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 Post subject: Markov chains - with one-step trainsition matricesPosted: Sun, 6 May 2012 01:26:18 UTC

Joined: Sun, 6 May 2012 01:08:03 UTC
Posts: 5
Hi all hoping you can help me with a couple of questions:

Firstly I have a two-state transition probability matrix:

where . I need to prove that

I know that to go from would be and the only other option is which is so is it simply that this probability is just . This seems far too straightforward!

Secondly for another two-state transition probability matrix:

Given the chain is in state at time , what is the probability it was in state at time ?

I wasn't sure at all about this either, the only thing I could think of was calculating which sounds reasonable but could be a complete fallacy Could anyone please give me a thumbs up (or down) on this too?

Many thanks!

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 Posted: Sun, 6 May 2012 02:13:02 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 13974
Location: Austin, TX
jjgoth wrote:
Hi all hoping you can help me with a couple of questions:

Firstly I have a two-state transition probability matrix:

where . I need to prove that

I know that to go from would be and the only other option is which is so is it simply that this probability is just . This seems far too straightforward!

Secondly for another two-state transition probability matrix:

Given the chain is in state at time , what is the probability it was in state at time ?

I wasn't sure at all about this either, the only thing I could think of was calculating which sounds reasonable but could be a complete fallacy Could anyone please give me a thumbs up (or down) on this too?

Many thanks!

That seems pretty reasonable to me, why do you think it isn't what you should do?

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 Posted: Sun, 6 May 2012 02:27:01 UTC

Joined: Sun, 6 May 2012 01:08:03 UTC
Posts: 5
It's just I haven't answered questions of this form so just used intuition so could easily be wrong! That's cool though if it's right, thanks

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 Posted: Sun, 6 May 2012 02:34:55 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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 Posted: Mon, 7 May 2012 10:23:38 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6782
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
jjgoth wrote:
Hi all hoping you can help me with a couple of questions:

Firstly I have a two-state transition probability matrix:

where . I need to prove that

I know that to go from would be and the only other option is which is so is it simply that this probability is just . This seems far too straightforward!

Secondly for another two-state transition probability matrix:

Given the chain is in state at time , what is the probability it was in state at time ?

I wasn't sure at all about this either, the only thing I could think of was calculating which sounds reasonable but could be a complete fallacy Could anyone please give me a thumbs up (or down) on this too?

Many thanks!

That seems pretty reasonable to me, why do you think it isn't what you should do?

The second one is wrong --- you don't even have a stochastic matrix, and you should use Bayes' theorem.

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 Posted: Mon, 7 May 2012 13:27:32 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 13974
Location: Austin, TX
outermeasure wrote:
jjgoth wrote:
Hi all hoping you can help me with a couple of questions:

Firstly I have a two-state transition probability matrix:

where . I need to prove that

I know that to go from would be and the only other option is which is so is it simply that this probability is just . This seems far too straightforward!

Secondly for another two-state transition probability matrix:

Given the chain is in state at time , what is the probability it was in state at time ?

I wasn't sure at all about this either, the only thing I could think of was calculating which sounds reasonable but could be a complete fallacy Could anyone please give me a thumbs up (or down) on this too?

Many thanks!

That seems pretty reasonable to me, why do you think it isn't what you should do?

The second one is wrong --- you don't even have a stochastic matrix, and you should use Bayes' theorem.

Right, didn't check on the second one, should have said that for the op's benefit.

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 Posted: Tue, 8 May 2012 06:27:43 UTC

Joined: Sun, 6 May 2012 01:08:03 UTC
Posts: 5
My bad, lack of sleep is getting to me...

Given the chain is in state at time , what is the probability it was in state at time ?

So using Bayes' theorem I get:

Is that better?

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 Posted: Tue, 8 May 2012 06:49:32 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6782
Location: On this day Taiwan becomes another Tiananmen under Dictator Ma.
jjgoth wrote:
My bad, lack of sleep is getting to me...

Given the chain is in state at time , what is the probability it was in state at time ?

So using Bayes' theorem I get:

Is that better?

No.

Unless you assume your starting distribution ..., you won't get . However, there is no reason why you would want, a priori, to have that distribution (and another problem is whether that is reachable from a distribution of , for all n). On the other hand, if you start with the invariant distribution, and solve for your reverse time Markov chain ...

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