AI Assistant
Transcript
00:04:370Paolo Guiotto: Okay, good morning.
00:11:590Paolo Guiotto: On the chest of this… this cat.
00:15:240Paolo Guiotto: Okay, today we start the second part.
00:20:980Paolo Guiotto: Which is, on probability.
00:26:630Paolo Guiotto: There are several connections between the first part, and first of all.
00:34:620Paolo Guiotto: The, the general definition of what… what is it in modern probability, a probability space.
00:42:620Paolo Guiotto: Actually, probability theory, historically, was,
00:49:280Paolo Guiotto: But the basic, let's say the basic, the main, let's say the core business of probability is, making predictions, under uncertainty.
01:04:540Paolo Guiotto: So, if you think the example of the coin tossing.
01:09:610Paolo Guiotto: You could use the law of physics, in principle, to describe the movement of the coin when you… when you flip the coin.
01:19:740Paolo Guiotto: and predict whether it falls on one side or on the other. But as you may imagine, this is extremely complex, because the physics of this system is complicated.
01:33:930Paolo Guiotto: And the equations are mathematically complicate. So…
01:39:650Paolo Guiotto: Probability in certain sense simplifies this problem, because it says you have 50% probability that it falls ahead, and the other 50% is tail.
01:52:570Paolo Guiotto: So, you lose the prediction, the certain prediction. You cannot say that it will be. Certainly, if this is the initial impulse deposition, you will have this outcome, or that outcome.
02:06:60Paolo Guiotto: You can say something which is a little bit less precise, much less precise, we may say, but in certain sense, in certain adult sense, it's much easier.
02:17:320Paolo Guiotto: And many phenomena of the real world are like that. If you think to forecast, to weather predictions, weather forecasts, you have a system which is the physics of the atmosphere.
02:33:230Paolo Guiotto: Which is, let's say, in a certain sense, well understood from physicists.
02:38:690Paolo Guiotto: But the description of the dynamics of the atmosphere is complicated. It's too complex, and even we don't have
02:47:520Paolo Guiotto: actually a complete information. For example, we should need to have… if we want to do a prediction, typically the evolution of the atmosphere, the speed, the pressure, and so on.
03:00:90Paolo Guiotto: Fulfilled a certain differential equations, certain time-dependent differential equations, so…
03:07:180Paolo Guiotto: Typically, the problem is, I know the state of the system today. I want to know what is the system, or the state of the system in 6 hours, in 12 hours, tomorrow, and so on.
03:19:330Paolo Guiotto: So it's typically a Cauchy problem, no? You give the initial condition, you want to determine the future of the atmosphere.
03:27:560Paolo Guiotto: that you don't have the exact initial condition, because you would need the pressure at each single point of the atmosphere. We can measure pressure only a few points on the surface.
03:39:380Paolo Guiotto: And do some kind of deduction about the pressure.
03:44:750Paolo Guiotto: On, high atmosphere, and we cannot measure speed, the air, the speed of the airflow at each point in certain moments, so we do not have this initial information.
03:59:960Paolo Guiotto: And moreover, the equation… the equation that described the evolution of a fluid are complicated, nonlinear equations, so
04:08:320Paolo Guiotto: We don't even have a mathematics to solve exactly those equations. So it means that we have to solve approximately, and then we introduce errors, and this, together with the lack of information, initial information, makes the prediction
04:25:420Paolo Guiotto: basically degrading as time goes by. So, maybe a few hours prediction is good, and then it becomes more and more, less precise.
04:39:870Paolo Guiotto: On the other hand, doing predictions like the probability that tomorrow is going to rain here and there is this or that, it's a less certain probability. If you look at the weather forecast, normally they predict something, but it hardly happens exactly, or not with the exact precision. Maybe it is raining here and not raining
05:03:660Paolo Guiotto: just 10 kilometers away from this point. So, you see, it's a different kind of,
05:10:570Paolo Guiotto: of, of, information that you can get from a system, but it's, it's, it, it is easier, let's say. It is, better feasible. So that's why, in many, many systems,
05:29:600Paolo Guiotto: to accept the idea that we do not have an exact prediction, but a probabilistic prediction of what is going to do. The system makes sense.
05:43:520Paolo Guiotto: And, moreover, Whatever is the system you imagine, a complex system depends on so many factors that,
05:54:670Paolo Guiotto: We always do simplifications, approximations of the real world.
06:00:270Paolo Guiotto: So, the equation of the atmosphere are never the exact physics equation, because we…
06:06:510Paolo Guiotto: perhaps we do not consider a certain number of other phenomena that could affect the atmosphere. We should consider everything that could affect the atmosphere, but that's practically impossible.
06:21:620Paolo Guiotto: So, it's not just, accepting the idea that we do less.
06:29:140Paolo Guiotto: Because we are not able to do more. But the idea is that complexity matches well with the uncertainty, and therefore with the need of having tools to do such kind of predictions.
06:45:450Paolo Guiotto: Now, probability remained until the beginning of the last century.
06:52:330Paolo Guiotto: Basically, a very, a very,
06:58:40Paolo Guiotto: a very particular, not even a real theory, a very particular field that arose a few centuries ago.
07:08:340Paolo Guiotto: to do predictions, basically, on, on, on games. So, to predict what is the probability that a certain, bet, win, and so on.
07:23:820Paolo Guiotto: Now, at the beginning of the last century, things changed a lot, because, first of all.
07:34:250Paolo Guiotto: Mathematician tried to…
07:37:620Paolo Guiotto: build a structure, for the so-called Brownian motion, which is a very important, concept in probability, and mostly in stochastics. If you will do stochastics.
07:57:390Paolo Guiotto: For finance, but also for modeling, physics systems.
08:03:410Paolo Guiotto: you will see that everything is based on this, on the Brownian motion.
08:08:540Paolo Guiotto: And it was just the study of the Brownian motion, the need to have a probabilistic structure for the Brownian motion that made mathematicians to think anew to probability.
08:21:350Paolo Guiotto: And this was not incidental at the beginning of last century, because at the beginning of last century, Lebeck introduced the measure theory we have seen at the beginning of this course, the concept of measure.
08:35:10Paolo Guiotto: And at the end, in the 30s of last century, Kolmogorov understood that probability should be a measure, a particular type of measure, that is the right set up to
08:48:410Paolo Guiotto: put probability theory, where we set up probability theory. But in the 30 years from the beginning of 20th century and 1932, which is the year when Kolmogorov
09:02:640Paolo Guiotto: introduced the modern probability theory. Many, many attempts were made of… where informally mathematicians were already using these concepts.
09:15:780Paolo Guiotto: And this was because they were trying to introduce a probability structure for the Brownian motion. Why for the Brownian motion is so important? Because in Brownian motion, you introduce a probability on a set where the elements are
09:33:350Paolo Guiotto: Paths that represented the evolution, the trajectories of these particles.
09:38:810Paolo Guiotto: So you want to assign a probability to a set made of trajectories. Now.
09:45:210Paolo Guiotto: trajectories means functions, so on a set which has an actual of an infinite dimensional set, so it's a very complex structure. So, that's where the need of a solid ground for probability was, was,
10:02:290Paolo Guiotto: came out. So, now, formally.
10:07:690Paolo Guiotto: So let's start, by… with the definition, and… Sorry.
10:14:430Paolo Guiotto: And a few examples.
10:17:610Paolo Guiotto: So, probability… Probability.
