Understanding Markov Processes and the Impact of Varying Variables

[hikaru1221]

Hi all,

This is not a homework question, but I think it's reasonable to post it here.
Say, for a Markov process, $Pr(Z_{i+1}|Z_{i},...,Z_0) = Pr(Z_{i+1}|Z_{i},Z_{i-1})$, people prove that if we consider $V_{i} = (Z_{i+1},Z_{i},Z_{i-1})$, it would be a 1st-order Markov process, i.e. $Pr(V_{i+1}|V_{i},...,V_0) = Pr(V_{i+1}|V_{i})$.

However if $V_{i} = (*,0,0)$ and $V_{i-1} = (1, 0, *)$, then $Pr(V_{i+1}|V_{i},V_{i-1}) = 0$, while if $V_{i-1} = (0, 0, *)$ then $Pr(V_{i+1}|V_{i},V_{i-1})$ may be non-zero. That is, it's not really reducible to the 1st-order. I understand that in a sense, the former case is rather invalid, as $Pr(V_{i} = (*, 0, 0) | V_{i-1} = (1,0,*) ) = 0$. Yet it seems not very reasonable to me to neglect the case when it comes to $Pr(V_{i+1}|V_{i},V_{i-1})$, as the variables can take in any possible value.

Did I miss something?

 

[mighty2000]

Hello,

Thank you for bringing up this interesting topic. It is true that in the case of V_{i} = (*,0,0) and V_{i-1} = (1,0,*), the probability of V_{i+1} given V_{i} and V_{i-1} is 0. This is because the Markov property assumes that the future state only depends on the current state, and in this case, the current state has no influence on the future state.

However, this does not mean that this case should be neglected. In fact, it is important to consider all possible cases and their probabilities in order to fully understand and analyze a Markov process. This includes the case of V_{i-1} = (0,0,*), where the probability of V_{i+1} given V_{i} and V_{i-1} may be non-zero.

It is also worth noting that while the Markov property assumes that the future state only depends on the current state, this does not mean that the current state is the only factor influencing the future state. Other external factors may also have an impact, and it is important to consider them in the analysis of a Markov process.

In summary, the case of V_{i} = (*,0,0) and V_{i-1} = (1,0,*) should not be neglected, but it should also be understood in the context of the Markov property and other external factors that may influence the process. I hope this helps clarify your understanding.