Basic Probability

[arabianights]

2nded

 

[TheBramble]

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.

 

[TheBramble]

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.

 

[Skill Leverage]

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.

 

[TheBramble]

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.

 

[TheBramble]

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.

 

[trendie]

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.

 

[trendie]

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.

 

[MrGecko]

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"

 

[Calinor]

Oh geez I really wish people would understand what probability is, and not just throw it around and wonder why it doesn't work.

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

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).

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.

 

[TheBramble]

If you do the coin experiment properly, the results will converge on whatever the theory says they will...

Are you sort of agreeing what I'm suggesting is a justifiable viewpoint with specific regard to the markets and the results of the behaviour we experience from them?

... 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).

'Probability in the context of the markets' is the one I am very much interested in regardless of it's heresy or otherwise with regard to orthodox probability theory.

 

[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?

Whoa! You’ll get me thrown out of town….

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.

Yeah. I offered a similar link the other day, something to do with Power of Consciousness (really can’t remember) but within a jokey context, although the original line of research in which it occurred is well documented. A random generator net set up between a number of academic institutions around the globe measuring – well, randomness. And how at times of global focus or coherence, there is a demonstrable shift away from randomness/toward non-randomness/toward coherence (whichever you prefer).

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?

Erm…

 

[TheBramble]

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.

It’s not just outliers. Outliers are the names we give to data that we’d like to ignore, so we do, we take them out of standard deviation calculations. Brush them under the probabilistic carpet. But they exist and they’ll take a chunk from/add a chunk to your trading capital regardless.

I’m not sure I’m agreeing 50/50 is normal, but…and I’m definitely not agreeing the remaining 90 tosses would distribute 45/45.

In one private exchange with another member on this topic I wondered if I should just have started off with what prompted my interest in this area, specifically premium on commodity futures options and how they don’t fit the models, rather than coins. Every man and his dog thinks they know probability when you discuss coins because that’s where most get their elementary (and only) education in probability. OK, maybe dice too.

If I’d have given my query within the context of what brought it about, perhaps more would have decided to think afresh rather than fall back to what they ‘believe’ is the case.

Case in point (I think…)

If as in the coin example, a Head is just as likely as a Tail for the next flip of a coin, we could just as well state that for any given price in any instrument, the next price could be either up, down or the same. I think we could say that.

But when you ask most who trade what the probability of any given, existing trend continuing, most will say there is a higher probability of it doing so than not. I’ve never seen any empirical data from anyone making this claim as to just how much higher that probability might be – but it’s given and very generally accepted. Isn’t this an example of where we take the reality of probability which we experience on a daily basis but which does not fit within classical probability?

 

[TheBramble]

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.

I’m not sure I said that. I suggested perhaps one’s starting point and expectation might have greater influence on trading longevity than any other factor or factors. More of a question?

And I didn’t simply mean that getting off the blocks sharply with a good system, sound risk & money management and the resulting larger wins than losses and more wins than losses lead to confidence to continue to use your system ‘as is’ without tweaking or second guessing that will lead to failure.

While I’m more than willing to accept the possibility that we (anyone) can alter the probability of success in any event with appropriate focus, preparation and expectation, I’m also more willing to accept there is a more empirical basis for this phenomenon.

If your expectations are of success, based on recent results or some other over-arching factor (your white-bearded mentor says with kindly and deeply assuring smile that you simply can’t lose…) you will trade your setups as they should be traded. You’ll allow them to do their work. Rather than have second thoughts and vague instincts that ‘this one doesn’t feel right’ cause you to fail to act when you should and miss a winner and then get in and double up on the next one which doesn’t fit, but you get another instinct about (to make up for your stupidity in missing the last one) – just in time for the inevitable loser. And a downward spiral ensues in this case.

I mention in a post above that I found it fascinating so many could lose and get taken out of the game, when it is essentially, a simple process. Sure you need a good system and emotional control and risk and money management – all those good things. But I don’t believe any but a minority don’t have a least a half decent semblance of that package at some point’ during their trading careers. That they fail to capitalise on that seems to me to be a factor of expectation, based on past events, which in turn drives their future probabilities of success.

Just one narrow area where I believe this issue is important a way away from the originating thoughts relating to options futures premium, but I hope my point has come through.

 

[Calinor]

I know you didn't say that. I'm saying what occurred to me when I read it. And yes I think you're right about the attitude and confidence in your system. If I had full confidence in my system, and didn't make any stupid discretionary trades, or discretionary get outs, I could make some decent money. I think I see now more what you're getting at.

 

[Skill Leverage]

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.

Completely, but coin-tossing in the theoretical context does not imply standing and actually flipping the coin; of course a machine or human behaviour could skew the results on way or the other. 'Coin-flipping' in the sense we are talking about is a real-world approximation to a random event with two potential outcomes of equal probability. Bramble is trying to pooh-pooh conventional probability by hashing together theoretical principles with real-world data, and this is impossible by definition.

The random walkers among us are completely wrong, and I completely agree that in the real world probabilities are not what they would appear to be theoretically; almost nothing in this world is. The idea however that a theoretical set of data has any impact whatsoever on the next set of data taken from the same experiment is ludicrous.

 

[Pacmanpth]

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.

More likely than not, what's really more random is which side of the coin people face up as they are about to flip the coin. I know I for one don't pay attention to which side is initially facing upwards.

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.

By the basic laws of economics, people are going to enter in to something until it no longer has profit potential (by basis of driving down/up costs or purchase price). It seems like if one method proved to be successful, more people would use that method until the number of people using it was large enough to the point where potential expected "edge" was 0 on average. People would begin to try to purchase increasingly in advance of the indicator shift in attempts to capture as much profits as they could until the indicator really didn't tell people anything at all. The thing is though, the parameters people use for a single indicator are so varied, with people using other indicators with various parameters on top of that...so even if you could figure out how that scenario would play out, it tells us nothing about the real world.