10:28:690Paolo Guiotto: space… So we started directly from the definition.
10:36:350Paolo Guiotto: which is in… which is accepted in modern probability. So, a probability space…
10:52:900Paolo Guiotto: is, A triplet, where we have Is that…
10:59:530Paolo Guiotto: a family of subsets of the set, and a probability measure.
11:06:300Paolo Guiotto: So, where?
11:11:150Paolo Guiotto: Omega is a set, In language of probability, it is called the sample space.
11:32:730Paolo Guiotto: The family F is a sigma algebra.
11:41:130Paolo Guiotto: contained in the subset of Omega.
11:45:340Paolo Guiotto: And, the elements of this sigma algebra are called events.
11:52:720Paolo Guiotto: D.
11:53:820Paolo Guiotto: elements…
11:58:770Paolo Guiotto: off… F.
12:05:650Paolo Guiotto: events.
12:09:920Paolo Guiotto: And P, finally, is a measure.
12:19:100Paolo Guiotto: on… Often.
12:23:560Paolo Guiotto: With the extra condition with respect to an usual measure, such that the probability of the full space is
12:35:80Paolo Guiotto: Wang.
12:36:270Paolo Guiotto: Now, in general, if E is an event.
12:42:70Paolo Guiotto: An element of the family of measurable sets.
12:46:720Paolo Guiotto: P of P is called…
12:55:810Paolo Guiotto: probability… of you.
13:00:170Paolo Guiotto: Now, why am I underlying these names? Because in probability.
13:06:980Paolo Guiotto: Probability arises as, initially as,
13:11:50Paolo Guiotto: an applied, say, branch of mathematics. You… you… initially, mathematicians, were interested mostly in certain, say, probability spaces, and
13:24:990Paolo Guiotto: The emphasis on what is the space, what are the events, what is the probability,
13:32:200Paolo Guiotto: It's important because it comes… normally, you introduce… the space itself is part of the modeling.
13:40:60Paolo Guiotto: Now, we have been used in the first part to work mainly with the RN measure, and, well, of course, there are not only these kind of measures that we could consider, there are many other important measures, even in RN and different from the Lebec measure.
13:59:220Paolo Guiotto: that can be introduced. For example, if you think about measuring the area of a surface in space, you cannot use the Lebag measure, because
14:10:420Paolo Guiotto: If you have a surface, which is a dimension 2 object, so imagine a plane, no, in space, and you use the measure of R3, that measure computes volumes.
14:21:250Paolo Guiotto: So this will yield value zero on surfaces. So it's completely useless, the three-dimensional measure for that set.
14:29:790Paolo Guiotto: So, building a surface measure that provides the area of objects, like surfaces in artery, is a different problem. This is a geometric measure, a surface measure.
14:47:290Paolo Guiotto: But that's just an example.
14:49:220Paolo Guiotto: Well, improbability, as we will see, as you will see by doing modeling, the space itself, the probability is an important part of the modeling, because you want to
15:03:130Paolo Guiotto: To build a system with certain features, and this will determine a probabilistic structure.
15:10:530Paolo Guiotto: So that's why these names, and in general, the interpretation of what we are doing is important here. And in fact, as you will see, this will also yield certain properties which are, let's say, relatively uncommon with ordinary measures. We will see later.
15:29:30Paolo Guiotto: Okay, let's introduce another terminology. We say that an event E
15:40:480Paolo Guiotto: is, certain.
15:46:990Paolo Guiotto: If the probability of E is 1,
15:52:940Paolo Guiotto: While we say that it is impossible.
15:59:470Paolo Guiotto: if, the probability of E is 0.
16:06:280Paolo Guiotto: Now, of course, impossible, one could think it's an event where there are no elements, so there is no possibility at all, no? But here, for probabilistic jargon, the impossible events are those with probability equal to zero.
16:23:950Paolo Guiotto: Okay, now, the scope of today is mainly to take confidence with this, which is not at all a new definition, because we already know what is a measure. The unique novelty is this fact here, okay?
16:40:240Paolo Guiotto: So, let's see some… Standard examples.
16:45:220Paolo Guiotto: And, a few generalized considerations, and finally, I would like, if we arrive there, to show you
16:56:690Paolo Guiotto: Exactly how this, modeling aspect of probability
17:06:810Paolo Guiotto: can be hard in a very simplified version of a Brownian motion model. We could even start with the Brownian motion, but it would be technically too hard to do right now.
17:21:390Paolo Guiotto: Okay, so let's see some examples, just to have in mind. Also, this is important, as usual, because when you have examples, you can build examples, depending on what you need to do.
17:35:890Paolo Guiotto: So, first of all, we have immediately a large class of probabilities if we start from the measure we know better, which is the Lebang measure. So take omega equal RD,
17:54:670Paolo Guiotto: equipped with the sigma algebra of Lebec measurable sets.
18:00:00Paolo Guiotto: and take a function f, function of X, which is positive.
18:07:870Paolo Guiotto: and such that integral F in L1, R.
18:15:460Paolo Guiotto: And such that the integral on R, RB
18:20:360Paolo Guiotto: RD of the function F be equal to 1.
18:27:640Paolo Guiotto: So you define this probability, P of E, by definition, is the integral on E of the function F.
18:38:570Paolo Guiotto: Now, this is a well-defined probability measure.
18:44:550Paolo Guiotto: So, omega, MFP… is it.
18:53:40Paolo Guiotto: probability… space.
18:58:910Paolo Guiotto: Now, the fact that it is a probability space is somehow non-trivial. It demands a little argument, which is based, in fact, at the end on the monotone convergence theorem, because if you want, let's check this.
19:23:310Paolo Guiotto: Probably, we already,
19:26:330Paolo Guiotto: We have already seen part of this, because we already have seen that you can take an integral positive function and defining
19:35:340Paolo Guiotto: integrals on subsets of that function, you get a function of a set.
19:42:160Paolo Guiotto: That is positive numbers, and that's a measure. However, let's redo this check. So, first of all, what we have to check is that P is well-defined, and it is a probability measure.
19:56:620Paolo Guiotto: B.
19:57:870Paolo Guiotto: Jeez.
19:59:400Paolo Guiotto: Well… defined, So… B is a measure.
20:09:290Paolo Guiotto: And three, it is a probability in the sense that P of The full space, omega.
20:15:880Paolo Guiotto: Which is, in this case, FD is equal to 1.
20:20:390Paolo Guiotto: Well, this is well-defined, because P over E
20:24:660Paolo Guiotto: is the integral on E of F. F is supposed to be L1 in the full space, RD, so this is the integral on RD.
20:36:180Paolo Guiotto: of F times the indicator of the set E, and of course, if F is in L1, also this one belongs to L1, because the integral honor B of the modulus of F indicator E
20:52:980Paolo Guiotto: Well, first of all, this is, the function f is positive, so everything here is positive, so this is the integral, on RD of F.
21:04:40Paolo Guiotto: indicator, and since the indicator is less than 1, this is less than the integral on RD of F, which is fine.
21:12:530Paolo Guiotto: So… This P is a well-defined.
21:17:800Paolo Guiotto: So, P of E is well-defined and positive.
21:25:110Paolo Guiotto: for every E.
21:27:550Paolo Guiotto: measurable set.
21:30:140Paolo Guiotto: Which is, our class F.
21:33:130Paolo Guiotto: It is a measure, so, P of,
21:39:20Paolo Guiotto: empty is equal to the integral of nothing on the empty set of F, and we know that when we integrate on a measure 0 set.
