Occupancy grid algorithms represent the map as a fine-grained grid over the continuous space of locations in the environment. The most common type of occupancy grid maps are 2d maps that describe a slice of the 3d world.
If we let denote the grid cell with index i (often in 2d maps, two indices are used to represent the two dimensions), then the notation represents the probability that cell i is occupied. The computational problem with estimating the posterior is the dimensionality of the problem: if the map contains 10,000 grid cells (a relatively small map), then the number of possible maps that can be represented by this gridding is . Thus calculating a posterior probability for all such maps is infeasible.
The standard approach, then, is to break the problem down into smaller problems of estimating for all grid cells . Each of these estimation problems is then a binary problem. This breakdown is convenient but does lose some of the structure of the problem, since it doesn't enable modelling dependencies between neighboring cells. Instead , the posterior of a map is approximated by factoring it into . Due to this factorization, a binary Bayes filter can be used to estimate the occupancy probability for each grid cell. It is frequent to use a log-odds representation of the probability that each grid cell is occupied.