src.util¶
Utilities for the model classes
Module Contents¶
- conditional_from_precision_sites_white(Kuu: tf.Tensor, Kff: tf.Tensor, Kuf: tf.Tensor, l: tf.Tensor, L: tf.Tensor = None, L2=None, jitter=1e-09)[source]¶
- Given a g₁ and g2, and distribution p and q such that
p(g₂) = N(g₂; 0, Kuu)
p(g₁) = N(g₁; 0, Kff) p(g₁ | g₂) = N(g₁; Kfu (Kuu⁻¹) g₂, Kff - Kfu (Kuu⁻¹) Kuf)
- And q(g₂) = N(g₂; μ, Σ) such that
Σ⁻¹ = Kuu⁻¹ + Kuu⁻¹LLᵀKuu⁻¹ Σ⁻¹μ = Kuu⁻¹l
- This method computes the mean and (co)variance of
q(g₁) = ∫ q(g₂) p(g₁ | g₂) dg₂ = N(g₂; μ*, Σ**)
with Σ** = k** - kfu Kuu⁻¹ kuf - kfu Kuu⁻¹ Σ Kuu⁻¹ kuf
= k** - kfu Kuu⁻¹kuf + kfu (Kuu + LLᵀ)⁻¹ kuf
- μ* = k*u Kuu⁻¹ m
= k*u Kuu⁻¹ Λ⁻¹ Kuu⁻¹ l = k*u (Kuu + LLᵀ)⁻¹ l
Inputs: :param Kuu: tensor M x M :param Kff: tensor N x 1 :param Kuf: tensor M x N :param L: tensor L x M x M :param l: tensor M x L
- conditional_from_precision_sites(Kuu: tf.Tensor, Kff: tf.Tensor, Kuf: tf.Tensor, l: tf.Tensor, L: tf.Tensor = None, L2=None)[source]¶
- Given a g₁ and g2, and distribution p and q such that
p(g₂) = N(g₂; 0, Kuu)
p(g₁) = N(g₁; 0, Kff) p(g₁ | g₂) = N(g₁; Kfu (Kuu⁻¹) g₂, Kff - Kfu (Kuu⁻¹) Kuf)
- And q(g₂) = N(g₂; μ, Σ) such that
Σ⁻¹ = Kuu⁻¹ + LLᵀ Σ⁻¹μ = l
- This method computes the mean and (co)variance of
q(g₁) = ∫ q(g₂) p(g₁ | g₂) dg₂ = N(g₂; μ*, Σ**)
with Σ** = k** - kfu Kuu⁻¹ kuf - kfu Kuu⁻¹ Σ Kuu⁻¹ kuf
= k** - kfu Kuu⁻¹kuf + kfu Kuu⁻¹ (Kuu⁻¹ + LLᵀ)⁻¹ Kuu⁻¹ kuf = k** - kfu Kuu⁻¹kuf + kfu Kuu⁻¹ (Kuu - Kuu L(I + LᵀKuuL)⁻¹Lᵀ Kuu) Kuu⁻¹ kuf = k** - kfu L(I + LᵀKuuL)⁻¹Lᵀ kuf = k** - kfu LW⁻¹Lᵀ kuf = k** - kfu L Lw⁻ᵀLw⁻¹ Lᵀ kuf = k** - (Lw⁻¹ Lᵀ kuf)ᵀ Lw⁻¹ Lᵀ kuf = k** - (D kuf)ᵀ D kuf
- μ* = k*u Kuu⁻¹ m
= k*u Kuu⁻¹ Σ l = k*u Kuu⁻¹ (Kuu⁻¹ + LLᵀ)⁻¹ l = k*u Kuu⁻¹ (Kuu - Kuu L(I + LᵀKuuL)⁻¹Lᵀ Kuu) l = k*u l - k*u L(I + LᵀKuuL)⁻¹Lᵀ Kuu l = k*u l - k*u LW⁻¹Lᵀ Kuu l = k*u l - (Lw⁻¹ Lᵀ kuf)ᵀ Lw⁻¹ Lᵀ Kuu l = k*u l - (D kuf)ᵀ D Kuu l
Inputs: :param Kuu: tensor M x M :param Kff: tensor N x 1 :param Kuf: tensor M x N :param L: tensor L x M x M :param l: tensor M x L
- project_diag_sites(Kuf: tf.Tensor, lambda_1: tf.Tensor, lambda_2: tf.Tensor, Kuu: Optional[tf.Tensor] = None, cholesky=True)[source]¶
From Kuu, Kuf, λ₁, λ₂, computes statistics L = Kuu⁻¹ (Σₙ Kufₙ λ₂ₙ Kfₙu) Kuu⁻¹ l = Kuu⁻¹ (Σₙ Kufₙ λ₁ₙ)
- Parameters
Kuf – L x M x N
lambda_2 – N x L
lambda_1 – N x L
Kuu – L x M x M
return: l : M x L L : L x M x M
- kl_from_precision_sites_white(A, l, L=None, L2=None)[source]¶
Computes the kl divergence KL[q(f)|p(f)]
where: q(f) = N(μ,Σ) and covariance of a Gaussian with natural parameters with
Σ = Λ⁻¹ = (A⁻¹ + A⁻¹LLᵀA⁻¹)⁻¹ = A(A + LLᵀ)⁻¹A μ = Λ⁻¹ A⁻¹l = Σ A⁻¹l
p(f) = N(0,A)
- kl_from_precision_sites(A, l, L=None, L2=None)[source]¶
Computes the kl divergence KL[q(f)|p(f)]
where: q(f) = N(μ,Σ) and covariance of a Gaussian with natural parameters with
Σ = Λ⁻¹ = (A⁻¹ + A⁻¹LLᵀA⁻¹)⁻¹ = A(A + LLᵀ)⁻¹A μ = Λ⁻¹ A⁻¹l = Σ A⁻¹l
p(f) = N(0,A)
- posterior_from_dense_site(K, lambda_1, lambda_2_sqrt)[source]¶
Returns the mean and cholesky factor of the density q(u) = p(u)t(u) = 𝓝(u; m, S) where p(u) = 𝓝(u; 0, K) and t(u) = exp(uᵀλ₁ - ½ uᵀΛ₂u)
P = -2Λ₂
S = (K⁻¹ + P)⁻¹ = (K⁻¹ + LLᵀ)⁻¹ = K - KLW⁻¹LᵀK , W = (I + LᵀKL) m = S λ₁
Input: :param: K : M x M :param: lambda_1: M x P :param: lambda_2: P x M x M
Output: m: M x P chol_S: P x M x M
- posterior_from_dense_site_white(K, lambda_1, lambda_2, jitter=1e-09)[source]¶
Returns the mean and cholesky factor of the density q(u) = p(u)t(u) = 𝓝(u; m, S) where p(u) = 𝓝(u; 0, K) and t(u) = exp(uᵀλ₁ - ½ uᵀΛ₂u)
S = Λ₂⁻¹ = (K⁻¹ + K⁻¹PK⁻¹)⁻¹ = = [K⁻¹(K + P)K⁻¹]⁻¹ = K(K + P)⁻¹K m = S K⁻¹λ₁ = K(K + LLᵀ)⁻¹λ₁
Input: :param: K : M x M :param: lambda_1: M x P :param: lambda_2: P x M x M
Output: m: M x P chol_S: P x M x M