21:49:270Paolo Guiotto: We have integral equals to 0.
21:52:910Paolo Guiotto: and the P of a disjoint union of sets, yen.
21:59:970Paolo Guiotto: This is the same of doing the integral on the disjoint union of the EN of, now.
22:09:560Paolo Guiotto: If the union is finite, we can say that the integral on the union… see, if the union is finite and disjoint is the sum of the integrals. Since this is not a finite union, we have to do a little work here, so we transform this integral
22:28:280Paolo Guiotto: of F times the indicator of the union.
22:32:350Paolo Guiotto: on Avdiv.
22:34:230Paolo Guiotto: Now, we notice that this is the same of the sum of the indicators, because the union is made of these joint sets. So, in fact, only one, or at most one of these indicators can be one, all the others must be 0.
22:49:670Paolo Guiotto: No? You cannot have even 2 equal to 1, because to be 2 equal to 1, it means that there is a point that is in EN and in EM for N different from N, but they are disjoint. And so this transforms into the integral
23:05:280Paolo Guiotto: of the sum of F indicators of yen.
23:09:810Paolo Guiotto: Now, these are positive functions, and we have that because of monoton convergence.
23:16:950Paolo Guiotto: When we have an infinite sum of positive things, we can carry outside this infinite sum, so we get sum of n of integral on RD of F indicator EN. This is because of monotone
23:34:450Paolo Guiotto: convergence.
23:36:740Paolo Guiotto: And here we read back that this is the sum of these P of N.
23:43:550Paolo Guiotto: So we proved the data.
23:45:600Paolo Guiotto: This fee is, comfortably additive.
23:51:600Paolo Guiotto: And number 3, the probability of the space omega, which is the probability of the set RD, V, R,
24:00:800Paolo Guiotto: is the integral on RD of F, but since we suppose that this is 1, we get that this will be 1.
24:11:560Paolo Guiotto: So, immediately, you have many, many examples, because if you take any function f, which is positive, with integral equal to 1, you have automatically well-associated probability measure through this formula.
24:30:100Paolo Guiotto: A second example that we already seen as an example of measure is the example of Dirac delta. So here, omega is an arbitrary set, F is parts of omega.
24:48:20Paolo Guiotto: and delta, say, X0,
24:52:830Paolo Guiotto: Well, we usually call the elements of omega with the letter LITEN Omega, so let's say omega 0.
24:59:720Paolo Guiotto: With omega-0, an element of omega.
25:03:100Paolo Guiotto: is this quantity, delta omega zero of E is 1 or 0, depending on omega 0 is in E or not. So, I say that this is 1 when omega 0 is in E, so when E contains his omega 0.
25:22:100Paolo Guiotto: And 0 when omega 0 is not in here.
25:26:130Paolo Guiotto: Also, this one is a trivial example of probability space.
25:33:460Paolo Guiotto: A third example is the example of a discrete
25:43:920Paolo Guiotto: probability… space.
25:48:650Paolo Guiotto: Let's say that this is the space of the…
25:52:30Paolo Guiotto: Ancient probability, because we may say that, all probability of games, card games, and so on, dice rolling, etc, coin tossing, can be described through this kind of,
26:08:40Paolo Guiotto: structure. So, we have a set omega, which is finite, or at most, if it is infinite, countable, okay?
26:16:780Paolo Guiotto: Find it.
26:20:460Paolo Guiotto: or countable.
26:26:210Paolo Guiotto: So, we take this second case, because let's say that it's a… it contains the other one, so let's say that it is a list of omega, indexed by a natural number.
26:43:560Paolo Guiotto: As a family, we take, the family of four possible subsets of Omega.
26:52:160Paolo Guiotto: And as probability, we assume that
26:57:170Paolo Guiotto: We have, a… a sequence, PN, of numbers.
27:05:460Paolo Guiotto: Index it again by integers.
27:08:550Paolo Guiotto: Which are positive.
27:17:30Paolo Guiotto: And such that the sum of the PN
27:20:500Paolo Guiotto: is equal to 1. The sound for n going from 0 to infinity of this P is equal to 1. As you can see, this sounds like the example is similar, same philosophy, like this one, no? We have a set, which is now a countable set.
27:37:500Paolo Guiotto: a sigma algebra, which is now part of the set itself, and then the function F is now replaced by this sequence PN, that must be positive, and the integral, the sum of all the PN is 1. So you see there is a certain
27:54:320Paolo Guiotto: Connection between these two.
27:57:900Paolo Guiotto: we define the probability of the set E,
28:01:730Paolo Guiotto: counting the PN, we may say, with sum.
28:07:300Paolo Guiotto: the PN for DN, such that… Omega N belongs to E.
28:16:190Paolo Guiotto: So, in other words, that number, PN, is the probability
28:22:890Paolo Guiotto: of the set of the event made by the single omega n. So it's, in other words, the interpretation of PN is that it is the probability that the little omega n happens.
28:39:230Paolo Guiotto: It is not necessarily true that singletones are events, okay? This is not necessarily true, but if they are, this means that this is… in this case, they are events, and this little PN are the probabilities. We can also write this
28:56:860Paolo Guiotto: sum, PN, then we put an indicator of, set E, Omega, no?
29:08:570Paolo Guiotto: Or perhaps better, it is the same sump, and we can do.
29:16:670Paolo Guiotto: Yeah, the… it's the same delta omega n of the set E.
29:24:190Paolo Guiotto: We can… it's just a restyling. Now, this is omega, with this F, with this P, is, probability…
29:38:200Paolo Guiotto: space, huh?
29:41:680Paolo Guiotto: So, also in this case, you should verify that, first of all, that POV is well-defined.
29:50:340Paolo Guiotto: No? And this is, easy because, P of E…
29:56:630Paolo Guiotto: Is, this sum is the sum of… it's similar to the…
30:01:220Paolo Guiotto: To the check we have seen above for the,
30:07:920Paolo Guiotto: for the… this probability space based on the Lebec measure, no? It's basically the same check, no? You have that this is well-defined, because this sum is less or equal than the sum of all the PN, since this is a number which is 0 or 1.
30:27:20Paolo Guiotto: It can be at most 1, so if we replace all this with 1, we have that this is no larger than this sum, which is equal to 1, so in particular, just find it.
30:39:210Paolo Guiotto: And, of course, we can also see that here immediately that P of omega is the sum PN delta omega n of omega. But this quantity in this case will be always equal to 1, because omega is made
30:57:520Paolo Guiotto: by the omega N, okay? So this is exactly the sum of the PN, and it is equal to 1. So we also check that this
31:06:770Paolo Guiotto: Probability of omega is 1.
31:08:980Paolo Guiotto: Number two, the probability of empty.
31:12:930Paolo Guiotto: It's trivially equal to zero, because if we put the empty set
31:17:340Paolo Guiotto: There is no omega n into the empty set, so that sum is a sum of zeros.
31:22:690Paolo Guiotto: And the countable additivity is similar with,
31:28:300Paolo Guiotto: a little, let's say, difficulty, technical difficulty, which is this one, no? You know that, delta omega n
31:38:810Paolo Guiotto: We said above, easily, these are…
31:42:350Paolo Guiotto: the Dirac delta, they are measures, probabilities. So, I can say that this is… sorry, let's use another index.
31:51:170Paolo Guiotto: Because this is the index for the union, which is different from the summation index.
