Suppose you have some location or small area, call it location A, and you have decided for this location the 1-in-100 year event for some magnitude in that area is ‘x’. That is to say, the probability of an event with magnitude exceeding ‘x’ in the next year at location A is 1/100. For clarity, I would rather state the exact definition, rather than say ‘1-in-100 year event’.
Now suppose you have a second location, call it location B, and you are worried about an event exceeding ‘x’ in the next year at either location A or location B. For simplicity suppose that ‘x’ is the 1-in-100 year event at location B as well, and suppose also that the magnitude of events at the two locations are probabilistically independent. In this case “an event exceeding ‘x’ in the next year at either A or B” is the logical complement of “no event exceeding ‘x’ in the next year at A, AND no event exceeding ‘x’ in the next year at B”; in logic this is known as De Morgan’s Law. This gives us the result:
Pr(an event exceeding ‘x’ in the next year at either A or B) = 1 – (1 – 1/100) * (1 – 1/100).
This argument generalises to any number of locations. Suppose our locations are numbered from 1 up to n, and let ‘p_i’ be the probability that the magnitude exceeds some threshold ‘x’ in the next year at location i. I will write ‘somewhere’ for ‘somewhere in the union of the n locations’. Then, assuming probabilistic independence as before,
Pr(an event exceeding ‘x’ in the next year somewhere) = 1 – (1 – p_1) * … * (1 – p_n).
If the sum of all of the p_i’s is less than about 0.1, then there is a good approximation to this value, namely
Pr(an event exceeding ‘x’ in the next year somewhere) = p_1 + … + p_n, approximately.
But don’t use this approximation if the result is more than about 0.1, use the proper formula instead.
One thing to remember is that if ‘x’ is the 1-in-100 year event for a single location, it is NOT the 1-in-100 year event for two or more locations. Suppose that you have ten locations, and x is the 1-in-100 year event for each location, and assume probabilistic independence as before. Then the probability of an event exceeding ‘x’ in the next year somewhere is 1/10. In other words, ‘x’ is the 1-in-10 year event over the union of the ten locations. Conversely, if you want the 1-in-100 year event over the union of the ten locations then you need to find the 1-in-1000 year event at an individual location.
These calculations all assumed that the magnitudes were probabilistically independent across locations. This was for simplicity: the probability calculus tells us exactly how to compute the probability of an event exceeding ‘x’ in the next year somewhere, for any joint distribution of the magnitudes at the locations. This is more complicated: ask your friendly statistician (who will tell you about the awesome inclusion/exclusion formula). The basic message doesn’t change, though. The probability of exceeding ‘x’ somewhere depends on the number of locations you are considering. Or, in terms of areas, the probability of exceeding ‘x’ somewhere depends on the size of the region you are considering.
Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.
As plant science PhD students, we feel it is vital to share our research with other scientists to generate new ideas for collaborative projects. For this reason we decided to organise the ‘Innovations in Plant Science to Feed a Changing World’ workshop, which was held in the University of Bristol Biological Sciences department in February 2017. The delegates included early-career scientists from Kyoto University, Heidelberg University and of course the University of Bristol.
Figure 1. The Conference Poster
The University of Bristol has a long-standing partnership with Kyoto University and more recently, Heidelberg University, as our plant science groups share overlapping research areas. The main aim of the workshop was to encourage novel collaboration opportunities between the plant science groups, which would give rise to future projects, publications and ultimately funding.
Last year, Kyoto University hosted a highly engaging and productive workshop (see Sarah Jose’s blog post last year) for early-career scientists from the three universities in this coalition. Following from the success of this workshop, we decided to organise the second workshop, where participants could build upon the partnerships forged at the last meeting, form new links and present their results in a friendly environment. So, for the past six months, a team of PhD students and post-docs has been busy organising the meeting that took place in February.