With support and resistance this is especially noticeable, just simply due to the potentially arbitrary placement of levels based on what kind of time frame you are looking at.

The statistical likelihood of significantly large groups of people using very similar systems I imagine is minuscule. Doesn't mean they work, just means that this dilution of profits due to natural market behavior likely isn't the cause of failure. Still, if people were to adopt the exact same system as someone successful, the benefits of the system should slowly fade.

 

[Kisakye]

0.25% ?!!!

 

[coolTrader]

what an interesting discussion. Just for tossing what is pressumed to be a fair coin.

Reading through I totally lost the trail of thought. Next I am seeing that the outcome of the coin tossing is not random but...

Then I hear the results being skewed if you get heads... and not tails after X tosses. Does that mean it is skewed? Or do we say what is the probabilty of getting 10consecutive heads and 45tails and 45 heads (order not important).
Then base on our probability we take an action or trade. How do we find out that probability? We use the Binomial distribution(theoretical) as it is a discrete variable.

For real data depending on what you are doing you can find that over time and with data size greater than a certain size say 1,000 data points the central limit theorem holds and we can fit our data using a normal distribution with a few tweaks for discreate variables.

It is quite intersting when you fit not just price but certain aspects of price into a model and they suprisingly behave in a random way following a theoretical distribution.

In order to get a better view of your data you can sometimes transform the data.

One of the good things about the prices you see in the market is it take into account all the random errors which again you could check for randomness etc..

You do not have to model the price directly as you will always have extreme outliers (which cannot be ignored dare I say). These outliers as we know are caused by extreme events PPI, CPI... and the likes of them...

One could have different ways to trade based on events etc...It goes on.
So what are we trying to achieve in this discussion?

 

[stevespray]

Bramble,

Just spent the best part of two days working my way through the thread trying to pick up on a number of topics raised. There are a number of points which I’d like to raise.

Firstly there perhaps needs to be clarification on a number of points which people, when writing, might assume others automatically consider being correct or matter of fact. The matter of the ‘coin tossing’ itself is one such instance. One or two have to my mind correctly pointed out that coin tossing itself might not be 100% random due to simple human mechanics – A human who is asked to repetitively toss a coin will develop a method and possibly a characteristic which may cause a potential imbalance in the results. It’s just the same as a professional golfer or a snooker player who constantly practices his shots – his / her aim is to program some kind of ‘repetition’ which, when called upon, the player can repeat ‘at will’.
As a result we perhaps need to clarify that in reference to a ‘coin toss’ we mean a method of deriving a result which is one of two possible outcomes which has no bias one way or the other (or does that contradict what’s already written?)
I guess what I mean is that we are trying to assume that there is no human influence over the ‘coin toss’ outcome?

If what I have presumed so far is true then I have to say that I am sceptical (to put it mildly) that ‘price has memory’ in terms of the tossing of coins. Of course I am willing to learn a new trick here!
A number of posts have been made which refer to some kind of ‘regression to the mean’ in terms of a longer series of coin tosses. Mathematically this isn’t correct although a regression can and does on occasion occur. If we spin our coin 50 times and find that we have 30 heads and 20 tails then this does not mean that the next 50 spins should yield 30 tails and only 20 heads – the probability is that there will be roughly 25 heads and 25 tails in the 50 spins. Statistically there is just as much likelihood of another 30 heads and only 20 tails as there is of 30 tails and 20 heads. On that basis I cannot consider that any kind of ‘price memory’ exists in terms of coin tossing.

I would agree that people in general under estimate probability in given situations – trading is clearly one of those areas. As a result loss occurs.

Having read the thread I am having trouble in relating ‘coin tossing’ directly to trading. The thread has made me think about certain aspects of trading, and indeed coin tossing, more deeply. The more I think about it the less relationship they appear to have to each other. Initially it was suggested that trading was a matter of simply predicting which way something would move pricewise. My 10 years plus of experience has taught me that trading is so much more than that.

The question was raised in an earlier post about why so many (95%) lose their money when trading when the ‘simple’ decision is either to buy or to sell ie 50% vs 50% in terms of probability. My feeling is that the 95% figure reflects the ‘compounding’ of more than one or two trading mistakes. They say that you only need a tiny edge over the market to make more money than you’ll ever need but the flip side of that is that you only need to compound a few small errors and most of your capital is gone. Of course another fact that cannot be overlooked is a simple study of human behaviour – if someone starts trading and is successful they will carry on trading until they are not – when someone perceives that they are not successful / runs out of capital / realises that they cannot deal with the emotional stresses then they stop trading and enter ‘the 95% club’.

Randomness of price movement is a seriously interesting area to examine. I generally find that the smaller the timeframe the more random the price action is. This I would speculate is because short term price movements are caused by day to day ‘technically’ based buying / selling where as the larger time frames (weekly / monthly) are dominated by more ‘fundamental’ based buying / selling.
I therefore find it quite hard to compare any kind of price movement to a series of ‘coin tosses’. Whether or not coin tosses are random or not does not seem even remotely significant to me. My feeling is that market movement is not random. That is not the same as saying that market movement is totally predictable. I do however have a number of ‘tools’ which I feel allow me personally to understand why certain things happen to prices. I study two forex markets fairly closely – EUR/USD and GBP/USD. Obviously these two markets are driven, in the longer terms, by completely fundamental news flow. However, in the short term, this isn’t always the case. One only has to examine the size of intraday swings to realise that these swings clock up tens if not hundreds of more pips in terms of size than the fundamental moves if studied on a weekly chart. This can only be caused by short term speculation and in my opinion this apparent randomness can be quantified if studied closely enough.

Steve.