and a losing trade as a Tail (or a 0) and you had a trading system with a W:L of 0.5, no commissions or costs, you risked the same amount on each trade and your average loss equaled your average win - what would classical probability tell you about the state of your trading capital over time?

Now take those same parameters above and tell me what the following sequence of trading results are likely to indicate over time in relation to your trading balance? Just a guess, higher, lower, same?
011101000111101011110110110011111111001111010

And now these.
100010111000010100001001001100000000110000101

Your outcome was different. Or you’re just being bloody awkward…

My suggestion is that the sequence of wins and losses will determine the future sequence of wins and losses not according to frequentist or classical probability theory, but to subjective probability. You’re more likely to lose after losing and more likely to win after winning – and the data don’t regress as often, or as quickly or even ever, necessarily.

Depending on just how seriously you wish to take this it could suggest that your prognosis for longevity in trading is far more a function of your starting state and expectation than any other factor, or combination of factors.

When you consider that trading ‘should’ be simple – you either get the direction right – or not – far fewer people should drop out than the 95% which is often quoted. (In fact, this figure is supported by research done by Odean. I’ll dig out the refs if anyone is terribly interested). If you also consider trade entry into a ranging market is not good news, but neither is it bad, it’s a mild loss and if you tie in Risk and Money Management – which most 2nd time on upward trades do – and still fail, there HAS to be something else operating to cause these rather skewed results. Sure a crap trading system will not stem the gradual bleed of capital with numerous directional trades into a ranging market or getting the direction dead wrong every time in atrending market. But statistically (sorry, can’t help it) even the biggest dork in creation would stand a better chance than 5% unless other factors were operating.
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Old Apr 9, 2009, 6:24am   #84
 
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The idea that the outcome of 50 coinflips has ramifications on future repetitions of the exercise is ridiculous.

...there is absolutely no reason whatsoever why an infinite number of coin tosses will approximate toward a 50/50 split. We’ve just been educated to believe this so strongly as obvious and a given, that any challenge seems nonsense

Is that REALLY your argument? It's nice and comforting to come up with ideas that can never be absolutely proven or disproven isn't it:

There is absolutely no reason whatsoever to believe that if we were to click our heels together an infinite number of times, we would transform into a Super Saiyan Level 3. We’ve just been educated to believe this so strongly as obvious and a given, that any challenge seems nonsense

Subjective probability is a parlour trick that can be very cute when looking at a general framework, but becomes pure fantasy when you try to analyse data or derive any kind of second-level mathematical functionality. If you did your 'research' outside of Wikipedia and 'I'm feeling Lucky' on Google you might come to grips with that fact.

Go and stand in the middle of your street wearing a pointy hat with a big D on it, and spend the next 10 years flipping a coin and scrawling the results on the road with a nice piece of chalk. Let us know what the deviation from 50% is, then go and do it again an infinite number of times and tell us whether any patterns emerge, due to the fact that the coin remembered how much it preferred the sun shining on its head than its ar$e.

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Old Apr 9, 2009, 6:44am   #85
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TheBramble started this thread Not one of your most stupid responses so I’ll do you he courtesy of a genuine reply.

It’s those with the deepest immersion in classical probability that will have the hardest time even considering this, let alone working with it. And if you don’t have the flexibility to even consider it as a hypothesis, then I can understand your position. Investment in what we think we already know is always strong. Sometimes it serves us – sometimes it doesn’t.

Considering the ramification in our pursuit of these issues (it really does directly and indirectly put a curve on everything) it seems a worthwhile exercise.

It was precisely because I needed to analyse the data I was getting that I ended up going in this direction – it’s not a purely intellectual exercise, although there’d be no shame in that. I genuinely want to be able to understand why the values I derive on a day-to-day basis for market related data are so out of whack, what that means, what does that mean about others’ interpretations of worth and value and how that can increase my edge.

The thing is John, it’s not like you are right or I am right – it is what it is whatever we ‘decide’ it’s how we think it is. I’m just commenting on reality as I see it and am increasingly surprised on how many have previously going down this path and yet so little (none in fact) room is made for it in elementary probability. I wonder why that might be?

So, yes, I am saying the result of 50 coin flips may have an impact on the future results of further coin flips. That we assume any subsequent coin flip is not totally random and is not necessarily independent of any prior coin flip is an idea I’d like to examine. If you don’t, that’s fine. If nobody else does either and I end up talking to myself (happens a lot) I’ll take the hint not that I’m on the wrong track – but that I’m in the wrong place to even try and discuss it.
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Old Apr 9, 2009, 6:50am   #86
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Is that REALLY your argument? It's nice and comforting to come up with ideas that can never be absolutely proven or disproven isn't it:
To address that little edit you put in while I was responding above, the comfort factor is important. In both directions. Classical probability can never prove that an infinite number of coin tosses will revert to the mean nor that any subsequent coin toss is totally independent of a prior coin toss.

