Archive for mathematics

Mathematical and Physical Sciences Open Day at Sussex

Posted in Biographical, Education with tags , , , , on October 4, 2014 by telescoper

It’s another open day at the University of Sussex so I’m on campus again to help out as best I can, although I have to admit that all the hard work is being done by others! It’s been extremely busy so far; in fact, I’m told that about 6000 visitors are on campus today. This a good sign for the forthcoming admissions round, probably buoyed by the improved position of the University of Sussex in the latest set of league tables and in excellent employment prospects for graduates.

Anyway the good folks of  the Department of Physics & Astronomy  and Department of Mathematics were here bright and early to get things ready:

IMG-20141004-00413

IMG-20141004-00414

All morning we’ve had a steady stream of visitors to the School of Mathematical and Physical Sciences (which comprises both Departments mentioned above). While I’m at it let me just give a special mention to Darren Baskill’s Outreach Team (seen in the team photograph below).
outreachThey have had absolutely amazing year, running a huge range of events and activities that have reached a staggering 14,000 people of all ages (including 12,000 of school age).

Anyway, I think I’ll toddle off and see if I can sit in on one of today’s lectures. It’s about time I learned something.

 

UPDATE: Here is Mark Hindmarsh about to get started on his lecture.

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You could have knocked me down with a feather when I saw that he had included a quote from this blog in his talk:

I’ve worked in some good physics departments in my time, but the Department of Sussex is completely unique both for the level of support it offers students and the fact that so many of the undergraduates are so highly motivated.

And, yes, I did mean every word of that.

Bayes, Laplace and Bayes’ Theorem

Posted in Bad Statistics with tags , , , , , , , , on October 1, 2014 by telescoper

A  couple of interesting pieces have appeared which discuss Bayesian reasoning in the popular media. One is by Jon Butterworth in his Grauniad science blog and the other is a feature article in the New York Times. I’m in early today because I have an all-day Teaching and Learning Strategy Meeting so before I disappear for that I thought I’d post a quick bit of background.

One way to get to Bayes’ Theorem is by starting with

P(A|C)P(B|AC)=P(B|C)P(A|BC)=P(AB|C)

where I refer to three logical propositions A, B and C and the vertical bar “|” denotes conditioning, i.e. P(A|B) means the probability of A being true given the assumed truth of B; “AB” means “A and B”, etc. This basically follows from the fact that “A and B” must always be equivalent to “B and A”.  Bayes’ theorem  then follows straightforwardly as

P(B|AC) = K^{-1}P(B|C)P(A|BC) = K^{-1} P(AB|C)

where

K=P(A|C).

Many versions of this, including the one in Jon Butterworth’s blog, exclude the third proposition and refer to A and B only. I prefer to keep an extra one in there to remind us that every statement about probability depends on information either known or assumed to be known; any proper statement of probability requires this information to be stated clearly and used appropriately but sadly this requirement is frequently ignored.

Although this is called Bayes’ theorem, the general form of it as stated here was actually first written down not by Bayes, but by Laplace. What Bayes did was derive the special case of this formula for “inverting” the binomial distribution. This distribution gives the probability of x successes in n independent “trials” each having the same probability of success, p; each “trial” has only two possible outcomes (“success” or “failure”). Trials like this are usually called Bernoulli trials, after Daniel Bernoulli. If we ask the question “what is the probability of exactly x successes from the possible n?”, the answer is given by the binomial distribution:

P_n(x|n,p)= C(n,x) p^x (1-p)^{n-x}

where

C(n,x)= \frac{n!}{x!(n-x)!}

is the number of distinct combinations of x objects that can be drawn from a pool of n.

You can probably see immediately how this arises. The probability of x consecutive successes is p multiplied by itself x times, or px. The probability of (n-x) successive failures is similarly (1-p)n-x. The last two terms basically therefore tell us the probability that we have exactly x successes (since there must be n-x failures). The combinatorial factor in front takes account of the fact that the ordering of successes and failures doesn’t matter.

The binomial distribution applies, for example, to repeated tosses of a coin, in which case p is taken to be 0.5 for a fair coin. A biased coin might have a different value of p, but as long as the tosses are independent the formula still applies. The binomial distribution also applies to problems involving drawing balls from urns: it works exactly if the balls are replaced in the urn after each draw, but it also applies approximately without replacement, as long as the number of draws is much smaller than the number of balls in the urn. I leave it as an exercise to calculate the expectation value of the binomial distribution, but the result is not surprising: E(X)=np. If you toss a fair coin ten times the expectation value for the number of heads is 10 times 0.5, which is five. No surprise there. After another bit of maths, the variance of the distribution can also be found. It is np(1-p).

So this gives us the probability of x given a fixed value of p. Bayes was interested in the inverse of this result, the probability of p given x. In other words, Bayes was interested in the answer to the question “If I perform n independent trials and get x successes, what is the probability distribution of p?”. This is a classic example of inverse reasoning, in that it involved turning something like P(A|BC) into something like P(B|AC), which is what is achieved by the theorem stated at the start of this post.

