Markov => Chapman Kolmogorov => Total Probability => Conditional Probability => Set Theory ... => ...% & $ # & ...... It was a wonderful two days trip to the basics maths, but your explanation was what best I've seen about the "why". That graphic on 3:40 was enought... Thanks a lot man, really.
Hi there Gareth, I am not getting quite right how did you get a 4x4 transition matrix from minute 8.00 approximately, given there are three states therefore the matrix I think should be 3x3. As long as I've seen from the figure the transition matrix is: T = [0.3 0.4 0.3; 0.3 0.2 0.5; 0.8 0.1 0.1] Is there any recommendation of another video or anything for me to understand this part? Thank you very much, it is a really helpful input.
Hi Mariano. Thanks for your question. I have perhaps been unclear here. The transition matrix in the later part is NOT the transition matrix for the three-state Markov chain discussed in the previous slides. It is the transition matrix for a different Markov chain. The Chapman-Kolmogorov relationship is a general result that can be applied to any transition matrix though. So mathematically what I have done is fine. In this last part, I am trying to explain how we can couch this result using matrix multiplication. If you are struggling with the Matrix multiplication, perhaps look for a video in which matrix multiplication or powers of matrices are explained. I hope this helps.
@@gtribello Ok Gareth, I was trying to retrieve the same results as you have shown in the matrix but is ok about the matrix multiplication. Thanks again for your answer.
I wanted to edit the last comment but I deleted it. Thank you for your response. Though I know the definition of conditional probability, in the video you say moving from c in T_2 to b in T_1 is independent of moving from a in T_1 to c in T_2. My doubt comes when I tried to take that into notation and how to describe the *event* per se of going from a in T_1 to c in T_2 (and the other). That would be a joint probability. Otherwise, how do we describe the event of the jump a->b? Is there where a filtration would come to the picture? By Markov, I know that P(T_3=b|T_2=c, T_1=a)=P(T_3=b|T_2=c), however I do not see clear how to go from that to conclude the *jumps* a->b and b->c are independent, which then leads to multiply P(T_3=b|T_2=c) with P(T_2=c|T_1=a). To be clear, I know the argument of going from P(T_3=b|T_1=a), because taking the intersection with the sample space does not affect the event, that probability is the same as the tum of P(T_3=b|T_2=k,T_1=a)P(T_2=k|T_1=a) and though I see this argument as a much more approachable, I cant figure out how to mathematize it a little more rigurous. Edit: to be concise. How would you write the event described by the probabilites P(T_3=b|T_2=c)P(T_2=c|T_1=a)? Would that be a conditional, joint event? (Is there really a "conditional event"? )
Hello. Sorry for taking a while to reply and also sorry that my last message was a little patronising. I am not sure if I know the answer to your question. If you want a really rigorous argument a textbook will likely do better than the answer I will give here. Rigour is not really my strong suit. With that caveat aside, what I think I am saying here is that: P(T_3=b and T_2=c | T_1=a) = P(T_3=b| T_2=c)P(T_2=c|T_1=a) That result is useful because if that holds the Chapman Kolmogorov relationship holds. We can thus calculate the n-step transition probability matrix by taking the nth power of the 1-step probability matrix, which is good because all sorts of useful facts about Markov chains drop out from this fact. In other words, Making the assumption above is useful because it makes the maths into something we can do. If you think this way, the question then is what does the assumption above mean. Well: (1) It means that the matrix of transition probabilities does not depend on time (2) The probability of being in state c at time T_2 does not depend on the state we were in at time T_0 or any earlier time. I don't know if flipping the ideas around in this way helps at all. What I can say with some certainty is that the random variables T_1, T_2 and T_3 are all NOT independent. So you are perhaps right when you say that the events are not independent. The result that I am using above to derive the Chapman Kolmogorov holds because of the law of total probability instead: en.wikipedia.org/wiki/Law_of_total_probability
They are really just two different ways of looking at the same underlying fact. If you want, you might say that the fact that the n-step transition matrix is the nth power of the transition matrix is a consequence of the Chapman Kolmogorov relation. I hope this helps.
This was very helpful. I would just like to add as constructive criticism to be careful to correcting yourself in the same sentence because it might generate confusion or some people (like myself) might lose the train of thought throughtout the explanation. An example of this can be heard at 0:54 - 0:55. Cheers.
Such an excellent and comprehensive explanation!!! Really well done!!!
I was unable to understand this by reading the book, though understood it completely now. Thank you.
Thank you so, so, so much.
That saved me from dying --- Many thanks
Me as well!
I understand everything! Thank you!
Finally it all makes sense! Thank you
Good guy Gareth making all our lives easy
This is best lecture I've ever seen
Excellent! Thanks!
Really quality explanation
I salute to your pronunciation and skill in explaining Gareth!
Thank you. This comment made my day.
thank you so much for the video
Markov => Chapman Kolmogorov => Total Probability => Conditional Probability => Set Theory ... => ...% & $ # & ...... It was a wonderful two days trip to the basics maths, but your explanation was what best I've seen about the "why". That graphic on 3:40 was enought... Thanks a lot man, really.
