Stats for nerds
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What Hacker News says
- Might consider, say,
Sidney Siegel, N. John Castellan, Jr. 'Nonparametric Statistics for the Behavioral Sciences, Second Edition', ISBN 0-07-057357-3, McGraw-Hill, New York, 1988.
So, "nonparametric" means make no assumptions about a probability distribution based on parameters. Or, call the material distribution-free.
E.g., get to see about resampling plans -- tiny assumptions, really simple, darned cleaver, quite generally useful, especially appropriate for computing. Might use resampling to get more information from the data from "A - B" tests.
- For statistical learning, the book I mentioned
Nonparametric Statistics for the Behavioral Sciences,
should make a good contribution to statistical learning. Some of the techniques are so robust, i.e., need such meager assumptions, that they should be especially welcome in automatically applied AI (artificial intelligence) applications.
Actually the book ISLR, Introduction to Statistical Learning, does claim to cover
"Resampling methods"
- There is a nice treatment of resampling, i.e., permutation tests, in (from my TeX format bibliography)
Sidney Siegel, {\it Nonparametric Statistics for the Behavioral Sciences,\/} McGraw-Hill, New York, 1956.\ \
Right, there was already a good book on such tests over 50 years ago.
Can also justify it with an independence, identically distributed assumption. But a weaker assumption of exchangeability can also work -- I published a paper with that.
The broad idea of such a statistical hypothesis test is to decide on the null hypothesis, null as in no effect (if looking for an effect, then want to reject the null hypothesis of no effect) and to make assumptions to permit calculating the probability of what you observe. If that probability is way too small then reject the null hypothesis and conclude that there was an effect. Right, it's fishy.
- The 2nd edition is never far from my desk:
Siegel, S., & Castellan, N. J. (1988). Nonparametric statistics for the behavioral sciences (2nd ed.) New York: McGraw-Hill.
- Okay, PG has an hypothesis test.
There's a large literature for that, e.g.,
E. L. Lehmann, Testing Statistical Hypotheses.
E. L. Lehmann, Nonparametrics: Statistical Methods Based on Ranks.
Sidney Siegel, Nonparametric Statistics for the Behavioral Sciences.
In this case, PG will be more interested in the non-parametric case, i.e., distribution-free where we make no assumptions about probability distributions.
We start an hypothesis test with an hypothesis, commonly called the null hypothesis which is an assumption that there is no effect or, in PG's case, no bias. Then with that assumption, we are able to do some probability calculations.
Then we look at the real data and calculate the probability of, say, the evidence of bias being as large as we observed. If that probability is small, say, less than 1%, then we reject the null hypothesis, that is, reject the assumption of no bias, and conclude that the null hypothesis is false and that there is bias. The role of the assumption about the sample is so that we know that the problem is bias and not something about the sample.
In hypothesis testing, about all that matters are just two numbers -- the probability of Type I error and that of Type II error. We want both probabilities to be as low as possible.
Type I Error: We reject the null hypothesis when it is true, e.g., we conclude bias when there is none.
Type II Error: We fail to reject (i.e., we accept) the null hypothesis when it is false.
When looking for bias, Type I error can be called a false alarm of bias, and Type II error can be called a missed detection of bias.
In PGs case, suppose we have 100 startups and five of those have women founders. Suppose for each of the startups we have the data from "their subsequent performance is measured".
Our null hypothesis is that the expected performance of the women is the same as that of the men.
So, let's find those two averages and take the difference, say, the average of the women less the average of the men.
PG says if this difference is positive, then there was bias, but PG has not given us any estimate of the probability of Type I error, that is, of the probability (or rate) of a false alarm.
I mean we don't want to get First Round Capital in trouble with Betty Friedan, Gloria Steinem, Marissa Mayer, Sheryl Sandberg, Hillary Clinton, Ivanka Trump, or Lady Gaga unjustly! :-).
Let's call this difference our test statistic.
So, let's find the probability of a false alarm:
So, let's put all 100 measurements in a pot, stir the pot vigorously (we can use a computer for this), pull out five numbers and average, pull out the other 95 numbers and average, take the difference in the two averages, that of the five less that of the 95, and do this, say, 1000 times. Ah, computers are cheap; let's be generous and do this 10,000 times.
For a random number, how about starting with a 32 bit integer, with appropriately long precision arithmetic multiply by 5^15, add 1, take modulo 2^47, and scale as we want?
