This is Professor Jeffrey Amundson. We know Jeff as Jeff from down the corridor who comes to our states meetings and likens the map every month.
But the world of pharmacology knows Jeff as a leading clinical pharmacologist
and editor of a cult and thinks that the name of the book is a definitive,
constantly updated, definitive index of drug reactions and interactions.
Is that a fair description or do you?
We've really enjoyed collaborating with Jeff on many of the projects we do in the primary care department, and we enjoy his talks.
Let's hope you do, too. Of course, very much. Thank you.
Well, I hope I hope the title I've given you gives me is not too much of a come on.
I hope it works. Yeah, there is a little known law of numbers.
My experience in talking to people about this has been that it is little known.
When I gave this talk to the Stats Coven, as it's called, in the Centre for Evidence based Medicine.
I think three people out of about 20 said they'd heard of it,
but maybe some of you have heard of it and are going to be disappointed that it's not little known.
So I apologise if that's the case in advance. Any mathematicians here?
Just one or two. Right.
Well, you'll be able to explain things that I can't explain. That's good.
Any statisticians, some nobody.
Nobody else admitting to be a statistician. Well, you'll be able to say, well, the probability is that any explanations given are true.
So that will be helpful as well. Now I usually start with some conflicts of interest.
Let's move this over here. I can't change because there's a slight change of pull or a slight change.
A laser pointer that's still there. Good. Okay, let's try that.
Okay. So I have apart from being a physician and a clinical pharmacologist and not a mathematician.
Not a statistician. Those are my disavow. I have and I'm a member of the Centre for Evidence based Medicine in Oxford.
I do have other things that go on you ought to know about.
I'm connected with the British Pharmacological Society, a British National Formulary,
British Pharmacopoeia Commission, the West Midlands Centre for Adverse Drug Reactions.
That's in Birmingham. And I am a colleague, as Richard said, of Richard Stephens.
And you just know where I stand now. Okay.
So I'm going to give you a to ask a question.
This is two sets of data. Supposedly the area in square kilometres of each of these countries.
And the list goes on. But 196 of them, I think, in this example.
So you've got all the areas of all the countries.
The question is, are these real data or fake data?
All right. So have a look at the data and decide whether you can answer that question.
To start with, don't say anything. Oh, and by the way, if any of you does know this little known law, don't say, oh, yeah, I don't know you.
Please keep quiet for the moment. I'll ask you all later.
Okay. So you've had a chance to look at these data and decide whether they're real or fake.
Good. Now, I've selected. Not at random.
This is not a random selection, as you will see nine countries from that list and there are their populations.
And if I truncated them the last seven digits, I'd have the leading digit.
In each case, they'd have 111, two, one, three, one, four and so on.
So all I'm interested in is the leading digit. Okay.
So the question is, if I take all 196 or whatever countries, what is the distribution of the numbers of leading digits?
So that's the question I have to ask of you. What is the distribution of the say, let's say, 100 leading digits?
Okay. So here we are. We have the leading digits.
And I'm going to suggest some possible distributions. And here they are for possible distributions.
The first distribution is completely equal.
A flat distribution, the same number of ones, the same number of twos, the same number of three, same number for us.
Okay. The second distribution is a lot of ones, fewer twos and so on, down to very few names.
And the third distribution is that one upside down. Lots of names, very few ones.
And the fourth distribution is a normal distribution bell shaped curve.
Okay. So give you a minute, just a second or two just to think about it.
Then I'm going to ask you to vote. This is not Brexit.
It won't be held by it.
Okay, everybody ready? So who votes for this distribution here?
All equal across the board. Nobody's interested in that one.
Well, good. Okay. How about the second distribution?
Lots of ones. Very few names. Mm hmm.
Got three for that one. Good. Okay. How about lots of names and very few ones?
Nobody willing to lose a few.
You know, there's not going to be a we're not going to cut your heads off if you get it wrong or whatever.
It doesn't matter. Have a go. Guess if you don't know. Have a guess.
And what about the normal distribution? Any offers for that?
One, two. You're not a very good audience. You haven't said you haven't participated.
Let's try it again. I want everybody to guess if you don't know.
All right, but we'll shoot you. The even distribution.
What about gas? Is one gas ready to gas is all right.
I'm watching to see if you put your hands up again and vote off and vote for.
Okay. Second distribution, lots of ones. Okay.
One, two, three, four, five, six. Very good. This one.
Lots of nines. One, two, three. And the normal distribution?
Yeah. Half a dozen or so. So I guess that's no better than chance in a small group.
