Those a’s and b’s to put inside (a + b) when doing (a + b) power (p+q)

My editorial

I am finishing compiling notes for that article on the role of monetary systems in the transition towards renewable energies, at least I hope I am. This is a bit of a strange frame of mind when I hope I am. Could I be hoping I am not? Interesting question. Anyway, one of the ways I make sure I understand what I am writing about is to take a classic, whom I previously kind of attached to this particular piece of science I am trying to make, and I kind of filter my own thoughts and findings through that particular classic’s thoughts and findings. This time, Thomas Bayes is my classic. Didn’t have much to do with renewable energies, you would say? Weeeell, he was a philosopher and a mathematician, but he lived (and died) in the 18th century, when Europe was being powered by wind and water, thus, as a matter of fact, he had much to do with renewable energies. At the end of the 18th century, in my homeland – Southern Poland, and back in the day is was Austrian Galicia – there was one watermill per 382 people, on average.

And so I am rereading the posthumous article, attributed to reverend Thomas Bayes, received by Mr John Canton, an editor of ‘Philosophical Transactions’ at the Royal Society. On the 23rd of December, 1763, John Canton read a letter, sent from Newington-Green, on the 10th of November, by Mr Richard Price. The letter was being accompanied by an attachment, in the form of a dissertation on ‘the doctrine of chances’, allegedly found by Mr Price in the notes of a defunct friend, Thomas Bayes. The friend had been defunct for two years, at the time, which is quite intriguing in itself. Anyway, Mr Richard Price presented the dissertation as Thomas Bayes’ work, and this is how Bayesian statistics were born  (Bayes, Price 1763[1]). Just as a reminder: in Thomas Bayes’ world, we are talking about having p successes and q failures in p + q trials, in the presence of one single success being probable at the rate ‘a’, and the probability of a single failure being ‘b’. The general way of thinking about it, in this specific universe, is that we take the sum of probabilities, like (a + b), and we give it some depth by elevating it to the power p + q. We create a space of probability through developing the Newtonian binomial (a + b)p+q.

At this point it is useful to dig a little bit into the logic of the Newtonian binomial. When I do (a + b)p+q , Isaac Newton tells me to kind of climb a ladder towards q, one step at a time, and so I am climbing that ladder of failure. First, I consider full success, so my p successes are exactly equal to my n trials, and my failure count is q = 0. In this most optimistic case, the number of different ways I can have that full score of successes is equal to the binomial coefficient (pq/q!) = (p0/0!) = 1/1 = 1. I have just one way of being successful in every trial I take, whatever the number of trials, and whatever the probability of a single success. The probability attached to that one-million-dollar shot is (pq/q!)*ap. See that second factor, the ap.? The more successes I want the least probability I have them all. A probability is a fraction smaller than 1. When I elevate it to any integer, it gets smaller. If the probability of a single success is like fifty-fifty, thus a = 0,5, and I want 5 successes on 5 trials, and I want no failures at all, I can expect those five bull’s eyes with a probability of (50/0!)*0,55 = 0,55 = 0,03125. Now, if I want 7 successes on 7 trials, zero failures, my seven-on-seven-shots-in-the-middle-probability is equal to (70/0!)*0,57 = 0,57 = 0,0078125. See? All I wanted was two more points scored, seven on seven instead of five on five, and this arrogant Newtonian-Bayesian approach sliced my odds by four times.

Now, I admit I can tolerate one failure over n trials, and the rest has to be just pure success, and so my q = 1. I repeat the same procedure: (p1/1!)*ap-1b1. With the data I have just invented, 4 successes on 5 trials, with 0,5 odds of having a single success, so with a = b = 0.5, I have (41/1!) = 4 ways of having that precise compound score. Those 4 ways give me, at the bottom line, a compound probability of (41/1!)*0,54*0,51 = 4*0,54*0,51 = 0,125. Let’s repeat, just to make it sink. Seven trials, two failures, five successes, one success being as probable as one failure, namely a = b = 0,5. How many ways of having 5 successes and 2 failures do I have over 7 trials? I have (52/2!) = 12,5 them ways. How can I possibly have 12,5 ways of doing something? This is precisely the corkscrewed mind of Thomas Bayes: I have between 12 and 13 ways of reaching that particular score. The ‘between’ has become a staple of the whole Bayesian theory.