31:57:170Paolo Guiotto: So, for the delta omega n, I can say that delta omega n of the joint union will be the sum of a K of delta omega n of EK.
32:10:40Paolo Guiotto: So at the end, of course, what I want to do is to put outside the sum of… on… on K. So I have here a sort of double sum. Some PN, sum of a K delta, omega n.
32:24:490Paolo Guiotto: decay. So, this is the step where we…
32:28:550Paolo Guiotto: are here with the integral, integral of the sum of F times the indicator. It's exactly the same thing here.
32:38:980Paolo Guiotto: And so, you, you will have that this goes out, In a similar way.
32:46:160Paolo Guiotto: And we get sum of NPN…
32:49:490Paolo Guiotto: delta, omega N EK, but this one is, by definition, the probability of EK.
33:02:820Paolo Guiotto: Excellent.
33:05:850Paolo Guiotto: So, I don't, you can read by yourself, because we are not going, in any case, to do…
33:13:230Paolo Guiotto: to work with the finite, sets, or in general, with the discrete probability space, which is this case. So you can read by yourself the examples, the specific examples. For example, I just mentioned one, for example.
33:33:720Paolo Guiotto: for… the… Let's say, one shot coin… tossing.
33:47:650Paolo Guiotto: Omega will be made by just two states. We can say head or tail.
33:56:750Paolo Guiotto: if the, you know, there are two states, so this is omega 1, this is omega 2,
34:04:850Paolo Guiotto: The numbers I need are two numbers, P1, P2, which are the probabilities of these two events.
34:16:00Paolo Guiotto: Now.
34:16:960Paolo Guiotto: Here, depends on, so, probability of, head, or probability of tail. It depends on what is the coin we have. If we have a standard coin.
34:29:750Paolo Guiotto: Which is a perfect coin. Normally, we have 50% of possibilities to have head and 50% to have tails, so the probability will be 1 half
34:41:230Paolo Guiotto: 1 half.
34:43:610Paolo Guiotto: If you want to model a different
34:49:230Paolo Guiotto: point, you could have here a probability P, and on the other side, you will have a probability 1 minus P.
34:56:449Paolo Guiotto: For example.
35:00:630Paolo Guiotto: Okay.
35:03:890Paolo Guiotto: Now, we will, see,
35:10:210Paolo Guiotto: in a few minutes, a more complex example of probability space, which is not a discrete probability space, and which is not even, say, a LeBague
35:27:300Paolo Guiotto: a back-based space, so it's something different.
35:32:820Paolo Guiotto: But before we do that, let's just,
35:39:820Paolo Guiotto: see a few general, properties about, probability. So, if you… general…
35:53:750Paolo Guiotto: Now, what… what, is true for any measure is still true for a probability measure, okay? So that must be clear. So, all…
36:06:520Paolo Guiotto: properties…
36:12:280Paolo Guiotto: known.
36:15:40Paolo Guiotto: Four.
36:16:730Paolo Guiotto: measures.
36:23:00Paolo Guiotto: all the… probabilities.
36:37:630Paolo Guiotto: measures.
36:40:560Paolo Guiotto: So, for example, an interesting factor could be the following.
36:51:520Paolo Guiotto: well, let's say, let's list what is, no, for example, the probability… it is comfortably additive is also finely additive, is…
37:04:600Paolo Guiotto: Finally.
37:08:980Paolo Guiotto: additive.
37:13:120Paolo Guiotto: So the probability of a disjoint union of a finite disjoint union, E1 union, E2 union yen, is the sum of the probabilities of E1
37:25:870Paolo Guiotto: plus E2, plus… Jana?
37:37:230Paolo Guiotto: You remind that, measures are always continuous from below, so P is… 20 miles.
37:47:390Paolo Guiotto: from… Below.
37:52:10Paolo Guiotto: So whenever we have a sequence of events increasing, so EN is contained into EN plus 1,
38:02:190Paolo Guiotto: for every N, then we have that there exists the limit
38:08:520Paolo Guiotto: of, probabilities of the sets EN,
38:13:70Paolo Guiotto: And this is the probability of a set E, which is the union of all the yen.
38:21:00Paolo Guiotto: And this is the so-called unit side.
38:24:370Paolo Guiotto: but, interestingly, probability… air probability is also continuous from above.
38:32:640Paolo Guiotto: B is… Always.
38:37:880Paolo Guiotto: Continos, from… above, huh?
38:45:440Paolo Guiotto: And this is because of…
38:49:210Paolo Guiotto: You remind continuity from above is valid if, for example, we have a decreasing sequence of sets, and the measure of the first set is finite.
39:03:140Paolo Guiotto: then it is true that the limit of the measures is the measure of the limit, where in this case, the limit set is the intersection. So… and this is true for a probability, because in any case, whatever is the set, the probability is always bounded by 1, okay?
39:21:420Paolo Guiotto: So, we have this, for every sequence EN of events, which is decreasing, so now EN
39:32:810Paolo Guiotto: contains yen plus 1.
39:36:140Paolo Guiotto: For every n… Then… The limit of, probabilities of the yen.
39:46:540Paolo Guiotto: is the probability of the intersection of the…
39:57:140Paolo Guiotto: A little fact that it comes from finite additivity, which is useful to know, is this little formula that if you do the probability of the complementary, this is 1 minus the probability of the event.
40:13:620Paolo Guiotto: And this follows easily because the probability of E union, disjoint union because, of course, these two are disjoint, Saturn.
40:24:980Paolo Guiotto: Now, the union is the full space, omega, so this quantity must be equal to 1.
40:31:720Paolo Guiotto: At the same time, this is the sum of the two probabilities, so P of E and P of E complementary, and therefore you get the conclusion.
40:45:190Paolo Guiotto: Okay, so just to show something new, let's say, which is peculiar for probabilities.
40:59:40Paolo Guiotto: let's prove this, which is also, by the way, an important fact that, in some sense, we will use later. This proposition.
41:09:210Paolo Guiotto: So, let… Omega… F… B.
41:16:510Paolo Guiotto: be a probability.
41:20:620Paolo Guiotto: space.
41:22:100Paolo Guiotto: Well, it turns out that the countable additivity is equivalent to a continuity… we may… we said usually continuity at zero of the probability. What do we mean is the following. So, the following…
41:42:790Paolo Guiotto: properties.
41:48:800Paolo Guiotto: R… equivalent.
41:52:380Paolo Guiotto: So, number one is P… Well, let's say…
41:59:220Paolo Guiotto: Well, to be precise, if I say probability space, I'm actually assuming that P is a probability, so let's say that, let's write this in this way. Let's, omega is set, and F is a sigma algebra, omega B is set.
42:18:700Paolo Guiotto: F contained in, parts of omega.
42:24:580Paolo Guiotto: B, A, Sigma Algebra.
42:33:220Paolo Guiotto: Offset, so…
42:40:30Paolo Guiotto: B, well, I have… so in this case, sorry, let's write the statement, because in this case, I have to say what is B. So B is sigma algebra.
42:49:970Paolo Guiotto: off… Sets.
42:54:340Paolo Guiotto: and P… BF function defined on F with values in 0, 1.
43:18:500Paolo Guiotto: Oh, I don't need… I don't think… I'm thinking if I need to say that P of MT is 0,
43:24:910Paolo Guiotto: That we should…
43:27:730Paolo Guiotto: put in parentheses, we will see later if it is needed, P of Mt equals 0. So I'm just saying P is a function.