I don't need to do a significance test, but actually, that's the answer.
Right. Does anybody know what this distribution is called?
Anybody come across it? No. Well, you know, because you know, so it is little known.
It appears it's called Bedford's law or Bedford's distribution.
And it's it is frankly surprising. When I first was asked this, I thought this is going to be even, isn't it?
If I hadn't called it a little known law of numbers and you thought there was something funny going on here,
you might all have voted for that, I guess. I don't know. That's what I thought when I first heard the problem.
I thought, surely they're all going to be normally they're all going to be the same amount.
Equal numbers of each. Turns out not.
And that's the distribution. The probability that any number will occur is the logarithm of that number, plus one over the number.
That's benefits, distribution, so called. And we'll I'll show you why.
Well, there it is. So that's where these numbers come from.
Log two over one, log three over two, log four over three. And there's the distribution that you saw before rounded off.
Okay. That's where it comes from. All right. Has anybody heard of Sticklers Law upon to me?
Sticklers Law Stigmas Law was propounded by Mr. Stigler.
Dr. Stigler. And he says, I've chosen for the title of this paper.
Stigler Low of Economy, which may appear to be a flagrant violation of the institutional norm of humility.
But actually, Stigler is calling the law.
STIGLITZ Law obeys Stiglitz law.
It's very self referential because Stigler is law is that no scientific discovery is named after its original discoverer,
and Stigler did not discover Stiglitz law. So stimulus law obeys Stigler law and Stigler.
This is a great paper that Stigler goes on to mention.
Okay, Merton. Robert Merton. Some of you may have heard a great sociologist of science.
He wrote some wonderful papers. He was the man who first pointed out the law of unexpected consequences or unintended consequences.
His paper on that is is very worth reading.
And so Stigler quotes Merton saying that Merton's famous hypothesis that all scientific discoveries are in principle multiple.
Actually, of course, it goes back beyond that. Before that, before Merton even.
This is George Sutton in a book written in 1936, The Study of the History of Science.
And he says creations absolutely de novo are very rare if they occur at all.
And of course, we know that things are discovered simultaneously by different people.
Russell and Wallace and Russell discovering the idea of evolution simultaneously live
in IT'S and Newton discovering the ideas of calculus simultaneously and much disputed.
And so I co I called this independently before I ever learnt about Stigler.
I called it the law of no man. No man is the Latin, of course, for a name.
So it seemed to me that was a good name for a name and it stands for non original mal appropriate eponymous nomenclature and it Stigler is low.
No entity is named after its discoverer.
Actually it comes from this man, Simon Newcomb, who showed it in 1881, and he published this paper in the American Journal of Mathematics.
And what he noticed and this is well known by those who know it,
is that if you had a book of logarithmic tables and we couldn't discover this today because we don't have books of logarithmic tables,
he found that the first pages were much more thumbed than the last pages.
It's a fascinating observation. And he said, Well, this is odd.
People are looking at the ones more than the nines.
And he looked into this distribution and he said the first significant figure is oftener one than any other digit,
and the frequency diminishes up to nine. Beautiful demonstration.
And then he goes on to ask other questions that I won't go into and look, he's got it all there.
The distributions that I pointed out in that column, they're exactly the distributions.
Of course, it's bound to be exact, because he derived it. And we have just shown you it's log D plus one over D.
And the law of probability, he says, of the occurrence of numbers is such that all men to say of their logarithms are equally probable.
So it's an logarithmic law. And a lot of things in science, in biology follow logarithmic distributions, of course.
And actually in passing, one can note that it's not just the first digit.
It can be the first pair of digits. Ten, 11, 12, 13.
Same low applies it's log D plus one over D.
Then you can go on calculating the probabilities and it all fits as I'll show you later.
So is Benford in the 1930 is 1938.
And he repeats this observation and derives the same law.
Whether he knew about the previous derivative, I don't know.
But he certainly quotes the grubby pages hypothesis that is well known from books of logarithms.
And this time what he does is he actually tests the law.
He takes a whole load of data and he tests the law and he finds it fit.
And I think so. He says the law of anomalous numbers he calls them, is thus a general probability law of widespread application.
And there are some of his data and this is prime numbers, and he's plotted the distribution of first digits,
leading digits in all the prime numbers in the top left.
It's the first 5.7 million.
The next one is the first 50 million, then 455 million and then 4 billion, 4 billion prime numbers.
And the law holds throughout.
Oh, in fact, it improves, as you can see, very slightly the more you have to analyse.
