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The probability of reaching the absorbing states from a particular transient state?

Plotting absorbing state probabilities from state 1Defining a function of a distributionNicely illustrating the evolution and end-state of a discrete-time Markov chainHow to obtain the number of Markov Chain transitions in a simulation?Arranging “ranked” nodes of a graph symmetricallyState “i” goes to state “j”: list accessible states in a Markov-chainPart specification error with InterpolatingFunction when generating a Markov Modulated Poisson ProcessUsing Mathematica to calculate expected time to absorptionHidden Markov Model: emissions probabilities dependent on observable parameterEstimate process parameters of geometric Brownian motion with a two-state Markov chainPlotting absorbing state probabilities from state 1

4

$begingroup$

Can I use the data available from MarkovProcessProperties to compute the probability of reaching each of the absorbing states from a particular transient state?

In an earlier post, kglr showed a solution involving the probabilities from State 1. Can that solution be amended easily to compute the probabilities from any of the transient states?

markov-chains markov-process

share|improve this question

edited 3 hours ago

user120911

asked 3 hours ago

user120911user120911

67228

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean something like this? StationaryDistribution[DiscreteMarkovProcess[{1,0,0},{{0,1/2,1/2},{0,1,0},{0,0,1}}]]
  $endgroup$
  – Sjoerd Smit
  3 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am looking for a solution like the one shown by kglr in the link, but which is more dynamic because it offers the possibility of specifying the particular transient state to be examined.
  $endgroup$
  – user120911
  2 hours ago
  

  
  
  
  

add a comment | 

4

$begingroup$

Can I use the data available from MarkovProcessProperties to compute the probability of reaching each of the absorbing states from a particular transient state?

In an earlier post, kglr showed a solution involving the probabilities from State 1. Can that solution be amended easily to compute the probabilities from any of the transient states?

markov-chains markov-process

share|improve this question

edited 3 hours ago

user120911

asked 3 hours ago

user120911user120911

67228

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you mean something like this? StationaryDistribution[DiscreteMarkovProcess[{1,0,0},{{0,1/2,1/2},{0,1,0},{0,0,1}}]]
  $endgroup$
  – Sjoerd Smit
  3 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I am looking for a solution like the one shown by kglr in the link, but which is more dynamic because it offers the possibility of specifying the particular transient state to be examined.
  $endgroup$
  – user120911
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Can I use the data available from MarkovProcessProperties to compute the probability of reaching each of the absorbing states from a particular transient state?

In an earlier post, kglr showed a solution involving the probabilities from State 1. Can that solution be amended easily to compute the probabilities from any of the transient states?

markov-chains markov-process

share|improve this question

edited 3 hours ago

user120911

asked 3 hours ago

user120911user120911

67228

$endgroup$

share|improve this question

edited 3 hours ago

user120911

asked 3 hours ago

user120911user120911

67228

share|improve this question

edited 3 hours ago

user120911

asked 3 hours ago

user120911user120911

67228

asked 3 hours ago

user120911user120911

67228

asked 3 hours ago

user120911user120911

67228

1 Answer
1

active

oldest

votes

5

$begingroup$

proc = DiscreteMarkovProcess[1, {{0., 0.5, 0., 0., 0.5, 0., 0., 0., 0., 0.}, 

  {0., 0., 0.5, 0., 0., 0.5, 0., 0., 0., 0.}, 

  {0., 0., 0., 0.5, 0., 0., 0.5, 0., 0., 0.}, 

  {0., 0., 0., 1., 0., 0., 0., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0.5, 0., 0.5, 0., 0.},

  {0., 0., 0., 0., 0., 0., 0.5, 0., 0.5, 0.}, 

  {0., 0., 0., 0., 0., 0., 1., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0.5, 0.5},

  {0., 0., 0., 0., 0., 0., 0., 0., 1., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0., 1.}}];



Graph[proc]

{tr, ab, ltm} = MarkovProcessProperties[proc, #] & /@ 

  { "TransientClasses", "AbsorbingClasses", "LimitTransitionMatrix"};



TeXForm @ TableForm[ltm[[Flatten@tr, Flatten@ab]], 

 TableHeadings -> {Flatten@tr, Flatten@ab}]

$begin{array}{ccccc}
& 4 & 7 & 9 & 10 \
3 & 0.5 & 0.5 & 0. & 0. \
6 & 0. & 0.5 & 0.5 & 0. \
2 & 0.25 & 0.5 & 0.25 & 0. \
8 & 0. & 0. & 0.5 & 0.5 \
5 & 0. & 0.25 & 0.5 & 0.25 \
1 & 0.125 & 0.375 & 0.375 & 0.125 \
end{array}$

share|improve this answer

edited 2 hours ago

answered 2 hours ago

kglrkglr

186k10202421

$endgroup$

add a comment | 

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5

$begingroup$

proc = DiscreteMarkovProcess[1, {{0., 0.5, 0., 0., 0.5, 0., 0., 0., 0., 0.}, 

  {0., 0., 0.5, 0., 0., 0.5, 0., 0., 0., 0.}, 

  {0., 0., 0., 0.5, 0., 0., 0.5, 0., 0., 0.}, 

  {0., 0., 0., 1., 0., 0., 0., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0.5, 0., 0.5, 0., 0.},

  {0., 0., 0., 0., 0., 0., 0.5, 0., 0.5, 0.}, 

  {0., 0., 0., 0., 0., 0., 1., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0.5, 0.5},

  {0., 0., 0., 0., 0., 0., 0., 0., 1., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0., 1.}}];



Graph[proc]

{tr, ab, ltm} = MarkovProcessProperties[proc, #] & /@ 

  { "TransientClasses", "AbsorbingClasses", "LimitTransitionMatrix"};



