Good. So any questions about why we're doing probabilities?

Why are we doing probabilities?

Okay, anybody with an answer?

Yes, but why are we interested in that?

Okay, why would we want to do that?

There's one canonical answer in this course to every question.

Ultimately, we want to model AI.

We've kind of done that for laughably simple environments with very simple agents last semester.

And now we want to build, we want to make AI possible in uncertain worlds, environments.

And that's why we need to model uncertain, incomplete knowledge.

Okay, so you can always answer, we want to do AI.

But of course, my next question will then be, and why does that help what we've done?

But the idea here really is we need to model incomplete knowledge about the state of the world, about our actions, even about utilities.

All of those are uncertain. And the mechanism for doing that is probabilities, I claim.

And you probably believe, because otherwise you wouldn't have been taught about probabilities in, what is it, Math 3 or something like that.

Okay, and we're going to start out easy.

Okay, so a probability theory is, and that's important, it's a language for talking about possible worlds.

And together with an inference method for quantifying the degree of belief in such assertions.

It's quite a mouthful.

What we're doing, this says, is we're making a theory about how different probabilities of things I can talk about in my representation language

in which I have, in which I make statements about my world.

And then quantify the degree of belief.

Okay, so there's a couple of ingredients here, which is we have a formal language for talking about the state of the world.

And a way of making inference from old states to new states.

And that's something we've seen before.

We've called it logic.

And it's really just like logic, probability theory, except that instead of having two truth values, true and false,

we have a quantitative theory of the degree of truth.

Everything else, the intuitions are the same.

Very important to understand.

So now the question is, what's our language? What do we use to talk about the world?

And what is the basics of our model of possible worlds?

Well, one of the things that you should become clear about is that possible worlds are mutually exclusive.

The reality is always only one of the possible worlds.

And importantly, one of the possible worlds must be the actual world.

And of course, the possible world must be exhaustive, we say.

We're covering the actual world.

So these two things, mutual exclusivity and exhaustivity, are actually the things that we should keep in mind when we think about possible worlds.

That's what we want about them.

So we have two indistinguishable dice, which means not one red, one green, which you, well, no, one blue,

which you can distinguish even if you're red-blind, green-blind.

Then we have 36 possible worlds, which is the full set of combinations of these.

So these outcomes, we're going to restrict ourselves,

and that's going to make our life mathematically much easier than if you hear a probability theory course in math.

We're going to restrict ourselves to situations where the outcome space is actually discrete and countable,

either finite or only has countably many things.

Dice are a good example of that. We have six possible outcomes.

Or if you have a coin toss, we restrict our outcome space to heads or tails.

Not, which a better model would have been, a different model is where does the coin actually land, which is kind of something in three space, which would be continuous.

Those kind of things we're not going to model, even though they are interesting, but you need a lot more math to that.