Now, I did an analysis of this sort on the country data that I showed you at the beginning.
And here they are. It's just 196 countries. But you can see the leading digits in the populations of the capitals of those countries.
Follows benefits low. So does the frequency of the leading digits of the populations of the countries as a whole,
not just the capitals, but the populations of the countries as a whole.
So do the frequencies of the leaving digits in the areas in square miles of those countries.
And if you did it in square kilometres be exactly the same or an Egyptian cubits.
But it it just it it scales.
Square kilometres. So now you are in a position to say whether these data are real or fake.
I'll give you a minute or two to think about it. Okay.
So I'm going to ask you if you think these data are fake.
Would anybody think that the first column are the fake data?
I'm not saying that, but they may both be fake. They will both be real.
Okay. So anybody think that the first column are fake data?
Guess if you want to know. No. No penalty. No, nobody for that.
What about the second column? You think those might be fair? Yeah.
Yeah. There's no ones.
They don't begin with one, any of them. But the first column, there are some.
See, they might be. They might not be.
It's a small sample, of course, and as I've said over analyse the whole data set, but here are the distributions.
There's the distribution for the first one. So you can't tell really.
You certainly couldn't tell they were fake. You couldn't you couldn't say these are fake data.
You can't, on the other hand, say these are real data. It looks as if they probably are, but it's not a big enough sample.
But you'd be damn sure those were fake, but pretty high probability that they are fake data.
Okay. There are some statistics that match Bedford's law and you can read them for yourselves.
Some interesting, interesting sets of data there.
These all obey benefits law. It's quite striking.
It really is amazing. Here's some more.
This is top left is the distribution of 122,000 cities in the world,
according to the heights above sea level in metres and on the right in Royal Egyptian Cubits, just for the [INAUDIBLE] of it.
A daily volume of shares on Nasdaq, import export volumes for seafood, fish and so on.
That's the URL of that particular website.
And I said The leading pairs of digits. There you are.
Nasdaq daily trading volumes. And as you can see, remember that the names start at 4.5.
So here we have the tens elevens twelves up to the 99.
All following Bamford's law with incredible precision accuracy.
Don't don't correct me. Okay.
It's quite amazing. So why is this?
Well, this is how I understand it. We've got a pencil, a little stubby pencil.
It's one unit in length. And I'm going to make the pencil grow.
That shouldn't be there. Sorry. It doesn't matter. And until it's two units in length.
Right. And it's going to grow at a uniform rate.
So the time it takes to grow from one unit to two units is the time it takes to double in length.
Now, if I measured the length of it every second, let's say I'd have a range of numbers, all of which began with a 11. to a one 1.0 to 1.03 1.99.
During that period, I would have 99 C or however many numbers beginning with A one.
But if it continues to grow, when it grows between two and three, it only has to grow by half the length.
And so there will be fewer twos during the same time because the proportion by which it's growing is not as large.
Similarly, when it grows to four units, it's only growing by a third.
So in the same interval, growing at the same rate, there will be fewer threes than there were twos in theatres than there were ones.
That explains it. And it's a logarithmic distribution.
So if you if you plot the log, if you plot the normal numbers on a logarithmic scale, that's 30% log, 2.3012.
So that's 30% of the way from 1 to 10 and so on.
Now, as you go up the scale, the space is become narrower and narrower.
Anybody who's ever used a log paper for drawing, as I do sometimes for drawing plasma concentration verses, time curves, which are log linear.
If you give a drug to someone, measure the plasma concentrations, the concentration falls exponentially.
And if you plot it on a logarithmic concentration scale against time, it's a straight line.
And the logarithmic concentrations look like that. They get smaller and smaller as you go up from 1 to 10.
But when you get to ten to get to 20, you get a double again.
Going from 10 to 11 is smaller again than this, but going from 10 to 20 is the same size as going from 1 to 2.
So here you have to increase by 100% here by 50%, here by 33%, and so on.
And so if we go back to our law, we see that's exactly what it is.
Three over two is one and a half. It's 50% more.
Four over three is one in a third, it's 33% more.
Just like the pencil growing. That is why benefits law applies.
Here's the paper. Can't remember where this comes from, but it is post email.
So it's relatively recent on first digits of squares and cubes.
Here's one on the first digit frequencies of prime numbers where we saw that Benford did that.
But this these pair did the rim and the zeros of the Riemann Zeta function,
which is to do with the distribution of primes and to do with the famous Riemann hypothesis so far unsolved or unproven.
