How to MCMC (or other simulation) given a non-stationary distribution?
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Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $pi$, and I know the transition probabilities are functions of some unknown parameters $P_{ito j}=p(pmb{alpha})$, where $pmbalpha$ is a vector of unknown parameters (pardon my poor probability notation here). I know that I can use Markov Chain Monte Carlo to simulate the stationary distribution, $pi$, to calculate the posterior for $pmbalpha$. However, I am curious if MCMC can still be used to get the posterior if you were given some non-stationary distribution instead? By non-stationary, I mean non-equilibrium distribution here.
In example, let $t>0$ be time and $T$ be the time it takes for the graph to reach the equilibrium (stationary) state. Now say your experiment/data takes too long to reach equilibrium and you can only sample the data at some time $t_0<T$. Can you still use that distribution to get the posterior and thereby infer $pi$?
stochastic-processes markov-chains machine-learning markov-process monte-carlo
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
$endgroup$
add a comment |
$begingroup$
Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $pi$, and I know the transition probabilities are functions of some unknown parameters $P_{ito j}=p(pmb{alpha})$, where $pmbalpha$ is a vector of unknown parameters (pardon my poor probability notation here). I know that I can use Markov Chain Monte Carlo to simulate the stationary distribution, $pi$, to calculate the posterior for $pmbalpha$. However, I am curious if MCMC can still be used to get the posterior if you were given some non-stationary distribution instead? By non-stationary, I mean non-equilibrium distribution here.
In example, let $t>0$ be time and $T$ be the time it takes for the graph to reach the equilibrium (stationary) state. Now say your experiment/data takes too long to reach equilibrium and you can only sample the data at some time $t_0<T$. Can you still use that distribution to get the posterior and thereby infer $pi$?
stochastic-processes markov-chains machine-learning markov-process monte-carlo
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
$endgroup$
1
$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
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– Alex
Jan 6 at 21:14
1
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@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37
add a comment |
$begingroup$
Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $pi$, and I know the transition probabilities are functions of some unknown parameters $P_{ito j}=p(pmb{alpha})$, where $pmbalpha$ is a vector of unknown parameters (pardon my poor probability notation here). I know that I can use Markov Chain Monte Carlo to simulate the stationary distribution, $pi$, to calculate the posterior for $pmbalpha$. However, I am curious if MCMC can still be used to get the posterior if you were given some non-stationary distribution instead? By non-stationary, I mean non-equilibrium distribution here.
In example, let $t>0$ be time and $T$ be the time it takes for the graph to reach the equilibrium (stationary) state. Now say your experiment/data takes too long to reach equilibrium and you can only sample the data at some time $t_0<T$. Can you still use that distribution to get the posterior and thereby infer $pi$?
stochastic-processes markov-chains machine-learning markov-process monte-carlo
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
$endgroup$
Say I was given some directed graph that satisfies the Markov property, has a stationary distribution, $pi$, and I know the transition probabilities are functions of some unknown parameters $P_{ito j}=p(pmb{alpha})$, where $pmbalpha$ is a vector of unknown parameters (pardon my poor probability notation here). I know that I can use Markov Chain Monte Carlo to simulate the stationary distribution, $pi$, to calculate the posterior for $pmbalpha$. However, I am curious if MCMC can still be used to get the posterior if you were given some non-stationary distribution instead? By non-stationary, I mean non-equilibrium distribution here.
In example, let $t>0$ be time and $T$ be the time it takes for the graph to reach the equilibrium (stationary) state. Now say your experiment/data takes too long to reach equilibrium and you can only sample the data at some time $t_0<T$. Can you still use that distribution to get the posterior and thereby infer $pi$?
stochastic-processes markov-chains machine-learning markov-process monte-carlo
stochastic-processes markov-chains machine-learning markov-process monte-carlo
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
asked Dec 26 '18 at 17:20
MasterYodaMasterYoda
94839
94839
1
$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
$endgroup$
– Alex
Jan 6 at 21:14
1
$begingroup$
@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37
add a comment |
1
$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
$endgroup$
– Alex
Jan 6 at 21:14
1
$begingroup$
@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37
1
1
$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
$endgroup$
– Alex
Jan 6 at 21:14
$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
$endgroup$
– Alex
Jan 6 at 21:14
1
1
$begingroup$
@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37
$begingroup$
@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37
add a comment |
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$begingroup$
Yes, any data from the Markov chain on the graph will be useful. E.g. if there are two states, $0$ and $1$ and you observe that the first $10$ observations are $0, 0, 0, 0, 0, 1, 0, 0, 0, 0$ then you have information that $P_{0 rightarrow 0}$ is probably close to $1$ (because we observe 7 $(0 rightarrow 0)$ transitions and only $1$ $(0 rightarrow 1)$ transition). Unless the Markov chain is not time-homogeneous then observations at any time in its run are as good as any other. Of course, the more transitions you observe and the more states that you visit, the more information you will have.
$endgroup$
– Alex
Jan 6 at 21:14
1
$begingroup$
@Alex That makes sense. Thanks for your response!
$endgroup$
– MasterYoda
Jan 7 at 4:37