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with a corresponding factor graph shown on the right. Observe that the factor graph has a cycle. If we merge (,) (,) into a single factor, the resulting factor graph will be a tree. This is an important distinction, as message passing algorithms are usually exact for trees, but only approximate for graphs with cycles.
where P 1 is a probability distribution without any indecomposable factor and P 2 is a distribution that is either degenerate or is representable as the convolution of a finite or countable set of indecomposable distributions. The factorization is not unique, in general.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
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Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is ƒ θ (x), then T is sufficient for θ if and only if nonnegative functions g and h can be found such that
A factor graph is an undirected bipartite graph connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing belief propagation. A clique tree or junction tree is a tree of cliques, used in the junction tree algorithm.
A simplified version of the LLL factorization algorithm is as follows: calculate a complex (or p-adic) root α of the polynomial () to high precision, then use the Lenstra–Lenstra–Lovász lattice basis reduction algorithm to find an approximate linear relation between 1, α, α 2, α 3, . . . with integer coefficients, which might be an ...
In mathematics and computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors.This decomposition is theoretically possible and is unique for polynomials with coefficients in any field, but rather strong restrictions on the field of the coefficients are needed to allow the computation of the factorization by means of an algorithm.