I wondered if I could get a set of data that might follow Bedford's law.
And so I thought, well, the simplest set of data I can access without going to an encyclopaedia,
the stuff on the countries, the populations and so on, was from Whitaker's Almanac that I analysed.
This is from my computer. If you look at your computer, you've got Doc Dark and PDF and so and they all have document sizes, so many killer bases.
So I thought, well, I'll take out all of the data in my computer and see if the sizes of the documents fit.
Then for as long they yeah. It's not significantly different but it's, it's not quite right is it.
It's not quite fitting.
So I thought well maybe I haven't got a big enough sample 330 with it should be enough, but maybe I don't have a big enough sample.
So I extended it to more and stayed the same. 850 stayed the same.
And there they all are. So this it doesn't quite fit.
So I presented the data to the stats cover and we discussed it and I thought, well,
maybe it's something to do with the fact that I'm mixing up that dark dot doc x PDF and so on.
Maybe I should separate them out. So I did this afternoon on the train.
That's what I did, and that's the PDFs.
It's not helping. And worse still, the dogs are actually significantly different by Chi Square.
They just did a simple chi square with eight degrees of freedom and it's less than O2.
So there is something not quite right about these data, though not quite conventional, shall we say.
I don't know what it is, but this man, Theodore Hill, is the world's one of the world's experts on this benefits law.
And this is what he says. If distributions are selected at random in any unbiased way, then I suspect that's important.
I think the PDF data may be reasonably random, but I think the doc data probably aren't because they're my word files.
And I write a blog every week and it's the same length. Every time I write an article for a journal, it's similar length every time.
So I suspect that there is bias in the doc data, which explains why they're so bad.
I wouldn't have expected bias in the PDF data, but maybe there is a source of bias there.
I don't know. It's just a hypothesis. Then he says the significant digit frequencies,
that's the first digit of the first two digits of the combined sample will converge to Bamford's distribution.
And that's what we saw in a way, although the doc data were rubbish if you like, it did not fit.
When I put them in with all the other data and I haven't analysed all the other, I haven't analysed the input and so when I put them all in,
it started to converge, although it clearly hasn't converged fully to Bedford's level.
So maybe that's what it is, but my sample is biased.
So here are some lists of numbers that don't obey benefits.
Low lottery numbers, probably just as well, because otherwise, otherwise,
we'd all be guessing to win the lottery with greater ease, the odds would show.
And incidentally, there's an interesting thing when I don't know anybody buy lottery tickets here.
Nobody foolish enough to say.
Yeah, but it would change your life. It would, wouldn't it?
That's an interesting argument. Okay. So let me ask you this. When is the best time to buy a lottery ticket?
Day before. Why? Why? Any better chances?
Greater chance of surviving? Correct. 20 minutes, actually.
20 minutes. 20 minutes. If you buy it before 20 minutes, sooner than 20 minutes, you're more likely to die than win.
That just illustrates how unlikely it is that you're going to win.
It's a beautiful statistic. I haven't worked that out myself.
I read it somewhere with existing telephone numbers. Well, you can read it for yourself.
Interestingly, although the Bose-Einstein distribution in quantum mechanics obeys Bentham's law, Boltzmann Gibbs does not,
which is a thermodynamic distribution and the Fermi direct distribution, which is also to do with energy levels at the quantum.
In in stable systems.
Don't obey this law. So I don't know anything about physics, so I don't know why.
And importantly, fabricated data.
Most people who make up data don't know Bedford's law.
If they did, they get away with it. And actually, probably many of them get away with it anyway.
But if you want to make up data, don't forget before it's law.
You'll be all right. On the other hand, I am reminded of the story of Gregor Mendel.
We all know about Gregor Mendel and the Sweet Peas and how he discovered the laws of genetics.
Well, a famous statistician called Ari Fisher, whom you'll know from Fisher's exact test,
analysed Mendel's data in the 1930s, I think, and concluded that Mendel's data were too good.
They were too close to the prediction to be about, you know, p less than or one to be other than possibly he thought engineered to fit the hypothesis.
But maybe Mendel was just lucky. Or maybe, as some have suggested, his his gardeners gave him the result they thought he wanted.
I had a technician like that once, but, you know, don't stick too slavishly to Bedford's law.
If you're going to fabricate data, put some random error in fluke.
Much you know is not on the syllabus and fabricate that to fabricate data.
We could have a whole week on that, I suspect. All right. And there's a book on this.
It says an introduction, but don't believe it. And it's been I did a systematic well, not a systematic review.