43:36:780Paolo Guiotto: defined on the sigma algebra, so as any measure, with values in 0, 1, and I don't… I'm not yet sure, because here the statement is… I'm reviewing the notes, so…
43:48:400Paolo Guiotto: I will do in the weekend, next week you will have notes available.
43:54:380Paolo Guiotto: But I'm not clear… let's see on the proof what happens. Now, the following, the following… properties…
44:13:330Paolo Guiotto: Equivalent.
44:17:270Paolo Guiotto: So, number one, B.
44:20:330Paolo Guiotto: is… Countably.
44:24:110Paolo Guiotto: Additive?
44:26:460Paolo Guiotto: So, if I have also… P of MT… well, let's add the… definitely this one.
44:32:710Paolo Guiotto: Such that P of MT is B0. So, together with that, this means that P,
44:46:70Paolo Guiotto: Should I need also P equal 1?
44:49:270Paolo Guiotto: No. Automatically, if you have,
44:52:720Paolo Guiotto: Yes, I don't need to put any… I'm thinking if I need also to put P of omega equals 1, no. But you have a P, which is a function defined on the sigma algebra, let's go here, with the P of F equals 0, okay? Now…
45:10:110Paolo Guiotto: If it is, countably additive, It is additive.
45:15:590Paolo Guiotto: So, if it is additive, in particular, this formula here falls, huh?
45:20:160Paolo Guiotto: the probability of the complementary is 1 minus the probability of the set, and if you know that probability of a set of empty set is 0, the empty set, you get probability of omega here equal 1 minus probability of empty set, so you automatically get probability of omega equal 1. So I don't need…
45:39:130Paolo Guiotto: To add this,
45:40:460Paolo Guiotto: And second, the second property is the following.
45:46:470Paolo Guiotto: is continuous.
45:52:400Paolo Guiotto: from… above.
45:57:460Paolo Guiotto: at empty set. What does it mean? That if you have a sequence of EN that goes down to empty set, so EN contains EN plus 1 for every N,
46:13:850Paolo Guiotto: And… The intersection of all the yen is empty.
46:20:530Paolo Guiotto: Then, the limit of the probabilities of the yen
46:28:90Paolo Guiotto: Is equal to the probability of empty sets, so 0.
46:33:860Paolo Guiotto: So the countable… the countable additivity is equivalent to this continuity property.
46:42:530Paolo Guiotto: Okay, let's see, the proof, which is, basically, a reworking of,
46:53:00Paolo Guiotto: what we already know. So, the implication 1 implies 2 is, is just a continuity from above that we have for any, probability, no? If, so 1 implies 2.
47:07:950Paolo Guiotto: if, one. Olds.
47:12:850Paolo Guiotto: then it means that P… is a probability.
47:20:120Paolo Guiotto: Measure.
47:22:400Paolo Guiotto: Because in this case, we would have that P of omega.
47:26:350Paolo Guiotto: would be equal to P of empty complementary, which is equal by additivity, so we have comfortable additivity, it's more than additivity. 1 minus P of empty, and P of empty was assumed to be 0, so we get 1.
47:43:50Paolo Guiotto: So it would be a measure.
47:45:690Paolo Guiotto: No? Because we would have a function defined on a sigma algebra with values in 0, 1, such that P of MT is 0. Second, P is count on the additive, so now it becomes a measure, and moreover, P of omega is equal to 1, since it means that this is a probability measure.
48:04:660Paolo Guiotto: In this case, as we said above, the continuity from above always, holds, so… Continuity.
48:17:320Paolo Guiotto: from… above.
48:22:630Paolo Guiotto: folds.
48:25:450Paolo Guiotto: And therefore, whatever is the family EN, which is a decreasing family, we have that the limit of the probabilities of the n
48:36:100Paolo Guiotto: is the probability of the limit set, which is in this case the intersection of the end, but if the intersection is empty, this is probability of empty, which is supposed to be equal to zero.
48:49:530Paolo Guiotto: And this means that the property 2 is verified.
48:55:10Paolo Guiotto: So, the property too old.
48:59:760Paolo Guiotto: So this… this first implication is just a particular case of what we already know.
49:06:350Paolo Guiotto: What is, new is the vice versa, the 2 implies 1.
49:12:620Paolo Guiotto: So that from the fact that we assume the continuity from above at empty set, that means when you decrease down to empty set, the probabilities must go down to zero.
49:25:710Paolo Guiotto: then P is comfortably additive. So, we have to prove comfortable additivity in this case. So, the thesis is the goal.
49:35:270Paolo Guiotto: is, prove that the DP of a disjoint union of the
49:40:770Paolo Guiotto: or set EK is sum of P .
49:46:700Paolo Guiotto: PK.
49:48:400Paolo Guiotto: This is the goal.
49:53:280Paolo Guiotto: Okay, of course, we have somehow to use this factor, the continuity from above, so we have to build a decreasing sequence to zero. So, how do we do this?
50:07:740Paolo Guiotto: Well… Define the set E be the union of the yen, of the UK.
50:19:140Paolo Guiotto: So now, I take E minus a partial union of EK,
50:27:60Paolo Guiotto: So, the union for K that goes from 0 to n, for example.
50:32:560Paolo Guiotto: Now, you see that the partial unions are doing what? They are increasing, because I add more and more sets.
50:41:290Paolo Guiotto: And I fill all the set E, no? The set E is made of the unit. The EK are not necessarily contained, no? So the K are made like that, they are just this joint set into this, so E0, E1, E2, but…
50:57:720Paolo Guiotto: Increasing the union of this, you increase the set, and when you take all of them, you get the set E.
51:03:310Paolo Guiotto: So if you define now this set here, let's use a different letter, FN,
51:10:100Paolo Guiotto: Now, since this is a sigma… the family is supposed to be a sigma alga, but this is an element of F, is an event with the new language. And moreover, since when you increase n, you increase the union, it means that you decrease the FN.
51:27:240Paolo Guiotto: Because it's a difference, no? It's an E- something. If you increase the something, you decrease the E-. So this sequence FN goes down, is decreasing.
51:37:940Paolo Guiotto: And goes down to what?
51:40:420Paolo Guiotto: When you do the, and…
51:44:670Paolo Guiotto: the intersection, if you want, intuitively should be clear, no? This sequence is… the partial unions are increasing to E.
51:54:810Paolo Guiotto: And since the FN is made by E minus partial unions, you expect that this guy will go to zero, to empty set. If you want to, we can formally see this, because if you do the intersections of the FN, we are doing intersection of these differences.
52:10:20Paolo Guiotto: But a difference is what? Is a set intersection, the complementary of what I'm subtracting, so of the disjoint union k from 0 to n of the K, complementary of this.
52:24:500Paolo Guiotto: Now, if you do all these operations, you get E, intersection. The complementary of a union is a
52:39:820Paolo Guiotto: intersection. Now, it won't be disjoint, of the complementaries.
52:46:870Paolo Guiotto: So, at the end, if you look at this, what you are doing is,
52:52:270Paolo Guiotto: intersection on N of E intersects. The intersection for K going from 0 to N of the K Complementary.
53:05:390Paolo Guiotto: Now, since we are doing intersection on all possible ends, this is, at the end, E intersection with the intersection
53:14:160Paolo Guiotto: from 0 to infinity of the EK complementary.
53:20:420Paolo Guiotto: But there is no point in these two, because the ESAT is the union, this joint union, by the way, of the UK.
