Remark 1: Earlier multi-dimensional Markov chain spatial models are not much practical for conditional simulation on sample data. For example, the Krumbein's 2-D transgression and regression Markov model (Krumbein 1968) is not a model for spatial imaging. The Lin-Harbaugh's multi-D Markov model (Lin and Harbaugh 1984) incorporates limited spatial correlation information and is not conditional on samples. The Luo's linear 2-D Markov chain model (1996) is preliminary and tends to generate fragmentary patterns. The coupled Markov chain (CMC) model of Elfeki and Dekking (2001) has a small class underestimation problem, even if we ignore the layer/patch inclination tendency caused by asymmetric neighborhoods and paths. With the initial purpose of solving the deficiencies of the CMC model and a long-time exploration, the Markov chain random field (MCRF) theory/model (initially called "spatial Markov chain" models) was later proposed. The MCRF model solved the small class underestimation problem and finally extended the 1-D Markov chain model into a theoretically-sound and practical geostatistical approach for simulating categorical fields. It considers a single locally-conditioned Markov chain in a multi-D space, and uses Bayes' theorem (or rule) to construct a local sequential Bayesian updating process at neighborhood nearest data level for estimating the local conditional probability distribution, rather than coupling two or more 1D Markov chains.
Remark 2: The conditional independence assumptuion for nearest sample data within a neighborhood, extended from the cardinal-neighbor conditional independence property of Pickard random fields (Pickard 1980), was used to simplify the full MCRF model (with multiple-point likelihoods) into a simplified MCRF model that consists of only transition probabilities with different lag distances, which can be estimated from sample data through transiogram modeling. Therefore, the spatial conditional independence assumption is important for the implementation of the MCRF model (or for obtaining the simplified MCRF model). The conditional independence assumption was previously mainly used in nonspatial statistics (e.g., Bayesian networks) and rarely used in spatial statistical and geostatistical models.
Remark 3: The property of Pickard random fields (PRFs) (Pickard 1980) that given the state of a central cell its adjacent neighbors in cardinal directions on a rectangular lattice are conditionally independent was expanded to the sparse data situation to simplify the full MCRF model and support the neighborhood choice in MCRF simulation algorithms (the fixed-path algorithms and the random-path quadrantal neighborhood algorithm with the simplified MCRF model). Although the MCRF model was not derived from the PRF model, one may regard MCRFs to be an expansion of PRFs toward the sparse data situation, given that PRFs are known to be stationary Markov fields on finite rectangular lattices, in which any node sequence along a monotone path is a stationary Markov chain. However, the MCRF model has no requirement for global stationarity.
Remark 4: Matheron defined geostatistics as "The application of the formalism of random functions to the reconnaissance and estimation of natural phenomena" (quoted in Journel and Huijbregts 1978). According to this definition of geostatistics, the MCRF approach is typically a geostatistical approach and the MCRF model is a fundamental geostatistical model at the neighborhood nearest data level, let alone its similarity to kriging to some extent in the way of dealing with point sample data and aiming to be used in geospatial data.
Remark 5: For a geostatistical or spatial statistical approach, what we care about are its practicality and theoretical rationality. The MCRF model has both practicality and theoretical rationality, which are sufficient for its usefulness. A locally-conditioned spatial Markov chain, more specifically, a spatial Markov chain (a stochastic spatial process with a discrete state space and the Markov property) with local updating on nearest data (in each neighborhood in some subset of the d-dimensional space), is a simple and understandable explanation of the MCRF model (for the whole random field). Of course, there were reasons for us to use the name of "Markov chain random field", as explained in Li (2007) and in the webpage "The single-Markov-chain random-field idea". From the Bayesian view, MCRFs are a special kind of (neighborhood-based, spatial) Bayes nets (or Bayesian networks, Bayes networks, belief networks, Bayes(ian) models).
Remark 6: One working on geostatistical simulation should understand the large differences between spatial data and nonspatial data and between spatial models and nonspatial models, and the difficulty in developing a nonspatial model/theory into a spatial model. The complexity of dependencies among spatial data, particularly that among sparsely and irregularly distributed spatial data points, is not something easy to handle quantitatively. Sincere earlier efforts, no matter whether they developed perfect/practical spatial models for real world applications or not, should be respected, but should not be deified.
References:
Elfeki, A.M. and Dekking, F.M. 2001, A Markov chain model for subsurface characterization: Theory and applications. Math. Geol., 33:569-589.
Journel, A.G. and Huijbregts, C.J. 1978, Mining geostatistics. Academic press.
Krumbein, W.C. 1968, Fortran IV computer program for simulation of transgression and regression with continuous time Markov models. Kansas Geological Survey, Computer Cont. 26, 38p.
Li, W. 2007, Markov chain random fields for estimation of categorical variables. Math. Geol., 39: 321-335.
Lin, C. and Harbaugh, J.W. 1984, Graphic Display of Two and Three Dimensional Markov Computer Models in Geology. John Wiley & Sons, Inc.
Luo, J. 1996, Transition probability approach to statistical analysis of spatial qualitative variables in geology. In: Forster A, Merriam DF (eds.) Geologic modeling and mapping (Proceedings of the 25th Anniversary Meeting of the International Association for Mathematical Geology, October 10-14, 1993, Prague, Czech Republic). Plenum Press, New York. p. 281-299.
Pickard, D.K. 1980, Unilateral Markov fields. Advances in Applied Probability 12(3): 655-671.