I kind of flipped through papers that mentioned benefits law.
And a PubMed search has been widely used actually, so it's not so little known among those who know it.
Here's the the Fermi directed Boltzmann,
Gibbs and Bose-Einstein distributions discrete dynamical systems whatever that means election or irregularities.
They analysed elections that were supposed to have been fraudulent and found that they did not.
The data from those elections did not fit Bedford's law, thus not confirming the fraudulence.
But you know, if you think it is in the first place and it doesn't meet benefits law,
you've got pretty good reason for suspecting strongly that it was fraudulent.
And recent Russian elections are mentioned there. One wonders about the American the right.
What shouldn't say that? That's why the public health surveillance systems which vary from country to country.
If the data don't obey Bedford's law,
it suggests that the surveillance system is not very good because in systems where the surveillance system is good, the data obey benefits law.
So you can pick out countries that are not good at surveying the population for adverse reactions.
For example, there's the scientific fraud.
Anaesthesia papers. Quality of occupational hygiene data.
Drug Discovery data might help you in your structure, activity, relationships.
Drugs are often discovered because they're not just discovered.
You discover an effect of a drug on a system, some system or other.
If you if you have a target system, say it's an HIV, the virus that causes AIDS,
and there's a part of that virus that you want to target a protein on the surface of the of the of the virus.
And you have 10,000 compounds on your shelves as drug companies.
Do you have a 25% chance of hitting that target just from the random collection of stuff you have on your shelves?
It's really quite good. Now, what you get may not be very good at hitting the target, but it will hit it.
And once you've got that compound, you can modify it, change its structure until you get a better compound.
And that's how some new drugs are discovered. And this is known as structure, activity, relationships,
because the activity varies with the structure and clever chemical pharmacologists can pharmaceutical chemists can do this.
They can say, well, if we put a missile group on or a missile group, that will change the activity and it will make it better at hitting this target.
And then you put that you put the structure of the target into a computer and show how it fits.
The molecule is very clever.
Anyway, the data you get, Fitbit fits well, and that's to do with toxicology, monitoring, radiation, all kinds of brain activity.
What is going on in your brain at the moment? If it doesn't fit then French law.
I have failed electroencephalography.
Incredible. Really. Articles and journals, all sorts of things.
This is a good one. If you look at multiple choice tests and the answers at the ends of the chapter,
you all had these books of mathematics or physics or whatever, and they have problems.
At the end of the chapter, at the end of the book, you get the answers to the problems.
Well, it turns out that the first digits that the answers follow Bedford's law.
So you might think that in a multiple choice question system, if you knew this, you might have a better chance of getting the right answers.
Because if you just guess a one, you're more likely to get it right.
But it turns out not to work. So that's just as well because our students want to cheat the system in that way, as it were.
Distinguishing noise from chaos. Well, there you go.
Skewness in the distributions in cardiac models, you know, all sorts of stuff.
And there is a generalised form of benefits which I think is quite useful.
And I won't go into it because I don't understand the mathematics,
but that shows you that the probability of any number D coming as the leading digit was equal log D plus one over D.
But this is a more general law for all sorts of distributions, and in this thing you have an alpha,
and if alpha is zero, then the whole thing collapses to the normal distribution.
If alpha is one, it becomes Benford. And for other alphas, I don't know.
But it gives you other distributions which are of interest, I guess, to mathematic mathematicians and others.
So that's all I can tell you about it. It's not something I've read or understand in detail,
and there are some distributions that you can see that are different depending on whether Alpha is one,
two or three, perhaps even fractional for all I know, I don't know.
I mean, you can get a fractional polynomials, so I don't see why you shouldn't have a fractional generalised bedford's law.
But anyway, it can be used for different. So the next thing I suppose I should do is try my data on one of these distributions and
see if they fit when I get them a minute off and explaining the uneven distribution,
the laws of Benford and if anybody knows that rule is zip's law is to do with the frequency of words in a corpus in a dictionary,
say or well, the corpus of texts. And the fifth, let's say most common word, is 20 times more common than the hundredth.
Most common word. It's not, but doesn't necessarily follow intuitively, but it turns out to be so.
The fifth most common word is 30 times more common than the 150th most common word.
There is a distribution of common ness of words in the corpus of language, which is also very interesting.
That's low. There's the online bibliography. Anybody that's interested, it's just put it in Bedford on Bedford online and you'll find it.
And there's a lot of literature on this, which is where I've gone to look for it.
I think that's it. Thank you.