Now, I return to my sheep, as the French say. My sheep are renewable (energies). Let’s say I have statistics telling me that in my home country, Poland, I have 12,52% of electricity being generated from renewable sources, A.D. 2014. If I think that generating a single kilowatt-hour the green way is a success, my probability of single success, so P(p=1) = a = 0,1252. The probability of a failure is P(q=1) = b = 1 – 0,1252 = 0,8748. How many kilowatt-hours do I generate? Maybe just enough for one person, which, once again averaged, was 2495,843402 kg of oil equivalent or 29026,65877 kilowatt hour per year per capita (multiplied the oil of by 11,63 to get the kilowatt hours). Here, Thomas Bayes reminds me gently: ‘Mr Wasniewski, I wrote about the probability of having just a few successes and a few failures over a few plus a few equals a few total number trials. More than 29 thousands of those kilowatt-hours or whatever it is you want, it is really hard to qualify under ‘a few’. Reduce.’ Good, so I reduce into megawatt hours, and that gives me like n = 29.

Now, according to Thomas Bayes’ logic, I create a space of probabilities by doing (0,1252 + 0,8748)29. The biggest mistake I could make at this point would be to assume that 0,1252 + 0,8748 = 1, which is true, of course, but most impractical for creating spaces of probability. The right way of thinking about it is that I have two distinct occurrences, one marked 0,1252, the other marked 0,8748, and I project those occurrences into a space made of 29 dimensions. In this interesting world, where you have between six and eight ways of being late or being tall, I have like patches of probability. Each of those patches reflects my preferences. You want to have 5 megawatt hours, out of those 29, generated from renewable sources, Mr Wasniewski? As you please, that will make you odds of ((529-5/(29-5)!)*0,12525*0,874829-5 = 1,19236E-13 of reaching this particular score. The problem, Mr Wasniewski, is that you have only 0,000000096 ways of reaching it, which is a bit impractical, as ways come. Could be impossible to do, as a matter of fact.

So, when I create my multiverse of probability the Thomas Bayes way, some patches of probability turn out to be just impracticable. If I have like only 0,000000096 ways of doing something, I have a locked box, with the key to the lock being locked inside the box. No point in bothering about it. When I settle for 10 megawatt hours successfully generated from renewable sources, against 19 megawatt hours coming from them fossil fuels, the situation changes. I have ((1029-10)/(29-10)!) = 82,20635247, or rather between 82 and 83, although closer to 82 ways of achieving this particular result. The cumulative probability of 10 successes, which I can score in those 82,20635247 ways, is equal to ((1029-10)/(29-10)!)*0,125210*0,874829-10 =  0,0000013. Looks a bit like the probability of meeting an alien civilisation whilst standing on my head at 5 a.m. in Lisbon, but mind you, this is just one patch of probability, and I have more than 82 ways of hitting it. My (0,1252 + 0,8748)29 multiverse contains 29! = 8,84176E+30 such patches of probability, some of them practicable, like 10 megawatt hours out of 29, others not quite, like 5 megawatt hours over 29. Although Thomas Bayes wanted to escape the de Moivre – Laplace world of great numbers, he didn’t truly manage to. As you can see, patches of probability on the sides of this multiverse, with very few successes or very few failures, seem blinking red, like the ‘Occupied’ sign on the door to restrooms. Only those kind of balanced ones, close to successes and failures scoring close to fifty-fifty, yield more than one way of hitting them. Close to the mean, man, you’re safe and feasible, but as you go away from the mean, you can become less than one, kind of.

Thus, if I want to use the original Bayesian method in my thinking about the transition towards renewable energies, it is better to consider those balanced cases, which I can express in the form of just a few successes and a few failures. As tail events enter into my scope of research, so when I am really honest about it, I have to settle for the classical approach based on the mean, expected values, de Moivre – Laplace way. I can change my optic to use the Bayesian method more efficiently, though. I consider 5 local projects, in 5 different towns, and I want to assess the odds of at least 3 of them succeeding. I create my multiverse of probabilities as (0,1252 + 0,8748)3+2=5, which has the advantage of containing just 5! = 120 distinct patches of probability. Kind of more affordable. Among those 120 patches of probability, my target, namely 3 successful local projects out of 5 initiated, amounts to (32/2!) = 4,5 ways of doing it (so between 4 and 5), and all those alternative ways yield a compound probability of (32/2!)*0,12523*0,87472 = 0,006758387. Definitely easier to wrap my mind around it.

I said, at the beginning of the today’s update, that I am using Thomas Bayes’ theory as a filter for my findings, just to check my logic. Now, I see that the results of my quantitative tests, those presented in previous updates, should be transformed into simple probabilities, those a’s and b’s to put inside (a + b) when doing (a + b)p+q. My preferences as for successes and failures should be kept simple and realistic, better below 10.

[1] Mr. Bayes, and Mr Price. “An essay towards solving a problem in the doctrine of chances. by the late rev. mr. bayes, frs communicated by mr. price, in a letter to john canton, amfrs.” Philosophical Transactions (1683-1775) (1763): 370-418

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