Index: The Book of Statistical ProofsProbability Distributions ▷ Multivariate continuous distributions ▷ Multivariate normal distribution ▷ Conditional distributions

Theorem: Let $x$ follow a multivariate normal distribution

\[\label{eq:mvn} x \sim \mathcal{N}(\mu, \Sigma) \; .\]

Then, the conditional distribution of any subset vector $x_1$, given the complement vector $x_2$, is also a multivariate normal distribution

\[\label{eq:mvn-cond} x_1|x_2 \sim \mathcal{N}(\mu_{1|2}, \Sigma_{1|2})\]

where the conditional mean and covariance are

\[\label{eq:mvn-cond-hyp} \begin{split} \mu_{1|2} &= \mu_1 + \Sigma_{12} \Sigma_{22}^{-1} (x_2 - \mu_2) \\ \Sigma_{1|2} &= \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} \end{split}\]

with block-wise mean and covariance defined as

\[\label{eq:mvn-joint-hyp} \begin{split} \mu &= \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \\ \Sigma &= \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix} \; . \end{split}\]

Proof: Without loss of generality, we assume that, in parallel to \eqref{eq:mvn-joint-hyp},

\[\label{eq:x} x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\]

where $x_1$ is an $n_1 \times 1$ vector, $x_2$ is an $n_2 \times 1$ vector and $x$ is an $n_1 + n_2 = n \times 1$ vector.

By construction, the joint distribution of $x_1$ and $x_2$ is:

\[\label{eq:mvn-joint} x_1,x_2 \sim \mathcal{N}(\mu, \Sigma) \; .\]

Moreover, the marginal distribution of $x_2$ follows from \eqref{eq:mvn} and \eqref{eq:mvn-joint-hyp} as

\[\label{eq:mvn-marg} x_2 \sim \mathcal{N}(\mu_2, \Sigma_{22}) \; .\]

According to the law of conditional probability, it holds that

\[\label{eq:mvn-cond-s1} p(x_1|x_2) = \frac{p(x_1,x_2)}{p(x_2)}\]

Applying \eqref{eq:mvn-joint} and \eqref{eq:mvn-marg} to \eqref{eq:mvn-cond-s1}, we have:

\[\label{eq:mvn-cond-s2} p(x_1|x_2) = \frac{\mathcal{N}(x; \mu, \Sigma)}{\mathcal{N}(x_2; \mu_2, \Sigma_{22})} \; .\]

Using the probability density function of the multivariate normal distribution, this becomes:

\[\label{eq:mvn-cond-s3} \begin{split} p(x_1|x_2) &= \frac{1/\sqrt{(2 \pi)^n |\Sigma|} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right]}{1/\sqrt{(2 \pi)^{n_2} |\Sigma_{22}|} \cdot \exp \left[ -\frac{1}{2} (x_2-\mu_2)^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right]} \\ &= \frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \exp \left[ -\frac{1}{2} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) + \frac{1}{2} (x_2-\mu_2)^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right] \; . \end{split}\]

Writing the inverse of $\Sigma$ as

\[\label{eq:Sigma-inv-def} \Sigma^{-1} = \begin{bmatrix} \Sigma^{11} & \Sigma^{12} \\ \Sigma^{21} & \Sigma^{22} \end{bmatrix}\]

and applying \eqref{eq:mvn-joint-hyp} to \eqref{eq:mvn-cond-s3}, we get:

\[\label{eq:mvn-cond-s4} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \\ &\exp \left[ -\frac{1}{2} \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} - \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \right)^\mathrm{T} \begin{bmatrix} \Sigma^{11} & \Sigma^{12} \\ \Sigma^{21} & \Sigma^{22} \end{bmatrix} \left( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} - \begin{bmatrix} \mu_1 \\ \mu_2 \end{bmatrix} \right) \right. \\ &\hphantom{\exp \left[\right.} \left. + \frac{1}{2} \, (x_2-\mu_2)^\mathrm{T} \, \Sigma_{22}^{-1} \, (x_2-\mu_2) \right] \; . \end{split}\]

Multiplying out within the exponent of \eqref{eq:mvn-cond-s4}, we have

\[\label{eq:mvn-cond-s5} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \\ &\exp \left[ -\frac{1}{2} \left( (x_1-\mu_1)^\mathrm{T} \Sigma^{11} (x_1-\mu_1) + 2 (x_1-\mu_1)^\mathrm{T} \Sigma^{12} (x_2-\mu_2) + (x_2-\mu_2)^\mathrm{T} \Sigma^{22} (x_2-\mu_2) \right) \right. \\ &\hphantom{\exp \left[\right.} \left. + \frac{1}{2} (x_2-\mu_2)^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right] \end{split}\]

where we have used the fact that ${\Sigma^{21}}^\mathrm{T} = \Sigma^{12}$, because $\Sigma^{-1}$ is a symmetric matrix.