As it turns out, organizing a three-day conference, even a relatively small one, is quite a lot of work. Getting venues, transfers, catering, accommodation and social activities booked all presented their own particular challenges. However, perhaps the most challenging task was designing the program for the workshop, which was set out into different themes to encompass the participants’ different subject areas.
All the organisation paid off when the visitors arrived, slightly (very) jet lagged from their long flights. Once the workshop had started, we were delighted with how smoothly the sessions ran and how engaging the talks were. Following the talks there were many discussions over coffee, during the poster session and break-out session. We also included a careers talk from Prof Tokitaka Oyama from Kyoto University, who shared his insights on how to succeed as a plant scientist. Another highlight was the keynote talk from Professor Keith Lindsey (University of Durham), who shared his fascinating work on modeling plant developmental biology.
In amongst all the science, we had time for an excursion to the University of Bristol Botanical Gardens where Nick Wray gave a fascinating tour, which was very enjoyable. We also visited the Wills memorial building tower and even had a go at ringing the bell!
Figure 2. Nick Wray (far right) led a fascinating tour of the University’s Botanic Garden for the visitors.
Although organising the workshop was a lot of work, it was definitely worth it. Our organisation, leadership and project management skills were trained and tested in the run-up to the workshop, but in the end, it went very well indeed. All the delegates thoroughly enjoyed their participation and a comment that was heard a few times was that delegates were impressed, not just with the quality of the science being presented, but also the quality of the scientific discussion particularly given that English was not the first language for the majority of the participants.
We hope that the links formed at the workshop will continue to develop into novel collaborative projects. – I (Donald) definitely benefited as the post-doc Massaki Okada even stayed on a few days to teach me some techniques.
We would like to thank our funders, the Bristol Centre for Agricultural Innovation and the New Phytologist Trust for their support. We’d also like to thank the other members of the organising committee whose hard work made this workshop so successful: Fiona Belbin, Deirdre McLachlan, Tsuyoshi Aoyama and Antony Dodd.
Probability is a quantification of uncertainty. We use probability words in our everyday discourse: impossible, very unlikely, 50:50, likely, 95% certain, almost certain, certain. This suggests a shared understanding of what probability is, and yet it has proved very hard to operationalise probability in a way that is widely accepted.
Uncertainty is subjective
Uncertainty is a property of the mind, and varies between people, according to their learning and experiences, way of thinking, disposition, and mood. Were we being scrupulous we would always say “my probability” or “your probability” but never “the probability”. When we use “the”, it is sometimes justified by convention, in situations of symmetry: tossing a coin, rolling a dice, drawing cards from a pack, balls from a lottery machine. This convention is wrong, but useful — were we to inspect a coin, a dice, a pack of cards, or a lottery machine, we would discover asymmetry.
Agreement about symmetry is an example of a wider phenomenon, namely consensus. If well-informed people agree on a probability, then we might say “the probability”. Probabilities in public discourse are often of this form, for example the IPCC’s “extremely likely” (at least 95% certain) that human activities are the main cause of global warming since the 1950s. Stated probabilities can never be defended as ‘objective’, because they are not. They are defensible when they represent a consensus of well-informed people. People wanting to disparage this type of stated probability will attack the notion of consensus amongst well-informed people, often by setting absurdly high standards for what we mean by ‘consensus’, closer to ‘unanimity’.
Abstraction in mathematics
Probability is a very good example of the development of abstraction in mathematics. Early writers on probability in the 17th century based their calculations strongly on their intuition. By the 19th century mathematicians were discovering that intuition was not good guide to the further development of their subject. Into the 20th century mathematics was increasingly defined by mathematicians as ‘the manipulation of symbols according to rules’, which is the modern definition. What was surprising and gratifying is that mathematical abstraction continued (and continues) to be useful in reasoning about the world. This is known as “the unreasonable effectiveness of mathematics”.
