Formal Methods II

On the other side of that, the frequencies aren't definitive.

EG
You're flipping a coin. It's got a bias. You want to find out what that bias is. So you toss the coin 10 times. Even if you got 10 tails, it wouldn't garuntee the that that is the bias. There is no garuntee bw/ the frequencies and the actual probability.

waffle waffle.

We believe things with varying degrees of strength.

Frequentist: P(h) is (somehow) related to the
frequency with which things like h turn out to be
true.

Subjective: P(h) expresses the agent’s strength
of belief in h.

Both interpretations are quite reasonable, and it's not really right to say one is right over the other.

There is all sorts of uncertainty. By moot point we're going to not address it - it's open (see Cognitive Psychology).


Venn’s identity “probability = frequency” clearly fails
in small samples.

Frequencies vary all over in small samples
But in large samples they vary hardly at all

In small samples you can't really be told very much!

As you build up 10,000 of these small samples, the law of large numbers will make the combination of the results seem like the actual probability.

Richard von Mises’ Theory
Finite set of outcomes

A place selection is essentially just a bit string of 01101001010101010... that is infinitely large.
It turns out it's inadequate because if we choose the outcomes...


So what's the point?
Why as CS people are we studying probability. Algorithms are deterministic!
Uncertainty is central to the human condition. And CS are intended to the service of humanity. CS to help us cope with uncertainty. Bring on probabilistic reasoning. The application side is one reason to look at probability.

Google and Amazon are ranked. If they're not, they're useless! A lot of problems cannot be solved definitely, or deterministically. You can do probabilistic, probability samples. in order to get probabilistic solutions.

It's about sorting and managing information. Low probablity, you have a lot of information. If the probability is high, you get no new information.