So, we get an empirical distribution of these differences, from the five less the 95. Looking at the distribution, we see what the probability is of getting a difference as high or high or higher than our test statistic. If that probability is low, say, 1% or less, then we reject the null hypothesis of no bias and conclude bias with our estimate of probability of Type I error 1% or less.
If with the 1% we reject, then it looks like First Round has done a transgression, will get retribution from Betty, et al., and needs to seek redemption and Betty, et al., are happy to have their suspicions confirmed. Else First Round looks like the good guys, are "certified statistically fair to women", may get more deal flow from women, and Betty, et al., can be happy that First Round is so nice!
Notice that either way Betty, et al., are "happy". That's called "happy women, happy life"! Or, heads, the women win, tails they lose, and in no event is there a huge crowd of angry women in front of First Round's offices with a bonfire of lingerie screaming "bias"!
When we reject the null hypothesis, we want to know that the reason was men versus women and not something else, e.g., a biased sample. So here is where we use our assumption of independence with the same mean.
Now we have a handle on Type I error.
Here we have done a non-parametric statistical hypothesis test, i.e., have made no assumptions, except the means, about the distributions of the male/female CEO performance measurements.
And we can select our desired false alarm rate in advance and get that rate almost exactly.
For Type II error, that is more difficult.
Bottom line, what we really want is, for whatever rate of false alarms we are willing to tolerate, the lowest rate of missed detections we can get.
Can we do that? With enough more data, yup. There is a classic result due to J. Neyman (long at Berkeley) and K. Pearson (early in statistics) that shows how.
How? Regard false alarm rate as money and think of investing in SF real estate. We put our money done on the opportunities with highest expected ROI until we have spent all our money. Done. For details, an unusually general proof can follow from the Hahn decomposition from the Radon-Nikodym theorem in measure theory, e.g., Rudin, Real and Complex Analysis. Right, in the discrete case, we have a knapsack problem, known to be in NP-complete.
What we have done with our pot stirring is called resampling, and for more such look for B. Efron, long at Yale, and P. Diaconis, once at Harvard, now long at Stanford.
Tom, with a reputation as a hacker, likes to work late, say, till 2 AM. So, we look at the intrusion alerts each minute between 2 AM and 3 AM (something like the performance of the women) and compare with those of the other minutes of 24 hours (like the performance of the men) much as above and ask if Tom is trying to hack the servers.
Or, we have a server farm and/or a network, and we want to detect problems never seen before, e.g., zero day problems. So, we have no data at all on the problems we are trying to detect because we have never seen any of those before.
So, to do a good job, let's pick some system we want to monitor and for that system, get data on, say, each of 10 variables at, say, 20 times a second. Now what?
Our work with bias in women venture applications used just one number for our measurement and test statistic. So we were uni-dimensional. Here we have 10 numbers and need to be multi-dimensional.
Well, in principle we should be able to do much better (pair of Type I and Type II error rates) with 10 numbers than just one. The usual ways will require us to have, with our null hypothesis, the probability distribution of the 10 numbers, but can only get something like that from smoking funny stuff -- not even big data is that big.
So, we want to need no assumptions about distribution, that is, be distribution-free.
So, we want some statistical a hypothesis test that is both multi-dimensional and distribution free.
Can we do that? Yup.
"You mean you can select false alarm rate in advance and get that rate essentially exactly, as in PG's bias example?" Yup.
"Could that be used in a real server farm or network to detect zero day problems -- security, performance, hard/software failures, system management errors?" Yup -- just what it was invented for.
"Attempted credit card fraud?" Ah, once a guy in an audience thought so!
How? Ah, sadly there is no more room in this post!
What else might we do with hypothesis tests? Well, look around at, right, big data or just small data.
Do we have a case of big data analytics or artificial intelligence (AI)?
Ah, I've given a sweetheart outline of statistical hypothesis testing, and now you are suggesting some things really low grade? Where did I go wrong to deserve such an insult?
- Sure, but we should take that off-line.
What is just crucial, really, what it's all about, that is, where the value is, necessarily, inescapably is just in the two rates, the false alarm rate and the detection rate. If can't say at least good things about false alarm rate, then have a definite maybe.
The OP struggled with the false alarm rate -- my work solved that problem.
Really, essentially inescapably and necessarily, the core of the work has to be a statistical hypothesis test.
So, what is that? We make an assumption, typically, that the system we are monitoring is healthy. Commonly that assumption is called the null hypothesis, that is, null in the sense of nothing wrong.