The abstract theory of probability was finally defined by the great 20th century mathematician Andrey Kolmogorov, in 1933: the recency of this date showing how difficult this was. Kolmogorov’s definition paid no heed at all to what ‘probability’ meant; only the rules for how probabilities behaved were important. Stripped to their essentials, these rules are:
1. If A is a proposition, then Pr(A) >= 0.
2. If A is certainly true, then Pr(A) = 1.
3. If A and B are mutually exclusive (i.e. they cannot both be true), then Pr(A or B) = Pr(A) + Pr(B).
The formal definition is based on advanced mathematical concepts that you might learn in the final year of a maths degree at a top university.
‘Probability theory’ is the study of functions ‘Pr’ which have the three properties listed above. Probability theorists are under no obligations to provide a meaning for ‘Pr’. This obligation falls in particular to applied statisticians (also physicists, computer scientists, and philosophers), who would like to use probability to make useful statements about the world.
Probability and betting
There are several interpretations of probability. Out of these, one interpretation has emerged to be both subjective and generic: probability is your fair price for a bet. If A is a proposition, then Pr(A) is the amount you would pay, in £, for a bet which pays £0 if A turns out to be false, and £1 if A turns out to be true. Under this interpretation rules 1 and 2 are implied by the reasonable preference for not losing money. Rule 3 is also implied by the same preference, although the proof is arcane, compared to simple betting. The overall theorem is called the Dutch Book Theorem: if probabilities are your fair prices for bets, then your bookmaker cannot make you a sure loser if and only if your probabilities obey the three rules.
This interpretation is at once liberating and threatening. It is liberating because it avoids the difficulties of other interpretations, and emphasises what we know to be true, that uncertainty is a property of the mind, and varies from person to person. It is threatening because it does not seem very scientific — betting being rather trivial — and because it does not conform to the way that scientists often use probabilities, although it does conform quite closely to the vernacular use of probabilities. Many scientists will deny that their probability is their fair price for a bet, although they will be hard-pressed to explain what it is, if not.
Blog post by Prof. Jonathan Rougier, Professor of Statistical Science. First blog in series here.
Second blog in series here. Third blog in series here.
Ever wondered what a career in aid and development is like? Or how the world’s current development programmes came into being? Look no further than this blog on Myles Wickstead who gave a Cabot Institute lecture and short interview on his reflections and experiences on a colourful career in aid and development.
Among Wickstead’s notable achievements are a position as head of British Development Division in Eastern Africa, coordinating a British Government White Paper on eliminating world poverty and now being an advisor to the charity Hand in Hand International.
An audio recording of Myles lecture can be found above. His talk focussed largely on the inception of the building blocks of international development; the UN, the World Bank and the International Monetary fund. He began by turning back time towards the end of the second World War, in which the atmosphere of global reconciliation bred the need for trans-border institutions such as the UN that had the oversight necessary for peace to prevail.
Many years later, the UN decided to introduce development goals with the aim of reducing global poverty within a given time frame. The first of these was the Millennium Development Goals which were drafted in the UN head quarters with little external solicitation. In fact, Wickstead reminisced that environmental goals were almost completely overlooked and only added when a member of the committee ran into the director of the UN environment department on the way to the copier room…
Wickstead went on to add that a large parts of the Millennium goals were generally quite successful although there was still plenty of scope to be more inclusive. He also dwelt on the new Sustainable Development Goals drafted by the UN in 2015 and the Paris climate summit which, Wickstead claims, represent a much more integrated approach to propel international development into the future.
Below is my interview with Myles in which I question him on his talk and ask him about his career in aid and development:
You mentioned a fair bit in your talk about the importance of tying in environment sustainability with aid and development. How do you see that working in practice in a developing country when sustainable practices can be sometimes be quite anti-economic?
