Finding a recursive formula for these probabilities in a semi-Markov chain
$begingroup$
For a semi-Markov chain, let $i,jin S$ where $S$ is the space of states. Let also $P=(p_{ij})_{i,jin S}$ be the matrix of transition probabilities. We define the following probabilities:
$e_{i,j}(n)=P(text{the chain enters state j at time n } |text{ the chain entered state i at time 0})$
and
$q_{i,j}(n)=P(text{the chain is at state j at time n }|text{ the chain entered state i at time 0})$.
I believe that it is almost obvious that $e_{i,j}(n)=sum_{kin S}q_{i,k}(n-1)p_{kj}$, hence $E(n)=Q(n-1)cdot P$, where $E(n)=(e_{i,j}(n))_{i,jin S}, Q(n)=(q_{i,j}(n))_{i,jin S}$. My intuition says that there is a recursive (with respect to $n$) formula for the probabilities $e_{i,j}(n)$ but I can't really formulate it. anyone has an idea?
stochastic-processes markov-chains hidden-markov-models
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
$endgroup$
add a comment |
$begingroup$
For a semi-Markov chain, let $i,jin S$ where $S$ is the space of states. Let also $P=(p_{ij})_{i,jin S}$ be the matrix of transition probabilities. We define the following probabilities:
$e_{i,j}(n)=P(text{the chain enters state j at time n } |text{ the chain entered state i at time 0})$
and
$q_{i,j}(n)=P(text{the chain is at state j at time n }|text{ the chain entered state i at time 0})$.
I believe that it is almost obvious that $e_{i,j}(n)=sum_{kin S}q_{i,k}(n-1)p_{kj}$, hence $E(n)=Q(n-1)cdot P$, where $E(n)=(e_{i,j}(n))_{i,jin S}, Q(n)=(q_{i,j}(n))_{i,jin S}$. My intuition says that there is a recursive (with respect to $n$) formula for the probabilities $e_{i,j}(n)$ but I can't really formulate it. anyone has an idea?
stochastic-processes markov-chains hidden-markov-models
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
$endgroup$
add a comment |
$begingroup$
For a semi-Markov chain, let $i,jin S$ where $S$ is the space of states. Let also $P=(p_{ij})_{i,jin S}$ be the matrix of transition probabilities. We define the following probabilities:
$e_{i,j}(n)=P(text{the chain enters state j at time n } |text{ the chain entered state i at time 0})$
and
$q_{i,j}(n)=P(text{the chain is at state j at time n }|text{ the chain entered state i at time 0})$.
I believe that it is almost obvious that $e_{i,j}(n)=sum_{kin S}q_{i,k}(n-1)p_{kj}$, hence $E(n)=Q(n-1)cdot P$, where $E(n)=(e_{i,j}(n))_{i,jin S}, Q(n)=(q_{i,j}(n))_{i,jin S}$. My intuition says that there is a recursive (with respect to $n$) formula for the probabilities $e_{i,j}(n)$ but I can't really formulate it. anyone has an idea?
stochastic-processes markov-chains hidden-markov-models
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
$endgroup$
For a semi-Markov chain, let $i,jin S$ where $S$ is the space of states. Let also $P=(p_{ij})_{i,jin S}$ be the matrix of transition probabilities. We define the following probabilities:
$e_{i,j}(n)=P(text{the chain enters state j at time n } |text{ the chain entered state i at time 0})$
and
$q_{i,j}(n)=P(text{the chain is at state j at time n }|text{ the chain entered state i at time 0})$.
I believe that it is almost obvious that $e_{i,j}(n)=sum_{kin S}q_{i,k}(n-1)p_{kj}$, hence $E(n)=Q(n-1)cdot P$, where $E(n)=(e_{i,j}(n))_{i,jin S}, Q(n)=(q_{i,j}(n))_{i,jin S}$. My intuition says that there is a recursive (with respect to $n$) formula for the probabilities $e_{i,j}(n)$ but I can't really formulate it. anyone has an idea?
stochastic-processes markov-chains hidden-markov-models
stochastic-processes markov-chains hidden-markov-models
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
asked Jan 14 at 17:08
JustDroppedInJustDroppedIn
2,350420
2,350420
add a comment |
add a comment |
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