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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
D’Angelo Alfredo 74480