53:29:580Paolo Guiotto: And that's all.
53:31:630Paolo Guiotto: If you are here, you are at least in one of the CK.
53:36:570Paolo Guiotto: Okay?
53:37:950Paolo Guiotto: But if you are…
53:44:130Paolo Guiotto: But if you are here, you are ignored thinking.
53:48:40Paolo Guiotto: It means that there is no oil that belongs to this and to this section. This means that everything is empty, as, of course, expected. Okay, so we have now our sequence of sets FN that goes down to empty.
54:08:930Paolo Guiotto: And therefore, we apply the hypothesis.
54:12:840Paolo Guiotto: Hypothesis says that whenever you have
54:17:380Paolo Guiotto: a sequence going down to empty, the sequence of probabilities will go down to zero. So we have that, the limit
54:27:180Paolo Guiotto: Yunen.
54:28:740Paolo Guiotto: of the probabilities of these FN
54:33:260Paolo Guiotto: is equal to 0. Now, let's see that this means that P is comfortably additive, because what is the probability of Fn?
54:42:430Paolo Guiotto: The probability of Fn is, by definition of what is FN, is probability of
54:49:130Paolo Guiotto: E minus the disjoint union, K goes from 0 to N of the K.
55:05:300Paolo Guiotto: I need any… I missed an iPod, is this okay?
55:10:60Paolo Guiotto: I will put…
55:11:980Paolo Guiotto: Or no, maybe not. Okay, now, this reminds that E is the union of all the EK, so this set here is contained into this one.
55:24:790Paolo Guiotto: So you are doing P of Set E minus something which is the union of the K.
55:35:860Paolo Guiotto: the measure… Here we need the… what is the… I missed something here in the statement, good.
55:44:960Paolo Guiotto: We need that PB additive, not countable additive.
55:51:850Paolo Guiotto: P. P. Additive.
55:58:360Paolo Guiotto: So, finally additive, not countably additive.
56:02:900Paolo Guiotto: Because here I have to say that, P of the difference when they are included is the difference of the P. So P of E minus P of the
56:14:830Paolo Guiotto: union for K going from 0 to n of day K.
56:21:160Paolo Guiotto: And since P is additive, this is… B.
56:26:570Paolo Guiotto: additive.
56:28:310Paolo Guiotto: also here, B.
56:30:300Paolo Guiotto: Additive.
56:32:110Paolo Guiotto: This is B of E… minus the sum for k going from 0 to n of P of DK.
56:44:30Paolo Guiotto: So, this is the P of Fn, and we know that this quantity is going to zero.
56:51:180Paolo Guiotto: So it means that this is going to zero.
56:55:240Paolo Guiotto: But then, this quantity can go to zero if and all if
57:01:700Paolo Guiotto: the sum for k going from 0 to n of the PEK
57:09:420Paolo Guiotto: the unique possibility is that this goes to P of E.
57:15:720Paolo Guiotto: But this means that, B of E…
57:20:390Paolo Guiotto: which is, he reminded that it is the disjoint union of the UK, Is equal to the limit
57:30:380Paolo Guiotto: for N going to plus infinity.
57:33:530Paolo Guiotto: of the sum for k going from 0 to n of PK.
57:39:860Paolo Guiotto: And the quantity you have at right inside is limit of finite sums. These are the partial sums of the series, and that's the definition of infinite sums, sum for n going from 0 to infinity of PEK.
57:57:940Paolo Guiotto: Sorry, K going from K. And this means that if you look at this, P of the union is equal to the infinite sum of the P,
58:09:280Paolo Guiotto: And this means that, the, P is, countably.
58:18:310Paolo Guiotto: Thank you.
58:21:880Paolo Guiotto: So, I'm sorry, the statement was corrected during the, the way. So, this, says, let's review.
58:32:640Paolo Guiotto: That if we have… A sector with a sigma algebra sets.
58:38:320Paolo Guiotto: a function that is a candidate to be a probability, defined on F with values in 0, 1, such that probability of empty is equal to 0.
58:47:720Paolo Guiotto: Then the T's two properties are equivalent. Number one, P is countably additive. Number two, if P is just additive, so not necessarily countably additive, so I don't know what happens to an infinite disjoint union, but I know what happens to a finite disjoint union. So I can say that P
59:05:830Paolo Guiotto: A union B, disjoint is P of A plus P of B, and then this extends to any finite unit, but not necessarily to any infinite unit.
59:17:360Paolo Guiotto: You can extend to infinite unions whenever you know this, that you have the continuity from above at empty. So if you have a sequence that descends down to empty set, the limit of P of E is 0. If you have these two, then P is also countably additive.
59:37:170Paolo Guiotto: We will, sue… we will see now exactly an example where this, is, useful.
59:48:560Paolo Guiotto: So this example is the so-called space… of sequences.
00:02:830Paolo Guiotto: As I told you, this is a sort of,
00:08:110Paolo Guiotto: primitive model of the Brownian motion.
00:12:890Paolo Guiotto: So, it's a primitive model because it's actually the model of a random work, even if we don't use this language here.
00:21:730Paolo Guiotto: So, imagine that we want to describe this movement. What is the Brownian motion? Well, physically, in physics, the Brownian motion is the movement of particles
00:32:880Paolo Guiotto: In… suspended in a fluid, which are subject to continuous collisions with the particles of fluid, and this change the trajectory basically at every second, at every moment.
00:49:700Paolo Guiotto: So the typical trajectory of this motion that was observed is a very regular trajectory. For example, if I'm observing plane, I would
00:58:680Paolo Guiotto: See a curve like that, okay?
01:01:590Paolo Guiotto: Now…
01:02:610Paolo Guiotto: let's downgrade this to a one dimension. So imagine that we move only along a straight line, and imagine that we start at a certain point.
01:13:200Paolo Guiotto: And it's like if we toss a coin, and we decide if we go left or right with 50% of probability. So, at every second, I would see this kind of movement. I start here, I toss a coin, let's say that I get to right, I go here, I toss another coin, I get right, I go here, then left, I go back here, then right, back here, right, another here, left.
01:38:190Paolo Guiotto: Left, the left, the left, the left, the left, the left, the right, and so on.
01:43:450Paolo Guiotto: So, a trajectory is actually described by an infinite sequence of head and tails.
01:52:00Paolo Guiotto: Okay? So, a trajectory is a list of things like this. Head, tail, tail, head, tail, head, head, head, and so on.
02:05:200Paolo Guiotto: And each sequence
02:09:750Paolo Guiotto: identify a particular motion, a particular trajectory. Of course, this is a trajectory made by jumps, okay? It's not the continuum trajectory we have for a particle, physical particle, but let's say that it is a simplified model of that, no?
02:30:120Paolo Guiotto: So, the, one element of this type, so one entire sequence of head and tails.
02:40:30Paolo Guiotto: is n little omega of our space.
02:44:150Paolo Guiotto: So, our Space Omega is going to be made
02:48:40Paolo Guiotto: of sequences, let's say, of… sequences omega n within natural.
02:56:430Paolo Guiotto: Such that omega n can be either head or tail.
03:05:990Paolo Guiotto: Now, this is not a discrete probabilistic space, because one list here is one single element of the space, you see?
03:14:720Paolo Guiotto: So, I have lots of these elements, how many?
03:19:270Paolo Guiotto: In fact, if I take this example.
03:23:20Paolo Guiotto: And you replace letters H and T with numbers 0, 1, digits 0, 1. These are binary sequences of 01s, and with these sequences, you can quotationally describe the set of the real numbers.
