I remember, in college, we were given a lecture on ‘Neonatal Mortality’. Our lecturer was an experienced professor of Pediatrics who was trying to equate the rate of neonatal deaths among countries as probabilities per births. She was inducing that this probability, portrayed, whether the country is taking proper measures to reduce the mortality rates or not. It was a room full of medical students, and medical doctors. At the end of the lecture I pointed out to the professor about the non-linearity in probability. I commented that one can not simply equate the deaths of 1 child per 10 to the deaths of 100 children per 1000. Although, the probability is the same for both cases (0.1) it cannot be used to make inferences about any country’s policies. Off course in the first case (1 death per 10) the chances of genetic cause of mortality and of communicable diseases (that spread via an organized network of infected individuals) is minimal and hence, the ‘rules’ of the game are different for different cases. I tried my best to explain my thesis as simply and as non-technically as possible. But I was laughed at and after the sheer humiliation of being told that I could not understand as I “… lack knowledge of basic probability”, I stopped arguing. It was a humiliation because before being a medical student, I was training to be an Actuary. Actuaries are professionals who study risk and uncertainty by applying mathematical and statistical methods. Probability theory, hence, was my mother tongue.
Non-linearity arises in the models when the ‘rules of the game’ changes as quantities or scenarios change. One cannot then simplify the model and generalize it on more complex scenarios. Paul Willmot gave a great example of it. He explained that if a bottle of beer costs 1$, it does not mean that the price of 100 beers is 100$! lets say you went to buy a drink and the shopkeeper anticipated that your ‘demand’ is of buying 100 bears. He can then ask you that ‘.. we are allowed to sell only 1 beer per person but if you want 100 beers you can, perhaps, pay 120$’. This is not uncommon in markets. But people fail to realize it. Non-linearity is one example of many mathematical errors one makes buy over or under simplifying the system being modeled. Today mathematical model plays a major role in policy making in everything, from Economics and Politics to Physics and Medicine. with the advent of quantitative analysis in almost every field and subjects like systems biology, there is high chance that we will be using more and more of mathematical tools, to understand and make decision about the world around us. Hence, it has become very important for us to understand models properly. And to do so, we must understand the aims of a model clearly. A model is just and analogy, where different levels of mathematical abstraction portray different levels of complexity and applicability. Like a famous adage in mathematics goes.. “All models are wrong, but some are useful”. Alex Mogilner et al. sheds light on the nature of mathematical models in biology:
“We argue that quantitative modeling is a versatile tool that has to fit the biological problem and can be judged by its usefulness rather than its comprehensiveness and sophistication” ~ Science(vol 336), 13th April 2012. pg: 175-179.
I have found this great manifesto written by concerned Quantitative financial analysts or Quants after the collapse of 2007 sub-prime mortgage market collapse. Like doctors, entrusted upon the lives of their fellow men, are required to undertake an oath, a code of ethics perfected by a greek doctor, Hippocrates, following oath has been designed for mathematical modelers that highlights the key points of good quantitative analysis practice: (The full Manifesto can be accessed here http://bit.ly/13cremc )
The Modelers’ Hippocratic Oath
~ I will remember that I didn’t make the world, and it doesn’t satisfy my equations. ~ Though I will use models boldly to estimate value, I will not be overly impressed by mathematics. ~ I will never sacrifice reality for elegance without explaining why I have done so. ~ Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights. ~ I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension.
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