That assumption leads to some mathematical assumptions that can be used as hypotheses in some theorems with proofs.
Then, that assumption with some real data let's us calculate the probability of getting data like we just observed.
If that probability is relatively small, then we have (A) the system is healthy and we just observed something quite rare or (B) the system is sick. If the probability is really small and/or the health of the system is really important, then we guess that case (A) is too rare to be believed, we reject the assumption, the null hypothesis, and conclude (B) that the system is sick.
That's a short description of statistical hypothesis testing for 100+ years a main pillar of pure and applied statistics and a lot in science, both social and physical.
But for monitoring computer server farms, this context very much need the test to be both multidimensional and distribution-free. The world wide collection of those was from tiny down to zero -- likely and apparently zero. So, I invented a nice, large collection of them.
The core of the work was really some novel derivations, right, with theorems and proofs, in, call it, applied probability.
Then I worked out some algorithms to make the computing fast.
The work is likely and apparently the best approach to do zero day monitoring for health and wellness in high end server farms and networks. And there may be some other applications.
Zero day is the case where we are looking for problems never seen before. So, it's sometimes called behavioral monitoring.
Sure, should monitor for all problems have seen before, and there we should be able to do relatively well on the two rates. But, after those old problems, we still need to look for new ones -- sorry 'bout that.
I did this work a long time ago and presented it, or put it in front of, a lot of people, but no one was much interested.
From time to time, people have, as in the OP, rediscovered the problem and done this and that with it. E.g., there was a project of profs Fox and Patterson at Stanford and Berkeley funded by Google, Microsoft, and Sun and in part considered monitoring -- looking at their work, I thought that mine was better.
Sure, there was
http://www.sans.org/resources/idfaq/behavior_based.php
where it was claimed that behavioral monitoring had high false alarm rates. Nope -- my work doesn't! Instead, with my work, get to select false alarm rates over a wide range, including quite small.
I have another project, and for it I recently completed writing the version 1.0 software -- 18,000 programming language statements in 80,000 lines of text (with comments, blank lines, etc). I want to go live ASAP. This project is much easier for people to like! E.g., don't have to get people interested in rates of false alarms and detections. If the project grows, then I will implement and deploy my anomaly detection ideas for my server farm.
Otherwise, for now, I'm no longer interested in that anomaly work mostly because no one else is. Others are welcome, from time to time, to rediscover the problem, struggle with false alarm rate, see no good, general results on detection rate, etc., think that the key is artificial intelligence or machine learning or some such (I didn't use either), etc. Uh, it's statistical hypothesis testing, guys. Sorry 'bout that.
For more, see, say some of the classics:
E. L. Lehmann, Testing Statistical Hypotheses.
E. L. Lehmann, Nonparametrics: Statistical Methods Based on Ranks.
Sidney Siegel, Nonparametric Statistics for the Behavioral Sciences.
Once I presented my work to Cisco. Their reaction was that my work was "just statistical". Well, what the heck do you expect? Want something deterministic, 0 or 1, for healthy or sick? Do you realize what the heck you are asking for?
Another time I presented the work and the reaction was, "What we really want is diagnosis." Gads. You are Sony -- or some other big organization, take your pick from various headlines from current to going way back -- and your server farm is under attack with gigabytes or terabytes on the way out the door to hackers, your whole company is at risk of going belly up, and you don't want to know that there's a serious problem until you can also be told just what the heck to do about it? Sure that's your view?
Gee, you don't want to know that your car's about out of gas unless you are also told that you can get more gas right away? Ever drive in rural Montana? Like long walks? What about engine oil pressure or coolant temperature -- don't want to know about those, either? Or your blood pressure? Not until can be told just what to do about it? Sure about that?
It's an important problem. I did the work. It's good work. But if people are not much interested, then so be it -- I'm not into pushing big rocks up long hills for no good reason.
Suspicion: A guy has a sore on his foot. He knows it's there. It hurts. But he lives with it. He's tried this and that, but he still has the sore. Someone has a good cure, but he is so used to putting up with the sore that he ignores the good solution. Besides, he didn't see a way to cure the sore so doesn't believe that anyone else could either.
Even worse, my solution is not really computer science but is, really, some applied math, actually applied probability, and such work, especially new such work, is rare in practice.
More generally, good, new stuff is rare, so rare that people are very reluctant to consider that they should pay any attention to it.
Fine: Let them live with a sore foot.
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