03:38:650Paolo Guiotto: Because to each real numbers, there is a unique binary sequence of certain cases, you know, sequences that ends with a list of ones.
03:48:350Paolo Guiotto: But let's forget that maybe that for each binary sequence, we have a real number. So, in this data.
03:56:980Paolo Guiotto: the set rate of all possible sequences of this is basically like R, so it's not a discrete set at all, okay? But here, the natural probability has nothing to do with the exact measure on the line.
04:13:280Paolo Guiotto: something different, what we are going to do. So now, the model can be actually generalized.
04:20:310Paolo Guiotto: In this way, we take S, a set, a finite set.
04:31:830Paolo Guiotto: So here in the example, S is the set made by H and T, okay?
04:38:360Paolo Guiotto: in the… Example, S is the set made by H and T.
04:48:260Paolo Guiotto: But this is the case when we have two states, we can have Any definite number of states.
04:56:460Paolo Guiotto: So, I don't know if we will need, right now, but… so this means that we have elements S1, S2, let's say, S capital N.
05:09:610Paolo Guiotto: And this will be called the state… space, huh?
05:15:880Paolo Guiotto: By the way, this…
05:20:500Paolo Guiotto: discrete evolution model. Discrete because not the space is discrete, but because of the time is discrete, okay?
05:30:470Paolo Guiotto: Now, the space omega, the sample space, is, by definition, set of sequences omega n, and natural.
05:42:270Paolo Guiotto: where the omega n are in the state space S.
05:50:300Paolo Guiotto: So, this is exactly in the case when S is head-tail, list, infinite list of these add tails.
06:01:520Paolo Guiotto: Now, we assume that, in this model, there exist probabilities for each of these states. So, the next ingredient is P,
06:15:820Paolo Guiotto: is a finite set of probabilities, so P1, P2, etc, PN.
06:25:20Paolo Guiotto: are numbers which are positive, and the sum of these, PN… well, let's, no, let's use N, reserve N for the index.
06:36:420Paolo Guiotto: In the sequence, let's call K, the index here, sum of pK, k from 1 to N equal to 1.
06:44:930Paolo Guiotto: So that… now we are going to introduce… they basically are the probability that,
06:50:630Paolo Guiotto: of the state PJ is the probability of the state SJ.
06:58:340Paolo Guiotto: Now, we introduce… to introduce the probability space, we need to introduce the family of sets we want to assign a probability.
07:07:360Paolo Guiotto: And fine… and finally, the probability measure.
07:11:660Paolo Guiotto: Now, what kind of sets are interesting for us?
07:15:910Paolo Guiotto: Let's introduce this family, which is called the family of cylinders.
07:26:640Paolo Guiotto: So, a cylinder.
07:29:630Paolo Guiotto: is a set of trajectories, so of this omega na, okay? So, an element of this will be simply denoted by a letter omega.
07:41:810Paolo Guiotto: So, it's a set of, let's call them trajectories in this state space.
07:47:600Paolo Guiotto: For WeChat, certain, times, times are referred to this index n.
07:55:530Paolo Guiotto: the… the state omega n belongs to, a subset of states that we determine,
08:05:790Paolo Guiotto: we can modify. So, let's, see what we mean. Well, now, to simplify, we can always reduce to this situation. We introduce this set, C, say…
08:22:20Paolo Guiotto: I use the letter K, so let's say…
08:29:120Paolo Guiotto: So, to keep uniform notation, let's change… I will call this J.
08:35:630Paolo Guiotto: So that's called CK… CK… where K is, natural.
08:49:229Paolo Guiotto: And, well, let's start, let's start,
08:55:550Paolo Guiotto: I… let's do by K. Let's start with the initial K, then I will formulate the general case. Maybe it's easier. So let's take the set C0…
09:06:840Paolo Guiotto: S1, where S1 is a subset of S. What is this? This is the set of omega in capital omega, so the trajectories, in this,
09:25:550Paolo Guiotto: the set of all possible trajectories such that the first element of this trajectory belongs to the set S1.
09:37:840Paolo Guiotto: So, for example, in the example,
09:43:180Paolo Guiotto: In the example where S is made by head and tail, so the case of the coin, for example, I have, C,
09:56:650Paolo Guiotto: 0, let's say, H, This could be a possibility, is the set of all possible
10:04:810Paolo Guiotto: Trajectories of possible sequences of states, where the first state
10:12:550Paolo Guiotto: is in the set made by this unique element, so the first state is just head. So these are all possible stories that starts with an head.
10:24:100Paolo Guiotto: Okay? So…
10:25:960Paolo Guiotto: After the first experiment, anything could be… could happen, but the first must be an head, no? Similarly, C0. T will be the set of all the trajectories in omega where omega zero is tailed.
10:48:560Paolo Guiotto: You could extend this, huh?
10:51:350Paolo Guiotto: For example, saying, let's put the C1
10:55:780Paolo Guiotto: Now, C1 won't mean omega 1, but here one will be. I put a condition on the first two, coordinates, so let's say S…
11:10:80Paolo Guiotto: S… well, maybe it would have been better to call it S0.
11:15:860Paolo Guiotto: for… the indexes. So, cross S1,
11:25:390Paolo Guiotto: Or maybe, let's,
11:28:840Paolo Guiotto: Later, we will introduce this notation, but for a moment, let's put a comma here, maybe.
11:35:520Paolo Guiotto: So this is the set of omega, so the trajectories, such that omega zero belongs to S0, omega 1 belongs to S1.
11:48:780Paolo Guiotto: So, for example, an object like, C1 H… age, huh?
12:01:800Paolo Guiotto: is the set of all the possible trajectories, such that the first outcome is head, as well as the second outcome is head. So all the stories that starts with head, head, for example, no?
12:20:490Paolo Guiotto: Now, why I was saying this? Because it's convenient then to introduce this notation. I could also identify this pair with S0, Cartesian product with S1, and write omega 0. Omega 1,
12:35:920Paolo Guiotto: this pair in the Cartesian product as 0 crosses 1, it's the same thing.
12:44:450Paolo Guiotto: So now let's, do the jump to the general definition. What is CK?
12:49:990Paolo Guiotto: let's say, a set SK, where this SK is a subset of S times S times S K times.
13:09:320Paolo Guiotto: So this is the set of paths, omega in capital omega, such that omega 0, omega 1, etc, omega K, this belongs to this set SK.
13:23:490Paolo Guiotto: So, it's a set of trajectories whose first, say, K plus 1?
13:31:400Paolo Guiotto: Values must be in that, set of states, okay?
13:39:110Paolo Guiotto: Now, why this is called the S-cylinder?
13:43:250Paolo Guiotto: Because imagine that you have a cylinder, it's a geometric cylinder. What is a geometric cylinder in space? There are three coordinates, X, Y, and Z, let's say, no? A cylinder, let's say, the empty cylinder, for example.
13:59:540Paolo Guiotto: In the empty cylinder, you have points here, X, Y, Z,
14:05:660Paolo Guiotto: Such that you have a condition for the first, two components, no? XYZ belongs to this cylinder.
14:16:170Paolo Guiotto: If and only if, for example, I don't know, let's imagine that this is the standard circle with radius 1. You say.
14:25:850Paolo Guiotto: plus Y squared must be equal to 1.
14:29:630Paolo Guiotto: And about the third coordinate, you don't have any condition that is unconstrained is real.
