In engineering/economic practices I encounter many applications that gaussian distribution is used to model the probability of events. I know that distributions like gaussian distribution have a definite variance and mean that gives us good information on the probability of events.
On the other hand mandelbrotian or fat-tailed distributions describe processes that outlier events are not as unlikely as processes that are modeled using gaussian/poisson/exponential distribution.
I see many processes that could be considered mandelbrotian but treated as gaussian (for example in economic predictions/forecast or some machine learning problems).
I wonder who gives us permission to use distributions like gaussian/poisson/exponential distribution with so much confidence in many applications?
Hossein,
Your surprise is quite understandable. Yes, you are right, we are very often in the state of sin using gaussian probability distribution (".Anyone using Monte Carlo is in a state of sin. /-/ J. von Neumann, 1951).
Then, how can this be justified?
- 1. Gaussian is very well known and popular distribution
- 2. Its analytical properties are quite simple (well, maybe with the exception of the erf function, not present in every scientific calculator)
- 3. It produces reasonable results if you are satisfied with, say, plus/minus 0.1 accuracy in probabilities.
Should we continue to do so? Certainly not when doing highly accurate/reliable reasoning. On the other hand, the fat-tailed distributions (why do you call them mandelbrotian?) introduced by Levy (gaussian distribution is present in this family) are not expressible in closed forms, except for the very few specific cases. Consequently, they are pretty troublesome at least in practical calculations. Besides, investigation of noise characteristics takes precious time (sometimes precious resources as well, like liquid helium, too) when doing "essential" measurements, hence it almost never happens. This comfort, however, should not be enjoyed in problems when human life or safety of large gas platform, bridges or buildings are at stake. Here we use more and more often the tools originating from interval analysis.
We also know that fat-tailed distributions do exist in Nature (see earthquakes, rainfalls and many others) but it is not so rare that their (true) second moments do not exist. Hence, no reliable predictions, even those coarse-grained, can be obtained in such cases.
Hossein,
Your surprise is quite understandable. Yes, you are right, we are very often in the state of sin using gaussian probability distribution (".Anyone using Monte Carlo is in a state of sin. /-/ J. von Neumann, 1951).
Then, how can this be justified?
- 1. Gaussian is very well known and popular distribution
- 2. Its analytical properties are quite simple (well, maybe with the exception of the erf function, not present in every scientific calculator)
- 3. It produces reasonable results if you are satisfied with, say, plus/minus 0.1 accuracy in probabilities.
Should we continue to do so? Certainly not when doing highly accurate/reliable reasoning. On the other hand, the fat-tailed distributions (why do you call them mandelbrotian?) introduced by Levy (gaussian distribution is present in this family) are not expressible in closed forms, except for the very few specific cases. Consequently, they are pretty troublesome at least in practical calculations. Besides, investigation of noise characteristics takes precious time (sometimes precious resources as well, like liquid helium, too) when doing "essential" measurements, hence it almost never happens. This comfort, however, should not be enjoyed in problems when human life or safety of large gas platform, bridges or buildings are at stake. Here we use more and more often the tools originating from interval analysis.
We also know that fat-tailed distributions do exist in Nature (see earthquakes, rainfalls and many others) but it is not so rare that their (true) second moments do not exist. Hence, no reliable predictions, even those coarse-grained, can be obtained in such cases.
We might regard continuous distributions as being the sums of simpler distributions. In health research, if we see fat tails (or just one) we tend to think that there is a process driving the majority of observations and different processes driving the extremes. For example, in pregnancy you have high outlying blood pressures due to disorders in the response of the body to the pregnant state. Indeed, even the extremes of height are driven by conditions that do not affect most of the distribution.
So really, finding a single distribution to fit a real life scenario is not as good a tactic as the one proposed centuries ago by Bacon : "We learn when we establish the general rule, but we learn again when we examine the exceptions"
Those fat tails may be trying to tell you something!
If you see a process that has fat tails, there is probably either one of the following conditions:
All three are very common. Gaussian distributions can be used to model some of them, some of the time, but will mislead when you can least afford it (like right before a financial crisis).
An example of #1 is the assumption that future profits or revenues are IID. Even if they are, a serious mistake is made by applying the statistical analysis to a growth process. If profits are a percentage of investment, and investment is increased or decreased by last year's profits, this is a common serial dependency. Many IID theorems are applied to such data, where they really don't apply.
An example of #3 is a fracture process that amplifies deformations beyond a certain limit. This will appear as a very fat tail. Of course, this is also an example of #2 so there is overlap.
A big problem with medical experiments is that the statistics of one test are correct only if it is the one test ever performed. If multiple tests are performed, e.g. if thousands of substances are examined to see if one produces cancer, or cures it, then the probability that some one of those tests is off the normal distribution and therefore misleading increases.
In this last respect, I suspect that even quantum mechanical variables may not be truly Gaussian. Because they violate the multiple tests rule. No mater how many times you conduct quantum tests, you never find violations of the conservation of momentum or energy. Even for smallish sample sizes. You have to get a really small sample and then it is not exploitable.
Maybe this will not help you, but it's what I've learned about the subject over many years. Any process will follow a statistical pattern ... until it doesn't.
Mandelbrotian or Gaussian? It really depends on what you are trying to do. Put differently, what is the precise question to be answered (if there is one) and the framework in which this information is to be used e.g. reproducibility, prediction or descriptive. It’s like asking “which is better, a hammer or a screwdriver?” while not stating what the DIY job is at hand. This question cannot be answered without being explicit regarding your objectives.
All distributions are theoretic. There are NO natural phenomena in the world which are Gaussian/Normally distributed (which assumes that the phenoma being studied is the result of independent additive forces), log-normally distributed (multiplicative forces) etc. Sometimes we see phenomena that appear to be Gaussian distributed, but they really aren’t. Distributions are just mathematical models.
So why is the Gaussian one so popular? This has nothing to do with the nature of the phenomena being studied! A fundamental theorem in statistics states that averages of any phenomena are normally distributed (under iid). Whether you are meassuring the spin of boson particles, the blood pressure of people, financial market prices etc. – averaging these (under independence assumptions) will give you a normal distribution. This result allows us to move away from an understanding of how the phenomena are generated, by generating new phenoma (averages) in whose behaviour we have a better understanding i.e. we know these follow the Gaussian distribution. Of course, there is no such thing as true independence in the real world either and how well this theorem works depends upon the extent of violations of this assumption. Which bring us back to the first point - what exactly is it you're trying to do, what are the questions, so that we can consider alternative approaches.
Hello Goerge,
I am not a specialist but if the Brownian motion has bee ruled out from mathematical finance, what model is currently used? Thanks for your answer.
George, thanks for your answer.
There is some thing I don't understand: Poisson processes are based on an exponential distribution of inter time events. The only thing a Poisson process says is that there exist an average so that we can evaluate it (to my mind, the standard deviation has no real useful meaning). So, in my point of view, the problem with the calibration is the non-stationarity of the Poisson process (i.e. the seasonality of the market data).
Up to your knowledge, is there other reasons making difficult the calibration of the average with real market data?
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