Insufficient Reason Principle PDF

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Insufficient Reason Principle Before starting to talk about the Principle of Insufficient Reason, it is better to clear up which are the main decisions a DM can come across. We have decisions where we know the consequences of every alernatives with sureness, and this are called decisions under certainty, then we may bump into situations under risk where the probabilities of the possible outcomes are known, and finally there are certain decisions where we are facing a problem with different alternatives but we don’t know how to choose the best way, better we are unable to assign probabilities to the states. To solve up a decision under ignorance, there are different principle we can apply, that help us to better implement our decision problem, in this case the Principle of Insufficient Reason. What about those events where we are unaware of what will going to occur? To simply answer to this question, let’s bring in a brief history related to this particular principle. It was first stated by the mathematician Jacob Bernoulli (1654–1705), better known for his contribution on probability calculus, and another important theorem as the ‘Law of large numbers’. An important contribution was given by Pierre Simon Laplace through the study of mathematics, statistics and physics, declaring this principle as ‘the theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.’ (Example underneath). This principle can be included in the epistemic probabilities, where different interpretations can co-exist together. One of the best definition of the principle was introduced by Keynes in 1921, arguing that ‘if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability’. Practically we know which are the possible states, but we are unable to know the credibility of each one, we may believe that everyone of them might occur with the same probability, so we try to assign them an equal probability. So it appears that all things must have an explanation! If the DM tries to assign equal probability, it will turn out to be that all the states can occur equally! In our life we are always looking for explanations, so this principle is supposed to clarify our life decisions in lack of information! It is proper to mention Baruch Spinoza, a famous philosopher who tried to explain some issues of the world dealing with ethics and politics. According to him the the existence of things must be explained, we all have a goal in our decisions’ life. Another important contribution was given in the eighteenth century by the philosopher David Hume, arguing that there must be a reason to ascribe for everything, this principle was also used to argue the existence of God (his hypothesis should be granted equal probability against its alternatives,i.e. other religious worship) a quite alike

description of this principle of indifference (also named like this), was also attributed to Gottfried Leibniz, a philosopher who had a strong position in mathematics and physics. It was the economist Keynes who renamed the PSR (Principle of Sufficient Reason) into the PIS (Principle of Insufficient Reason), where ‘every fact has a sufficient reason for why it is the way it is and not otherwise’. Trying not to deal too much with the philosophical aspect, we can assume the Principle of insufficient Reason turns a decision problem from uncertainty/ignorance to an under risk. How? Trying to detect the possible probabilities and values through the assignment of an equal percentage to all the states. Basically the states represents the number n of alternative hypotheses (symmetric states) iff none of them is more likely to occur than another. In terms of mathematics this is the formula:

As we can see the first one (ai) is better than the other (aj) if and only if the average of the first one is greater than the other one. Example 

I’m about to go to Stockholm, but I’m not sure of what will be the weather like over there.

The possibilities are “rain”, “snow”, and “no precipitation”. Is it better to take the umbrella, a k -way or none of them?

UMBRELLA K-WAY NONE

RAIN (1/3) 20 5 -20

SNOW (1/3) 5 20 -20

NO PREP (1/3) -20 5 20

1/3 (20) + 1/3 (5) + 1/3 (-20) = 1,67 1/3 (5) + 1/3 (20) + 1/3 (5) = 10 1/3 (-20) + 1/3 (-20) + 1/3 (20) = - 6,67 The probability drops to 1/3. (3 states = rain, snow, no precipitation), so it is better to take a k-way just to be safe.

Another classic example can includes the flipping coins, that can land on heads or tails, so there’s no reason we can assume a favourable side to be occured, that’s why we assign equal propability, that is ½. Reference Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search, William A. Dembski. Keynes, J. M. "Fundamental Ideas.", A Treatise on Probability. Macmillan, 1921. Peterson Martin, An Introduction to Decision Theory, 2009.

Webliography http://patrick.maher1.net/318/lectures/keynes2.pdf https://en.wikipedia.org/wiki/Principle_of_sufficient_reason http://rationallyspeaking.blogspot.it/2005/10/principle-of-insufficient-reason.html

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