Bayes got the correct answer for his problem, eventually, but by very convoluted reasoning. In my opinion it is quite difficult to justify the name Bayes’ theorem based on what he actually did, although Laplace did specifically acknowledge this contribution when he derived the general result later, which is no doubt why the theorem is always named in Bayes’ honour.

 

This is not the only example in science where the wrong person’s name is attached to a result or discovery. Stigler’s Law of Eponymy strikes again!

So who was the mysterious mathematician behind this result? Thomas Bayes was born in 1702, son of Joshua Bayes, who was a Fellow of the Royal Society (FRS) and one of the very first nonconformist ministers to be ordained in England. Thomas was himself ordained and for a while worked with his father in the Presbyterian Meeting House in Leather Lane, near Holborn in London. In 1720 he was a minister in Tunbridge Wells, in Kent. He retired from the church in 1752 and died in 1761. Thomas Bayes didn’t publish a single paper on mathematics in his own name during his lifetime but was elected a Fellow of the Royal Society (FRS) in 1742.

The paper containing the theorem that now bears his name was published posthumously in the Philosophical Transactions of the Royal Society of London in 1763. In his great Philosophical Essay on Probabilities Laplace wrote:

Bayes, in the Transactions Philosophiques of the Year 1763, sought directly the probability that the possibilities indicated by past experiences are comprised within given limits; and he has arrived at this in a refined and very ingenious manner, although a little perplexing.

The reasoning in the 1763 paper is indeed perplexing, and I remain convinced that the general form we now we refer to as Bayes’ Theorem should really be called Laplace’s Theorem. Nevertheless, Bayes did establish an extremely important principle that is reflected in the title of the New York Times piece I referred to at the start of this piece. In a nutshell this is that probabilities of future events can be updated on the basis of past measurements or, as I prefer to put it, “one person’s posterior is another’s prior”.

 

 

 

Newcastle Joins the Resurgence of UK Physics

Posted in Education, Science Politics, The Universe and Stuff with tags , , , on August 17, 2014 by telescoper

I’ve posted a couple of times about how Physics seems to undergoing a considerable resurgence in popularity at undergraduate level across the United Kingdom, with e.g. Lincoln University setting up a new programme. Now there’s further evidence in that Newcastle University has now decided to re-open its Physics course for 2015 entry.

The University of Newcastle had an undergraduate course in Physics until 2004 when it decided to close it down, apparently owing to lack of demand. They did carry on doing some physics research (in nanoscience, biophysics, optics and astronomy) but not within a standalone physics department. The mid-2000s were tough for UK physics,  and many departments were on the brink at that time. Reading, for example, closed its Physics department in 2006; there is talk that they might be starting again too.

The background to the Newcastle decision is that admissions to physics departments across the country are growing at a healthy rate, a fact that could not have been imagined just ten years ago. Times were tough here at Sussex until relatively recently, but now we’re expanding on the back of increased student numbers and research successes. Indeed having just been through a very busy clearing and confirmation period at Sussex University, it is notable that its the science Schools that have generally done best.  Sussex has traditionally been viewed as basically a Liberal Arts College with some science departments; over 70% of the students here at present are not studying science subjects. With Mathematics this year overtaking English as the most popular A-level choice, this may well change the complexion of Sussex University relatively rapidly.

I’ve always felt that it’s a scandal that there are only around 40 UK “universities” with physics departments Call me old-fashioned, but I think a university without a physics department is not a university at all; it’s particularly strange that a Russell Group university such as Newcastle should not offer a physics degree. I believe in the value of physics for its own sake as well as for the numerous wider benefits it offers society in terms of new technologies and skills. Although the opening of a new physics department will create more competition for the rest of us, I think it’s a very good thing for the subject and for the Higher Education sector general.

That said, it won’t be an easy task to restart an undergraduate physics programme in Newcastle, especially if it is intended to have as large an intake as most successful existing departments (i.e. well over 100 each year). Students will be applying in late 2014 or early 2015 for entry in September 2015. The problem is that the new course won’t figure in any of the league tables on which most potential students based their choice of university. They won’t have an NSS score either. Also their courses  will probably need some time before it can be accredited by the Institute of Physics (as most UK physics courses are).

There’s a lot of ground to make up, and my guess is that it will take some years to built up a significant intake.The University bosses will therefore have to be patient and be prepared to invest heavily in this initiative until it can break even. The decision a decade ago to cut physics doesn’t exactly inspire confidence that they will be prepared to do this, but times have changed and so have the people at the helm so maybe that’s an unfair comment.