TeXForm @ TableForm[ltm[[Flatten@tr, Flatten@ab]], 

 TableHeadings -> {Flatten@tr, Flatten@ab}]

$begin{array}{ccccc}
& 4 & 7 & 9 & 10 \
3 & 0.5 & 0.5 & 0. & 0. \
6 & 0. & 0.5 & 0.5 & 0. \
2 & 0.25 & 0.5 & 0.25 & 0. \
8 & 0. & 0. & 0.5 & 0.5 \
5 & 0. & 0.25 & 0.5 & 0.25 \
1 & 0.125 & 0.375 & 0.375 & 0.125 \
end{array}$

share|improve this answer

edited 2 hours ago

answered 2 hours ago

kglrkglr

186k10202421

$endgroup$

add a comment | 

5

$begingroup$

proc = DiscreteMarkovProcess[1, {{0., 0.5, 0., 0., 0.5, 0., 0., 0., 0., 0.}, 

  {0., 0., 0.5, 0., 0., 0.5, 0., 0., 0., 0.}, 

  {0., 0., 0., 0.5, 0., 0., 0.5, 0., 0., 0.}, 

  {0., 0., 0., 1., 0., 0., 0., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0.5, 0., 0.5, 0., 0.},

  {0., 0., 0., 0., 0., 0., 0.5, 0., 0.5, 0.}, 

  {0., 0., 0., 0., 0., 0., 1., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0.5, 0.5},

  {0., 0., 0., 0., 0., 0., 0., 0., 1., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0., 1.}}];



Graph[proc]

{tr, ab, ltm} = MarkovProcessProperties[proc, #] & /@ 

  { "TransientClasses", "AbsorbingClasses", "LimitTransitionMatrix"};



TeXForm @ TableForm[ltm[[Flatten@tr, Flatten@ab]], 

 TableHeadings -> {Flatten@tr, Flatten@ab}]

$begin{array}{ccccc}
& 4 & 7 & 9 & 10 \
3 & 0.5 & 0.5 & 0. & 0. \
6 & 0. & 0.5 & 0.5 & 0. \
2 & 0.25 & 0.5 & 0.25 & 0. \
8 & 0. & 0. & 0.5 & 0.5 \
5 & 0. & 0.25 & 0.5 & 0.25 \
1 & 0.125 & 0.375 & 0.375 & 0.125 \
end{array}$

share|improve this answer

edited 2 hours ago

answered 2 hours ago

kglrkglr

186k10202421

$endgroup$

add a comment | 

$begingroup$

proc = DiscreteMarkovProcess[1, {{0., 0.5, 0., 0., 0.5, 0., 0., 0., 0., 0.}, 

  {0., 0., 0.5, 0., 0., 0.5, 0., 0., 0., 0.}, 

  {0., 0., 0., 0.5, 0., 0., 0.5, 0., 0., 0.}, 

  {0., 0., 0., 1., 0., 0., 0., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0.5, 0., 0.5, 0., 0.},

  {0., 0., 0., 0., 0., 0., 0.5, 0., 0.5, 0.}, 

  {0., 0., 0., 0., 0., 0., 1., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0.5, 0.5},

  {0., 0., 0., 0., 0., 0., 0., 0., 1., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0., 1.}}];



Graph[proc]

{tr, ab, ltm} = MarkovProcessProperties[proc, #] & /@ 

  { "TransientClasses", "AbsorbingClasses", "LimitTransitionMatrix"};



TeXForm @ TableForm[ltm[[Flatten@tr, Flatten@ab]], 

 TableHeadings -> {Flatten@tr, Flatten@ab}]

$begin{array}{ccccc}
& 4 & 7 & 9 & 10 \
3 & 0.5 & 0.5 & 0. & 0. \
6 & 0. & 0.5 & 0.5 & 0. \
2 & 0.25 & 0.5 & 0.25 & 0. \
8 & 0. & 0. & 0.5 & 0.5 \
5 & 0. & 0.25 & 0.5 & 0.25 \
1 & 0.125 & 0.375 & 0.375 & 0.125 \
end{array}$

share|improve this answer

edited 2 hours ago

answered 2 hours ago

kglrkglr

186k10202421

$endgroup$

proc = DiscreteMarkovProcess[1, {{0., 0.5, 0., 0., 0.5, 0., 0., 0., 0., 0.}, 

  {0., 0., 0.5, 0., 0., 0.5, 0., 0., 0., 0.}, 

  {0., 0., 0., 0.5, 0., 0., 0.5, 0., 0., 0.}, 

  {0., 0., 0., 1., 0., 0., 0., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0.5, 0., 0.5, 0., 0.},

  {0., 0., 0., 0., 0., 0., 0.5, 0., 0.5, 0.}, 

  {0., 0., 0., 0., 0., 0., 1., 0., 0., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0.5, 0.5},

  {0., 0., 0., 0., 0., 0., 0., 0., 1., 0.}, 

  {0., 0., 0., 0., 0., 0., 0., 0., 0., 1.}}];



Graph[proc]

{tr, ab, ltm} = MarkovProcessProperties[proc, #] & /@ 

  { "TransientClasses", "AbsorbingClasses", "LimitTransitionMatrix"};



TeXForm @ TableForm[ltm[[Flatten@tr, Flatten@ab]], 

 TableHeadings -> {Flatten@tr, Flatten@ab}]

share|improve this answer

edited 2 hours ago

answered 2 hours ago

kglrkglr

186k10202421

answered 2 hours ago

kglrkglr

186k10202421

answered 2 hours ago

kglrkglr

186k10202421