Yes, the two things are brought together in the Global Goals for Sustainable Development agreed in New York in September 2015. Let’s take an example of a country that’s well-endowed with forest resources. They could get rich quickly by chopping down the trees and selling the wood. You can’t expect those countries to simply say ‘we are not going to chop our forest down’. Firstly you need them to realise that for the long-term sustainability of their country they need to preserve the forest. But second, because maintaining the forests helps protect us all from climate change they rightly expect some compensation from the international community to do so. There are (albeit imperfect) mechanisms in place for this. Despite this I do, on the whole, think they are being successfully implemented: take Brazil for example.
There are also examples where – often without the consent of the government – indigenous forests are being cut down to make way for palm oil plantations, with devastating consequences not only for the trees but the wildlife. In these situations, governments need to be encouraged to take firm action against the individuals or companies concerned, again with support from the international community as and if appropriate.
I work on volcanic hazard in Ethiopia and one of the things I’ve noticed is the more wealthy urban areas are developing fast with an expanding middle class, but the more rural areas are still subject to a lot of extreme poverty. What part should external aid play in helping this wealth filter down?
It’s a very important question and one I touched on when talking about the Millennium Development Goals (MDGs) which were in place from 2000-2015. That period saw extraordinary progress, including halving the proportion of people living in absolute poverty, but many people (for example, those with disabilities or from ethnic minorities) were left out. It is also the case that urban areas, with generally better infrastructure and more job opportunities, tended to make faster progress. A lot of people in rural areas were in very much the same position in 2015 as in 1990. Within Ethiopia, a combination of rapid economic growth – supported both by investment and aid – and good policies mean that the benefits are now being felt more widely.
The role of Chinese investment in infrastructure, particularly roads, has I think been quite a positive one. The Government of Ethiopia has a very clear five year growth and investment plan and they expect their partners to deliver; I remember one case of former Ethiopian Prime Minister Meles requiring a Chinese company to rebuild a road they had just built as it was not up to standard; I am sure they were equally exacting of other companies from other countries. Not all African governments have that kind of determination but on the whole I think Chinese engagement has been a good thing. And the fact that Africa was largely unaffected by the global recession following the crash of 2008 was not only because it was not as connected to the international financial system as other parts of the world, but also because China and other countries in Asia continued to buy its raw materials.
What influenced your decision to have a career in Aid and development?
I had lived and travelled overseas a little. My father was a marine biologist and as a technical expert worked for the predecessors of DFID and lived and worked overseas in places like Singapore, Tanzania, and Jamaica. So I probably got some of the wish to live and work overseas from him – though alas didn’t inherit the science gene, which passed me by!
I went through the civil service fast stream process, and having successfully negotiated that had to make a choice about which Department I wanted to join. It was then the Ministry of Overseas Development; a few years later became the Overseas Development Administration of the FCO; and in 1997 became a fully-fledged Department of State with its own Cabinet Minister. Interestingly, DFID remains the most popular choice of all government departments for fast-steam applicants.
Is there a defining moment in your career you want to mention?
I have been extraordinarily lucky in the choices that I have made – or have been made for me – in terms of where I was at particular time. To have had the chance to run a regional office in Africa; to have been on the Board of the World Bank; to have worked closely with Ministers both as a Private Secretary and in coordinating the 1997 White Paper (the first in 24 years); and to be Ambassador to Ethiopia and the African Union – it was a huge privilege (and very hard work!) to be given these responsibilities. I ran the Commission for Africa Secretariat in 2004/5, and I suppose one of the great moments was going to present a copy of the Commission’s Report ‘Our Common Interest’ in 2005 to Nelson Mandela.
Someone asked earlier today- how do you keep positive despite the gloomy state of much of the world? My answer would be that the world has made extraordinary progress over the past quarter of a century in pulling people out of poverty, and that we have a real chance of completing the task, in line with the Sustainable Development Goals, by 2030. Of course there have been setbacks along the way, and there will be more – conflict and environmental challenges to name but two. But with political will, and by maintaining a positive focus, I believe we can aspire to a better world both for ourselves and for future generations.
Blog post by Prof. Jonathan Rougier, Professor of Statistical Science.