14:36:810Paolo Guiotto: So, a cylinder is typically a set where you have a certain number of an array of points where the first dot coordinates are constrained by some condition.
14:51:280Paolo Guiotto: So they must be, you can say, the point XY must be in a circle, so it's like here, only because, you know, 1 must be in something.
15:01:570Paolo Guiotto: And then, all the remaining coordinates, you see that here there is omega K1, omega D plus 2, omega K plus 3, and so on. No, these coordinates are constrained, so it means that they can be whatever in the state space S, no? So I could say also that
15:19:830Paolo Guiotto: the… set the cylinder CK.
15:25:750Paolo Guiotto: SK…
15:27:520Paolo Guiotto: could be said that it is the set of omegas in the set omega, where omega 0, omega 1, etc. Omega k, this is constrained into the set as K, but the remaining omega k plus 1
15:47:520Paolo Guiotto: He's in S, so this is like he's in R, he's in the state space, he's… can take all possible values, you see?
15:56:920Paolo Guiotto: So DSK is a subset of S times S times SK times, so K plus 1 times S to K plus 1.
16:09:400Paolo Guiotto: No? So it's a subset that…
16:12:50Paolo Guiotto: this is possibly made of singletons, like here, no? Omega zero is just H, omega 1 is just H, etc, cannot be everything, while all the others, omega k plus 1, omega k plus 2, and so on.
16:28:900Paolo Guiotto: These are in S, so we may say omega J, omega omega H is in S for every H greater or equal than K plus 1. So it means that there is no
16:44:670Paolo Guiotto: It's completely useless to write these conditions, because it's like if I'm not posing any condition. So, this is an explanation why these sets are called cylinders.
16:59:90Paolo Guiotto: Now, let's take the family of all the cylinders.
17:05:840Paolo Guiotto: The idea is now that we want to define a probability to assign a number to each of these sets. What is the natural number?
17:15:320Paolo Guiotto: So… The probability.
17:21:960Paolo Guiotto: Of a cylinder.
17:24:510Paolo Guiotto: K… SK…
17:31:610Paolo Guiotto: is what? Now, Let's stay on this formula.
17:38:80Paolo Guiotto: The cylinder is made by infinite sequences where the first K plus 1 values are determined, and the other ones are not determined.
17:51:520Paolo Guiotto: So, we have a distribution that says that
18:00:480Paolo Guiotto: The distribution of probabilities, let's say that for this state, the probability is P1, for this state is P2, for this state is P3, and so for this one is the N.
18:10:670Paolo Guiotto: So, like a hand and tape, there are two states. This one has probably one half, this one has probably one half.
18:18:100Paolo Guiotto: Now, let's imagine that we want to give, just to start.
18:22:140Paolo Guiotto: The probability do this set, you know?
18:24:920Paolo Guiotto: What should be a reasonable probability?
18:29:590Paolo Guiotto: Now, this is made of infinite boosts of elegant days, but with the mission that the faster…
18:40:220Paolo Guiotto: Outcome is bad.
18:42:140Paolo Guiotto: So, we base that 50% of these sequence we had initially had, and the other 50% initially take.
18:52:560Paolo Guiotto: The problem is that I cannot say it's just one half, because these sequels are infinitely very.
18:59:100Paolo Guiotto: You see that I inflict many lists that start with H. So everything that has H, then whatever, no?
19:06:390Paolo Guiotto: So, let's say, let's do the usual probabilistic approach. We count the set of all the possible cases.
19:13:900Paolo Guiotto: And then we divide over all the possibilities, because these are both infinity. But I can expect that if this, let's say, has probability 1 half, also this set should have probability 1 half, as well as this one.
19:29:400Paolo Guiotto: Let's, let's see what should be on this other case. Imagine that we have now two positions, no? Yeah. I need that. First, I have a head, and then second, I have again a head. So, we assume that we are tossing twice a coin. So, first time, I have a half possibility.
19:49:330Paolo Guiotto: And of these half positivities, I have another half of possibilities at the second time. So at the end, I expect that this reasonably should have already 1 fourth, which is one half times 1 half.
20:02:960Paolo Guiotto: This is because there are two states with equal probability. What if Ed has probability 25% and tail 75%?
20:14:80Paolo Guiotto: I would say that for this set, for the first position, I have 25, so 25 is 1 over 4, but the second is another 25,
20:24:150Paolo Guiotto: But it must be combined with this one, so I expect 1 over 4 times 1 over 4, so 1 over 4 squared.
20:31:90Paolo Guiotto: So I just multiply the probability. And what if I add and second take? The first one, one of the four. The second one, three of the four. I have the total is 1 fourth times 3 fourths.
20:46:90Paolo Guiotto: Now, if I have a family number of states, each with a certain probability.
20:52:30Paolo Guiotto: What I have to do, reasonably, is to put this…
20:56:520Paolo Guiotto: So, I have to sum… the probabilities, P… Yes, P,
21:14:570Paolo Guiotto: How do we buy it? PS1… P. S.2.
21:20:490Paolo Guiotto: Etc. B.
21:23:200Paolo Guiotto: S, PS0, or PS1, PS2, PSK.
21:28:410Paolo Guiotto: on the possible states S0, S1, SK, that belongs to that set SK.
21:38:710Paolo Guiotto: So, if we are in the case of the coin, these are only about 1 half times 1 half, no?
21:46:690Paolo Guiotto: We have several states, so now the first omega can be in a, in a, in a state that has, you know, that must be
21:57:390Paolo Guiotto: You gotta be the others in this set, no? So there will be a certain number of states that verify this condition. I take their probabilities and my sum. And for the second coordinate, I multiply with the relative probability, and so on, and I define this world here.
22:15:480Paolo Guiotto: Now, since this product of numbers between 0 and 1, it is positive, and that quantity would mean 0 and 1, so it makes sense as a probability.
22:28:740Paolo Guiotto: Now, what is the problem? The problem is that, number one.
22:33:220Paolo Guiotto: we don't see now, because time is over, is that a cylinder can be written in infinitely many ways. The same cylinder can be written in infinitely many ways, so
22:45:500Paolo Guiotto: There is potentially here a problem of good position. So, I add that if I represent the same cylinder in a different way, I could have a different value for the probability. Of course, this
22:57:810Paolo Guiotto: is not a good thing, and we hope that it won't happen, and in fact, if one dies. The second problem is that the family of the cylinder
23:08:810Paolo Guiotto: is, not a survival.
23:12:560Paolo Guiotto: So,
23:14:980Paolo Guiotto: We cannot take this as a definition because of this measure, because the family of sex where it is defined is not sigmatic.
23:24:10Paolo Guiotto: However, it can be proved that this P is additive, finitely additive, so when you take a finite number of cylinders, and you do, the artist joint, and you do the movement, this is really the sum of them.
23:40:930Paolo Guiotto: And it verifies this property that we have a… seen here.
23:46:620Paolo Guiotto: So… When we have a sequence.
23:51:130Paolo Guiotto: of cylinders that goes down to the empty sector, the probability of the probabilities goes to is zero.
24:00:40Paolo Guiotto: Okay.
24:01:230Paolo Guiotto: So this will, will make possible to extend this to a true probability measure.
24:07:260Paolo Guiotto: Okay, let's stop here for today, and
24:11:900Paolo Guiotto: We will continue the discussion on Monday on this, on this set.
24:23:600Paolo Guiotto: And, please, .