Any more than I can prove my current hypothesis.

But subjective probability does allow for a hypothesis where frequentist does not. Frequentist says it is either A or B – that you can’t get two different answers to the same question.

Perhaps the comfort is really yours in what you think you already know rather than face the dangers and risks (and effort) of considering the possibility you’ll need to review your thinking?

Appreciate your comments. I imagine they speak for many people – in essence, if not in style or manner.
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Old Apr 9, 2009, 7:33am   #87
 
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TheBramble,

Are your veering towards a sort of "creating your own future" type of idea?
The idea that your state of mind, and confidence in your success is manifested through what you do?

I am reminded of the engineering professor who did micro-PK experiments, and showed, that an intention to skew the outcome of a random event (radioactive decay) can have statistically valid outcomes, ie, if you focus of heads, then the "random" distribution is skewed toawrds heads, etc.
(I cant find his name, as all the Google hits come up with Dean Radin. (and it wasnt him))

Are you saying your state of mind, and confidence can skew the resutls in your favour?
Or against you, in contradiction to the classical answers?

EDIT: Robert Jahn of Princeton. Thats the guys name.
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Old Apr 9, 2009, 7:39am   #88
 
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I can see how the random distribution can be skewed by an outlier event.

If you had a normal run of heads/tails, you could expect it to be close to 50/50.

But if you had 10 consecutive heads, for example, perhaps an unusual event, the remaining 90 flips would distribute 45/45.

But the unusual (10 heads + 45 heads) and 45 tails, would skew it 55/45.

then, the "norm" has been shifted by 10 heads. EDIT: of course, should be 5 heads.
this bias would remain until an equally unusual event. it could even result in another consecutive 10 heads, biasing it even further.

If you had 50 consecutive heads, then the normal distribution of 25/25, the shift would be 75/25.
the next 100 flips might revert to "normal" 50/50, but the unusual event would remain.
it may be even more unusual for a correcting 50 tails, so the distribution is skewed.

the above is a very simplistic attempt to understand that reversion to mean may not work.
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Old Apr 9, 2009, 7:55am   #89
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Central limit theorem - Wikipedia, the free encyclopedia

I think ADB hit the nail on the head with one of his earlier posts... it's the difference between probability theory and inferential statistics. If you do the coin experiment properly, the results will converge on whatever the theory says they will...

... however, that is something different to "probability" in the context of the markets - it is a well known fact that "outlier"events (black swan etc) occur more than the statistics indicate they should (partly because of the jumps that happen in asset prices, and partly because markets are driven by people who are prone to shat themselves when there really is no need).

Definition of "Leptokurtosis"
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Old Apr 9, 2009, 8:01am   #90
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Oh geez I really wish people would understand what probability is, and not just throw it around and wonder why it doesn't work.
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It’s those with the deepest immersion in classical probability that will have the hardest time even considering this
As someone with possibly the deepest immersion in probability at the board, I can say that I have no problem considering this. Coin tossing is not random. You can build a machine, which in the right conditions can repeatedly toss a coin to come up heads. It is just a matter of the coin, and the force applied. When a human does it, we are not so precise, so it seems like it is approximately 0.5 probability. Although it would not be a surprise if the real probability may not be that. Something close to that but not exactly 0.5.

Mathematics is not the real world. Nor does it say it is. It is simply a model for the real world. So if it takes P(heads)=0.5 and then gives you some results about that, then that doesn't mean the probability of getting heads in the real world is 0.5. This applies to all mathematics. I think it is best summed up in the following quote
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As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
Calculating probabilities with coin tosses can give you a very good approximation to the real world, but never assume it is the real world. You will just confuse yourself, and you really haven't understood probability then. On your subjective probability. As I have understood what you wrote, the matter you discuss is already in classical probability. The information from the past, affecting the future is simply conditional probability/expcetation. In other words, if I don't assume coin tosses are independent, and I don't even assume they are probability 0.5, then I can still calculate conditional probabilities based on information (this is always what we're conditioning on - information).
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Classical probability can never prove that an infinite number of coin tosses
It can in a model of the real world. But of course not in the real world. There is no such thing as 'proof' in the real world anyway. There is simply experiments which back up a theory until disproved by a counter example. Separate model and real world. If I have misunderstood what you meant by subjective probability, then fair enough.

There is something valuable to me though. You mentioned that how you started out will affect your future in trading. I'm taking from that, that those who have been successful (with a method of their choosing) will still be around and trading that method, so in a sense, that method is more likely to work if I use it. Perhaps Support and Resistance only works because those that used it survived, became rich and kept using it so that now the best method might be to use that. If in 50 years time, the only methods between now and then that were successful were thos eusing Stochastics, would stochastics be the main thing then to get an edge? Interesting.

Last edited by Calinor; Apr 9, 2009 at 8:12am.
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