To see this, we rewrite (10) as marginal density: p ϕ ij = ∬ p ϕ ij | h i, w ij N h i | 0, I N w ij | 0, I d h i d w ij, (11 where the conditional density is given by p ϕ ij | h i, w ij = N ϕ | μ + F h i + G w ij, Σ. (12).
then the marginal density p(x k |z1 k) can be approximately expressed by a mixture of densities: p ( x k | z 1 : k ) ≈ ∑ j = 1 N w k j p ( x k | s k j, θ j, z 1 : k ).
We note that the one-point function is related to the marginal density as p 1=R 1/n, while the two-point function gives the JPDF of two eigenvalues as p 2(λ 1,λ 2)=R 2(λ 1,λ 2)/(n(n−1)).
If the marginal posterior density p(s k,θ|z1 k) is represented by a set of weighted samples { s k j, θ j, w k j } j = 1 N, i.e., p ( s k, θ | z 1 : k ) ≈ ∑ j = 1 N w k j δ ( s k − s k j ) δ ( θ − θ j ), (9).
After sampling jointly for y and z, z can be ignored and the marginal density p y) obtained [ 28].
Under assumption of statistical independence, the joint likelihood function leads to product of marginal density functions, defined as (15) P (X ∣ C i ) = ∏ k = 1 d P (x k ∣ C i ) = P (x 1 ∣ C i ) × P (x 2 ∣ C i ) × P (x 3 ∣ C i ) ⋯ × P (x d ∣ C i ).
The score function O l) for a particular subnetwork l is calculated by the formula (14) O (l ) = ∑ x ∈ l ∑ y ∈ z p (x, y ) l o g p (x, y ) p (x ) p (y ) where p x, y) is the joint probability density function of subnetwork l and a set of output labels z (metastatic or non-metastatic); p(x) and p y) are marginal density functions.
Thus p c (u| y) can be interpreted as the density of birth times conditional on y being observed in [0; τ0), and p m (y) as the marginal density for the observed y weighted with w i = G τ0 - y) - G(- y), i.e., the probability of birth occurring within the interval [- y; τ0 - y). When g is known, then so is p c, as are the weights in p m.
Given two random variables, x and y, MI is computed by: 1 where p x, y), p(x), and p y) are joint density function and marginal density functions of x and y, respectively.
The marginal probabilities p X (F X ), p Y (F Y ) and the joint probability density p XY (F X,F Y ) of the WB audio features and the HF spectral envelope coefficients are modeled by a GMM with 128 mixtures and with full covariance matrices.
Fig. 2 Marginal density of eigenvalues of the quotient ensemble.
