更改

The [[simplification  of  Markov Chains]], also referred to as [[coarse-graining of Markov chains]], is closely related to causal emergence. The coarse-graining process in causal emergence is essentially a model reduction of a Markov chain. Coarse-graining Markov processes <ref>Zhang A, Wang M. Spectral state compression of markov processes[J]. IEEE transactions on information theory, 2019, 66(5): 3202-3231.</ref> is an important problem for modeling a state transition system. It reduces the complexity of Markov chains by merging multiple states into one state.

The [[simplification  of  Markov Chains]], also referred to as [[coarse-graining of Markov chains]], is closely related to causal emergence. The coarse-graining process in causal emergence is essentially a model reduction of a Markov chain. Coarse-graining Markov processes <ref>Zhang A, Wang M. Spectral state compression of markov processes[J]. IEEE transactions on information theory, 2019, 66(5): 3202-3231.</ref> is an important problem for modeling a state transition system. It reduces the complexity of Markov chains by merging multiple states into one state.

There are mainly three meanings of coarsening. First, when we study a very large-scale system, it is unnecessary to focus on the changes of each microscopic state. Coarse-graining filters out noise and heterogeneity to derive mesoscale or macroscopic laws from microscopic data. Second, some state transition probabilities are very similar, so they can be regarded as the same kind of state. Clustering (partitioning) this kind of states (also called lumping the states) to obtain a new smaller Markov chain can reduce the redundancy of system representation. Third, in reinforcement learning using Markov decision processes, coarse-graining the Markov chain can reduce the size of the state space and improve training efficiency. In much of the literature, coarse-graining and dimension reduction are equivalent <ref>Coarse graining. ''Encyclopedia of Mathematics.'' URL: <nowiki>http://encyclopediaofmath.org/index.php?title=Coarse_graining&oldid=16170</nowiki></ref>.

There are mainly three meanings of simplification. First, when we study a very large-scale system, it is unnecessary to focus on the changes of each microscopic state. Coarse-graining filters out noise and heterogeneity to derive mesoscale or macroscopic laws from microscopic data. Second, some state transition probabilities are very similar, so they can be regarded as the same kind of state. Clustering (partitioning) this kind of states (also called lumping the states) to obtain a new smaller Markov chain can reduce the redundancy of system representation. Third, in reinforcement learning using Markov decision processes, coarse-graining the Markov chain can reduce the size of the state space and improve training efficiency. In much of the literature, coarse-graining and dimension reduction are equivalent <ref>Coarse graining. ''Encyclopedia of Mathematics.'' URL: <nowiki>http://encyclopediaofmath.org/index.php?title=Coarse_graining&oldid=16170</nowiki></ref>.

Among them, there are two types of partitioning (clustering, grouping) of the state space: hard partitioning and soft partitioning. Soft partitioning involves breaking up microscopic states, reconstructing macroscopic states, and allowing their superposition. Hard partitioning is a strict grouping of microscopic states, dividing several microscopic states into one group without allowing overlap and superposition  (see [[coarse-graining of Markov chains]]).

Among them, there are two types of partitioning of the state space: hard partitioning and soft partitioning. Soft partitioning involves breaking up microscopic states, reconstructing macroscopic states, and allowing their superposition. Hard partitioning is a strict grouping of microscopic states, dividing several microscopic states into one group without allowing overlap and superposition  (see [[coarse-graining of Markov chains]]).

The coarse-graining of Markov chains not only needs to be done on the state space, but also on the transition matrix. This involves simplifying the original transitional probability matrix (TPM) according to the state grouping to obtain a new, smaller TPM. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of states, TPM, and state vector at the same time. Thus, this raises the question of how to calculate TPM in the new Markov chain after states grouping. Another question is whether the normalization condition can be guaranteed.

The coarse-graining of Markov chains not only needs to be done on the state space, but also on the transition matrix. This involves simplifying the original transitional probability matrix (TPM) according to the state grouping to obtain a new, smaller TPM. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of states, TPM, and state vector at the same time. Thus, this raises the question of how to calculate TPM in the new Markov chain after states grouping. Another question is whether the normalization condition can be guaranteed.