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→Non-commutativity
====Non-commutativity====
====Non-commutativity====
If the two dissimilar row vectors are forcibly averaged, the resulting coarse-grained TPM may break the commutativity between the abstraction operation (i.e., coarse-graining) and marginalization (i.e., the time evolution operator). For example, assume that <math>A_{m\times n}</math> is a state coarse-graining operation (combining n states into m states). In this case, the coarse-graining strategy is defined as the one that maximizes the effective information of the macroscopic state transition matrix. <math>(\cdot) \times (\cdot)</math> is a time coarse-graining operation (combining two time steps into one). In this way, [math]A_{m\times n}(TPM_{n\times n})[/math] is to perform coarse-graining on a [math]n\times n[/math] TPM, and the coarse-graining process is simplified as the product of matrix [math]A[/math] and matrix [math]TPM[/math].
If the two dissimilar row vectors are forcibly averaged, the resulting coarse-grained TPM may break the commutativity between the abstraction operation (i.e., coarse-graining) and marginalization (i.e., the time evolution operator). For example, assume that <math>A_{m\times n}</math> is a state coarse-graining operation (combining n states into m states). In this case, the coarse-graining strategy is defined as the one that maximizes the EI of the macroscopic state transition matrix. <math>(\cdot) \times (\cdot)</math> is a time coarse-graining operation (combining two time steps into one). In this way, [math]A_{m\times n}(TPM_{n\times n})[/math] is to perform coarse-graining on a [math]n\times n[/math] TPM, and the coarse-graining process is simplified as the product of matrix [math]A[/math] and matrix [math]TPM[/math].
However, as pointed out in the literature <ref name=":6" />, the above problem can be alleviated by considering the error factor of the model while maximizing EI in the continuous variable space.
However, as pointed out in the literature <ref name=":6" />, the above problem can be alleviated by considering the error factor of the model while maximizing EI in the continuous variable space.
====Other problems and remedies====
====Other problems and remedies====