Let the labeled samples [z.sub.i] = ([x.sub.i], [y.sub.i]) [member of] X x Y, i [member of] [N.sub.m], be

independent random variables drawn from the probability space (X x Y, P), and the unlabeled samples [x.sub.m+i] [member of] X, i [member of] [N.sub.u], be

independent random variables drawn from the probability space (X, [P.sub.X]).

For instance, the hepatobiliary training data is represented as [D.sub.m] = ([Y.sub.1], [Z.sub.1]), ..., ([Y.sub.m], [Z.sub.m]) where Y and Z are

independent random variables which are the same as the autonomous sample pair (Y, Z).

Let ([OMEGA], S, P) be a probability space and [([X.sub.n]).sub.n[member of]N]] be a sequence of

independent random variables such that [EX.sub.n] = 0 for all n [member of] N and [[summation].sup.[infinity].sub.n=1] var [X.sub.n] < [infinity].

where f is the joint probability density function (PDF) of random vector [xi] which for

independent random variables [[xi].sub.i] can be written as the multiplication of the individual PDF for each variable [f.sub.i]([[xi].sub.i]); i.e.,

Furthermore, he studies some properties of this distribution and presents its stochastic representation as the product of two

independent random variables [square root of [bar.T]] and V, where T ~ [[chi square].sub.(3)] and V is a discrete random variable such that P(V = [+ or -]1) = 1/2; that is, X = [square root of [bar.T]]V has the distribution BN.

Firstly, this procedure is established in the next paragraphs for

independent random variables, based on the relationships between the random variables and random vector to.

[20.] Lugannani, R., and Rice, S., "Saddlepoint Approximation for the Distribution of the Sum of

Independent Random Variables," Advances in Applied Probability, 475-490, 1980.

Kruglov, "On one identity for distribution of sums of

independent random variables," Theory of Probability and its Applications, vol.

The principle of ICA estimation is based on the central limit theorem (CLT), which states that the sum of

independent random variables tends to be distributed towards Gaussian, regardless of the underlying distribution.

where [u.sub.1] and [u.sub.2] are two

independent random variables of uniform distribution over [0, 1].

For cases where [x.sub.n] is a discrete structure, such as a permutation or a graph, and where the input values are realizations of

independent random variables with the same distribution, the output sequence is a Markov chain X = ([X.sub.n])n[member of]N that is adapted to a combinatorial family F in the sense that [X.sub.n] takes its values in the subset [F.sub.n] [subset] F of objects with base parameter n.

where [[epsilon].sub.t] ~ WN(0, [[sigma].sup.2]) represents a white noise, a series of

independent random variables, identically distributed: E[[epsilon].sub.t]=0, [for all]t Var[[epsilon].sub.t] = [[sigma].sup.2], [for all]t [[epsilon].sub.t] si [[epsilon].sub.t + k] independent variables, [for all]k [not equal to] 0.