Amazing 🤗
I am guessing the 4x4 matrix is not part of the previous explanation? I thought that transition matrix's rows and columns add up to one.
Thank you for the video that was brilliant
Thx, It was very useful
Bloody legend, thanks mate.
Hi there Gareth,
I am not getting quite right how did you get a 4x4 transition matrix from minute 8.00 approximately, given there are three states therefore the matrix I think should be 3x3.
As long as I've seen from the figure the transition matrix is:
T = [0.3 0.4 0.3; 0.3 0.2 0.5; 0.8 0.1 0.1]
Is there any recommendation of another video or anything for me to understand this part?
Thank you very much, it is a really helpful input.
Hi Mariano. Thanks for your question. I have perhaps been unclear here. The transition matrix in the later part is NOT the transition matrix for the three-state Markov chain discussed in the previous slides. It is the transition matrix for a different Markov chain. The Chapman-Kolmogorov relationship is a general result that can be applied to any transition matrix though. So mathematically what I have done is fine. In this last part, I am trying to explain how we can couch this result using matrix multiplication. If you are struggling with the Matrix multiplication, perhaps look for a video in which matrix multiplication or powers of matrices are explained. I hope this helps.
@@gtribello Ok Gareth, I was trying to retrieve the same results as you have shown in the matrix but is ok about the matrix multiplication. Thanks again for your answer.
Thank you so much Sir. Your explanation was easily understandable!
great explanation
thank you sir, great video
Very clear explanation. Thanks :)
Professional instructor
Thank you very much for the clarification!
Very clear explanation. Thanks.
I wanted to edit the last comment but I deleted it. Thank you for your response. Though I know the definition of conditional probability, in the video you say moving from c in T_2 to b in T_1 is independent of moving from a in T_1 to c in T_2. My doubt comes when I tried to take that into notation and how to describe the *event* per se of going from a in T_1 to c in T_2 (and the other). That would be a joint probability. Otherwise, how do we describe the event of the jump a->b? Is there where a filtration would come to the picture?
By Markov, I know that P(T_3=b|T_2=c, T_1=a)=P(T_3=b|T_2=c), however I do not see clear how to go from that to conclude the *jumps* a->b and b->c are independent, which then leads to multiply P(T_3=b|T_2=c) with P(T_2=c|T_1=a).
To be clear, I know the argument of going from P(T_3=b|T_1=a), because taking the intersection with the sample space does not affect the event, that probability is the same as the tum of P(T_3=b|T_2=k,T_1=a)P(T_2=k|T_1=a) and though I see this argument as a much more approachable, I cant figure out how to mathematize it a little more rigurous.
Edit: to be concise. How would you write the event described by the probabilites P(T_3=b|T_2=c)P(T_2=c|T_1=a)? Would that be a conditional, joint event? (Is there really a "conditional event"?
)
Hello. Sorry for taking a while to reply and also sorry that my last message was a little patronising.
I am not sure if I know the answer to your question. If you want a really rigorous argument a textbook will likely do better than the answer I will give here. Rigour is not really my strong suit. With that caveat aside, what I think I am saying here is that:
P(T_3=b and T_2=c | T_1=a) = P(T_3=b| T_2=c)P(T_2=c|T_1=a)
That result is useful because if that holds the Chapman Kolmogorov relationship holds. We can thus calculate the n-step transition probability matrix by taking the nth power of the 1-step probability matrix, which is good because all sorts of useful facts about Markov chains drop out from this fact. In other words, Making the assumption above is useful because it makes the maths into something we can do. If you think this way, the question then is what does the assumption above mean. Well:
(1) It means that the matrix of transition probabilities does not depend on time
(2) The probability of being in state c at time T_2 does not depend on the state we were in at time T_0 or any earlier time.
I don't know if flipping the ideas around in this way helps at all. What I can say with some certainty is that the random variables T_1, T_2 and T_3 are all NOT independent. So you are perhaps right when you say that the events are not independent. The result that I am using above to derive the Chapman Kolmogorov holds because of the law of total probability instead:
en.wikipedia.org/wiki/Law_of_total_probability
@@gtribello Thank you very much. No worries, both of your responses have been very useful.
Thanks you very much sir I am very easily understood.
Hi, what is the difference between Chapman Kolmogorov equation and n-step transition?
They are really just two different ways of looking at the same underlying fact. If you want, you might say that the fact that the n-step transition matrix is the nth power of the transition matrix is a consequence of the Chapman Kolmogorov relation. I hope this helps.
@@gtribello thank you :-)
Very clear and helpful -- thank you
Thnks
This was very helpful. I would just like to add as constructive criticism to be careful to correcting yourself in the same sentence because it might generate confusion or some people (like myself) might lose the train of thought throughtout the explanation. An example of this can be heard at 0:54 - 0:55. Cheers.
Thanks
Better explanations than that in at my Uni..thnx a lot
thank u ~
poorly done