Conditional Laws of Multivariate Skew-Normal Vectors

Why Conditioning Changes the Skew-Normal Law

For a Gaussian vector, conditioning on one block changes the location and covariance of the remaining block but leaves the family Gaussian. A skew-normal vector is different: its asymmetry comes from selecting a joint Gaussian vector according to the sign of a latent variable. Observing part of the vector updates the distribution of that latent variable as well as the Gaussian block. The resulting conditional law is generally an extended skew-normal, with an additional extension parameter.

The derivation below makes this update explicit. It also separates the parameters of the conditional normal kernel from the actual conditional mean and covariance, which need not equal those kernel parameters.

Latent Gaussian Representation

To obtain the density, condition the joint Gaussian law on \(X=x\). Then \(H\mid X=x\) is normal with mean \(q^\mathsf{T}\Omega^{-1}(x-\xi)\) and variance \(1-q^\mathsf{T}\Omega^{-1}q\). Thus the probability of selection at \(x\) is the displayed \(\Phi\) term. Since \(P(H>0)=1/2\), conditioning on selection multiplies the joint marginal density of \(X\) by 2.

For comparison with the usual shape-vector notation, let \(\omega\) be diagonal with entries \(\sqrt{\Omega_{jj}}\). The shape vector in the density’s standardized argument is \[ \alpha=\frac{\omega\Omega^{-1}q}{\sqrt{1-q^\mathsf{T}\Omega^{-1}q}}. \] Indeed, \(\alpha^\mathsf{T}\omega^{-1}(x-\xi)\) equals the argument of \(\Phi\) above. The latent covariance vector \(q\) is often more convenient for conditioning.

Conditional Law After Observing a Block

Partition \(X=(X_1^\mathsf{T},X_2^\mathsf{T})^\mathsf{T}\) and conformably partition \(\xi,\Omega,q\). Write \(X_2=x_2\) for the observed block. Define the Gaussian conditional parameters and the updated latent quantities by

These are obtained by conditioning the joint Gaussian vector \((X_1,X_2,H)\) on \(X_2=x_2\). In particular, \(X_1\mid X_2=x_2\) has mean \(\mu\) and covariance \(\Sigma\); \(H\mid X_2=x_2\) has mean \(m\) and variance \(s^2\); and their conditional cross-covariance is \(c\). The positive definiteness of the original joint covariance ensures \(s^2>0\) and \(v>0\).

Proof. Conditional on \(X_2=x_2\), the pair \((X_1,H)\) is jointly Gaussian with the means, covariance and cross-covariance just given. Conditional again on \(X_1=y\), the latent variable \(H\) has mean \(m+c^\mathsf{T}\Sigma^{-1}(y-\mu)\) and variance \(v=s^2-c^\mathsf{T}\Sigma^{-1}c\). Therefore its probability of being positive is the numerator’s \(\Phi\) term. Dividing the joint density of \(X_1=y\) and \(H>0\), conditional on \(X_2=x_2\), by \(P(H>0\mid X_2=x_2)=\Phi(m/s)\) gives the stated conditional density.

For the parameter form, put \(z=\omega_c^{-1}(y-\mu)\). Then \(c^\mathsf{T}\Sigma^{-1}(y-\mu)/\sqrt v=\alpha_c^\mathsf{T}z\). Also, \[ 1+\alpha_c^\mathsf{T}R_c\alpha_c =1+\frac{c^\mathsf{T}\Sigma^{-1}c}{v} =\frac{s^2}{v}, \] so \(\tau_c/\sqrt{1+\alpha_c^\mathsf{T}R_c\alpha_c}=m/s\), as required. \(\square\)

Conditional Moments Are Not Kernel Parameters

The kernel parameters \(\mu\) and \(\Sigma\) describe the normal density multiplied by the selection factor; they are not usually the conditional mean and covariance. Let \(a=m/s\) and \(r=\phi(a)/\Phi(a)\), where \(\phi\) is the standard normal density. The latent variable \(H\), conditional on \(X_2=x_2\) and \(H>0\), is a normal variable with mean \(m\) and variance \(s^2\), truncated below at zero. Its mean is \(m+sr\), and its variance is \(s^2(1-ar-r^2)\).

Derivation. Given \(X_2=x_2\), Gaussian regression writes \(X_1=\mu+(c/s^2)(H-m)+\varepsilon\), where \(\varepsilon\) is independent of \(H\) and has covariance \(\Sigma-cc^\mathsf{T}/s^2\). Taking the expectation after truncating \(H\) gives \(\mu+(c/s^2)sr=\mu+(c/s)r\). For the covariance, add the covariance of the regression term to that of \(\varepsilon\): \[ \Sigma-\frac{cc^\mathsf{T}}{s^2} +\frac{cc^\mathsf{T}}{s^4}s^2(1-ar-r^2) =\Sigma-\frac{cc^\mathsf{T}}{s^2}(ar+r^2). \] This proves both formulas. \(\square\)

Common Pitfall and Practical Check

A common mistake is to treat the conditional skew-normal as an ordinary skew-normal with only a changed location and shape, setting the extension parameter to zero. That is valid in special cases, such as \(m=0\), but not generally: after observing \(X_2=x_2\), the latent Gaussian variable has conditional mean \(m\), which need not be zero. Omitting this updated truncation point changes both the density’s normalizer and its moments.

A second mistake is to report \(\mu\) and \(\Sigma\) as the actual conditional mean and covariance. They are the Gaussian-kernel parameters. The selection factor shifts the mean and reduces or otherwise modifies the covariance according to the truncated-latent moments. A reliable calculation therefore checks the updated \(m,s^2,c,v\), retains \(\Phi(m/s)\) in the normalizer, and applies the moment formulas only afterward.