data/en/crtw_101.txt

One very interesting aspect of statistics is the use of probability in
explaining scientific methodology. In particular, the _Bayesian approach_
provides a powerful framework to explain confirmation and many other aspects
of scientific reasoning.

On the Bayesian approach, probability is used to measure _degrees of belief_.
Belief does not come in an all-or-nothing manner. If it has been raining
heavily the past week, and the clouds have not cleared, you might believe it
is going to rain today as well. But you might not be certain that your belief
is true, as it is possible that today turns out to be a sunny day. Still, you
might decide to bring an umbrealla when you leave home, since you think it is
more likely to rain than not. The Bayesian framework is a theory about how we
should adjust our degrees of belief in a rational manner. In this theory, the
probability of a statement, P(S), indicates the _degree of belief_ an agent
has in the truth of the statement S. If you are certain that S is true, then
P(S)=1. If you are certain that it is false, then P(S)=0. If you think S is
just as likely to be false as it is to be true, then P(S)=0.5.

One important aspect of the theory is that rational degrees of belief should
obey the laws of probability theory. For example, one law of probability is
that P(S) = 1 - P(not-S). In other words, if you are absolutely certain that S
is true, then P(S) should be 1 and P(not-S)=0. It can be shown that if your
system of degree of belief deviates from the laws of probability, and you are
willing bet according to your beliefs, then you will be willing to enter into
bets where you will lose money no matter what.

What is interesting, in the present context, is that

Here, P(H) measures your degree of belief in a hypothesis when you do not know
the evidence E, and the conditional probability P(H|E) measures your degree of
belief in H when E is known. We might then adopt these definitions :

  1. E _confirms_ or supports H when **P(H|E) > P(H)**.
  2. E _disconfirms_ H when **P(H|E) < P(H)**.
  3. 3\. E is _neutral_ with respect to H when **P(H|E) = P(H)**.

As an illustration, consider definition #1. Suppose you are asked whether Mary
is married or not. Not knowing her very well, you don't really know. So if H
is the statement "Mary is married", then P(H) is around 0.5. Now suppose you
observe that she has got kids and has a ring on her finger, and living with a
man. This would provide evidence supporting H, even though it does not prove
that H is true. The evidence increases your confidence in H, so indeed P(H|E)
> P(H). On the other hand, knowing that Mary likes ice-cream probably does not
make a difference to your degree of belief in H. So P(H|E) is just the same as
P(H), as in definition #3.

One possible measure of the amount of confirmation is the value of
P(H|E)-P(H). The higher the value, the bigger the confirmation. The famous
Bayes theorem says :

P(H|E) = P(E|H)xP(H)/P(E)

So, using Bayes theorem, the amount of confirmation of hypothesis H by
evidence E

= P(H|E) - P(H)  
= P(E|H) x P(H)/P(E) - P(H)  
= P(H) { P(E|H) / P(E) - 1 }

Notice that all else being equal, the degree of confirmation increases when
P(E) decreases. In other words, if the evidence is rather unlikely to happen,
this provides a higher amount of confirmation. This accords with the intuition
that surprising predictions provide more confirmation than commonplace
predictions. So this intuition can actually be justified within the Bayesian
framework. Bayesianism is the project of trying to make sense of scientific
reasoning and confirmation using the Bayesian framework. This approach holds a
lot of promise, but this is not to say that it is uncontroversial.

If you want to read more on this topic, here are some advanced readings:

  * Howson and Urbach. _Scientific Reasoning : The Bayesian Approach_. 
  * Entry on "Bayesian Epistemology" in _The Stanford Encyclopedia of Philosophy_.

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