The inverse of a block matrix is

\[\label{eq:Block-inv} \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} (A-BD^{-1}C)^{-1} & -(A-BD^{-1}C)^{-1}BD^{-1} \\ -D^{-1}C(A-BD^{-1}C)^{-1} & D^{-1}+D^{-1}C(A-BD^{-1}C)^{-1}BD^{-1} \end{bmatrix} \; ,\]

thus the inverse of $\Sigma$ in \eqref{eq:Sigma-inv-def} is

\[\label{eq:Sigma-inv} \begin{bmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{bmatrix}^{-1} = \begin{bmatrix} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} & -(\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} \\ -\Sigma_{22}^{-1} \Sigma_{21} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} & \Sigma_{22}^{-1} + \Sigma_{22}^{-1} \Sigma_{21} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} \end{bmatrix} \; .\]

Plugging this into \eqref{eq:mvn-cond-s5}, we have:

\[\label{eq:mvn-cond-s6} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \\ &\exp \left[ -\frac{1}{2} \left( (x_1-\mu_1)^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} (x_1-\mu_1) \right. \right. - \\ &\hphantom{\exp \left[ -\frac{1}{2} \left( \right. \right.} 2 (x_1-\mu_1)^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} (x_2-\mu_2) + \\ &\hphantom{\exp \left[ -\frac{1}{2} \left( \right. \right.} \left. (x_2-\mu_2)^\mathrm{T} \left[ \Sigma_{22}^{-1} + \Sigma_{22}^{-1} \Sigma_{21} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} \right] (x_2-\mu_2) \right) \\ &\hphantom{\exp \left[\right.} \left. + \frac{1}{2} \left( (x_2-\mu_2)^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right] \; . \end{split}\]

Eliminating some terms, we have:

\[\label{eq:mvn-cond-s7} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \\ &\exp \left[ -\frac{1}{2} \left( (x_1-\mu_1)^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} (x_1-\mu_1) \right. \right. - \\ &\hphantom{\exp \left[ -\frac{1}{2} \left( \right. \right.} 2 (x_1-\mu_1)^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} (x_2-\mu_2) + \\ &\hphantom{\exp \left[ -\frac{1}{2} \left( \right. \right.} \left. \left. (x_2-\mu_2)^\mathrm{T} \Sigma_{22}^{-1} \Sigma_{21} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \Sigma_{12} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right] \; . \end{split}\]

Rearranging the terms, we have

\[\label{eq:mvn-cond-s8} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \exp \left[ -\frac{1}{2} \cdot \right. \\ &\! \left. \left[ (x_1-\mu_1) - \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right]^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \left[ (x_1-\mu_1) - \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right] \right] \\ = &\frac{1}{\sqrt{(2 \pi)^{n-n_2}}} \cdot \sqrt{\frac{|\Sigma_{22}|}{|\Sigma|}} \cdot \exp \left[ -\frac{1}{2} \cdot \right. \\ &\! \left. \left[ x_1 - \left( \mu_1 + \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right]^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \left[ x_1 - \left( \mu_1 + \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right] \right] \end{split}\]

where we have used the fact that $\Sigma_{21}^\mathrm{T} = \Sigma_{12}$, because $\Sigma$ is a covariance matrix.

The determinant of a block matrix is

\[\label{eq:Block-det} \begin{vmatrix} A & B \\ C & D \end{vmatrix} = |D| \cdot | A - B D^{-1} C | \; ,\]

such that we have for $\Sigma$ that

\[\label{eq:Sigma-det} \begin{vmatrix} \Sigma_{11} & \Sigma_{12} \\ \Sigma_{21} & \Sigma_{22} \end{vmatrix} = |\Sigma_{22}| \cdot | \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} | \; .\]

With this and $n - n_2 = n_1$, we finally arrive at

\[\label{eq:mvn-cond-s9} \begin{split} p(x_1|x_2) = &\frac{1}{\sqrt{(2 \pi)^{n_1} | \Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21} |}} \cdot \exp \left[ -\frac{1}{2} \cdot \right. \\ &\! \left. \left[ x_1 - \left( \mu_1 + \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right]^\mathrm{T} (\Sigma_{11} - \Sigma_{12} \Sigma_{22}^{-1} \Sigma_{21})^{-1} \left[ x_1 - \left( \mu_1 + \Sigma_{12}^\mathrm{T} \Sigma_{22}^{-1} (x_2-\mu_2) \right) \right] \right] \end{split}\]

which is the probability density function of a multivariate normal distribution

\[\label{eq:mvn-cond-s10} p(x_1|x_2) = \mathcal{N}(x_1; \mu_{1|2}, \Sigma_{1|2})\]

with the mean $\mu_{1 \vert 2}$ and variance $\Sigma_{1 \vert 2}$ given by \eqref{eq:mvn-cond-hyp}.


Metadata: ID: P88 | shortcut: mvn-cond | author: JoramSoch | date: 2020-03-20, 08:44.