There are also difficulties on the research side (which is also vital for a proper undergraduate teaching programme), there are also difficulties. Grant funding is already spread very thin, and there is little sign of any improvement for the foreseeable future  in the “flat cash” situation we’re currently in. There’s also the stifling effect of theResearch Excellence Framework I’ve blogged about before. I don’t know whether Newcastle University intends to expand its staff numbers in Physics or just to rearrange existing staff into a new department, but if they do the former they will have to succeed against well-established competitors in an increasingly tight funding regime. A great deal of thought will have to go into deciding which areas of research to develop, especially as their main regional competitor, Durham University, is very strong in physics.

On the other hand, there are some positives, not least of which is that Newcastle is and has always been a very popular city for students (being of course the finest city in the whole world). These days funding follows students, so that could be a very powerful card if played wisely.

Anyway, these are all problems for other people to deal with. What I really wanted to do was to wish this new venture well and to congratulate Newcastle on rejoining the ranks of proper universities (i.e. ones with physics departments). Any others thinking of joining the club?

A Keno Game Problem

Posted in Cute Problems with tags , , , , on July 25, 2014 by telescoper

It’s been a while since I posted anything in the Cute Problems category so, given that I’ve got an unexpected gap of half an hour today, I thought I’d return to one of my side interests, the mathematics and games and gambling.

There is a variety of gambling games called Keno games in which a player selects (or is given) a set of numbers, some or all of which the player hopes to match with numbers drawn without replacement from a larger set of numbers. A common example of this type of game is Bingo. These games mostly originate in the 19th Century when travelling carnivals and funfairs often involved booths in which customers could gamble in various ways; similar things happen today, though perhaps with more sophisticated games.

In modern Casino Keno (sometimes called Race Horse Keno) a player receives a card with the numbers from 1 to 80 marked on it. He or she then marks a selection between 1 and 15 numbers and indicates the amount of a proposed bet; if n numbers are marked then the game is called `n-spot Keno’. Obviously, in 1-spot Keno, only one number is marked. Twenty numbers are then drawn without replacement from a set comprising the integers 1 to 80, using some form of randomizing device. If an appropriate proportion of the marked numbers are in fact drawn the player gets a payoff calculated by the House. Below you can see the usual payoffs for 10-spot Keno:

tabke
If fewer than five of your numbers are drawn, you lose your £1 stake. The expected gain on a £1 bet can be calculated by working out the probability of each of the outcomes listed above multiplied by the corresponding payoff, adding these together and then subtracting the probability of losing your stake (which corresponds to a gain of -£1). If this overall expected gain is negative (which it will be for any competently run casino) then the expected loss is called the house edge. In other words, if you can expect to lose £X on a £1 bet then X is the house edge.

What is the house edge for 10-spot Keno?

Answers through the comments box please!

Time for a Factorial Moment…

Posted in Bad Statistics with tags , , on July 22, 2014 by telescoper

Another very busy and very hot day so no time for a proper blog post. I suggest we all take a short break and enjoy a Factorial Moment:

Factorial Moment

I remember many moons ago spending ages calculating the factorial moments of the Poisson-Lognormal distribution, only to find that they were well known. If only I’d had Google then…

Mathematics and Meningococcal Meningitis

Posted in Education, Science Politics with tags , , , , on June 9, 2014 by telescoper

Last week I attended a very enjoyable and informative event entitled Excellence with Impact that showcased some of the research that the University of Sussex submitted to the 2014 Research Excellence Framework. One of the case studies came from the Department of Mathematics which is part of the School of Mathematical and Physical Sciences (of which I am Head) so I thought I would showcase it here too:

The description from Youtube reads

Meningococcal meningitis is a debilitating and deadly disease, causing an estimated 10,000 deaths annually in endemic areas of sub-Saharan Africa. A novel mathematical model developed by Sussex researcher Dr Konstantin Blyuss and colleagues has helped explain the patterns of the dynamics of meningococcal meningitis in endemic areas. This model is now being used by epidemiologists and clinical scientists to design and deliver efficient public-health policies to combat this devastating disease.

You can find out more by following this link.

Lincoln – Green Shoots for Maths and Physics?

Posted in Education with tags , , , , on March 3, 2014 by telescoper

I noticed over the weekend that there’s a job being advertised at the University of Lincoln designated Founding Head of the School of Mathematics and Physics. It seems the powers that be at Lincoln University (which is in the Midlands) have decided to set up an entire new activity in Mathematics and Physics. I’m pointing this out not because of any personal connection with the position, but because it’s refreshing to see a new(ish) Higher Education Institute apparently willing to take the plunge and invest in a new venture, particularly because it includes Physics. It wasn’t at all long ago that UK Physics departments were being closed down – the University of Reading being a prominent example, in 2006. I think Reading is thinking of starting up Physics again, in fact. Perhaps these are the green shoots that presage a new spring for Physics in this country? I do hope so.

It won’t be an easy task to start up a new department from scratch in Lincoln: grant funding is tight and the competition for students among established institutions is already so intense that it will be very difficult for a brand new outfit to break through. Nevertheless, I think it’s a praiseworthy initiative and I wish it well.

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