Editing Gibbs sampling (section)

=== Relation of conditional distribution and joint distribution ===
Furthermore, the conditional distribution of one variable given all others is proportional to the joint distribution, i.e., for all possible value <math>(x_i)_{1\leq i\leq n}</math> of <math>\mathbf{X}</math>:

:<math>P(X_j=x_j\mid (X_i=x_i)_{i\neq j}) = \frac{P((X_i=x_i)_i)}{P((X_i=x_i)_{i\neq j})} \propto P((X_i=x_i)_i)</math>

"Proportional to" in this case means that the denominator is not a function of <math>x_j</math> and thus is the same for all values of <math>x_j</math>; it forms part of the [[normalization constant]] for the distribution over <math>x_j</math>.  In practice, to determine the nature of the conditional distribution of a factor <math>x_j</math>, it is easiest to factor the joint distribution according to the individual conditional distributions defined by the [[graphical model]] over the variables, ignore all factors that are not functions of <math>x_j</math> (all of which, together with the denominator above, constitute the normalization constant), and then reinstate the normalization constant at the end, as necessary.  In practice, this means doing one of three things:
# If the distribution is discrete, the individual probabilities of all possible values of <math>x_j</math> are computed, and then summed to find the normalization constant.
# If the distribution is continuous and of a known form, the normalization constant will also be known.
# In other cases, the normalization constant can usually be ignored, as most sampling methods do not require it.