.. _bda-missing-data-in-the-multivariate-normal-and-t-models:

========================================================================
Missing data in the multivariate normal and t models
========================================================================

**Part 4 · Stage 14 · 🛡️ Robustness & Missing Data**  ·  Lesson 120 of 144  ·  *advanced*

:doc:`◀ Previous · Multiple imputation <119-multiple-imputation>`   ·   :doc:`Next · Example: multiple imputation for a series of polls ▶ <121-example-multiple-imputation-for-a-series-of-polls>`   ·   :doc:`↑ Section <index>`


.. important::

   **✨ AI-generated content.** This page was written with the assistance of an
   AI language model and is provided as a learning aid. Despite careful
   review, it may still contain mistakes, omissions, or out-of-date
   information. Whether you are new to the topic, a team lead, or a senior
   practitioner, treat it as a starting point rather than an authoritative
   reference: read it critically and independently verify anything you act on
   (code, commands, figures, and factual claims) against official
   documentation and primary sources before relying on it.

Imputation from a joint model
-------------------------------

The cleanest setting for missing-data methods is a set of continuous variables modelled **jointly** as
multivariate normal. There the imputation is not a heuristic but a **direct consequence** of the model:
the conditional distribution of the missing entries given the observed ones is available in closed form,
so imputing is exact.

The multivariate normal case
------------------------------

Model the complete rows as :math:`y_i \sim \mathrm{N}(\mu, \Sigma)`. For a row with some entries missing,
partition into observed and missing parts; the conditional of the missing given the observed is again
normal, with the familiar regression form:

.. math::

   y_{\text{mis}} \mid y_{\text{obs}} \sim \mathrm{N}\bigl(
     \mu_m + \Sigma_{mo}\Sigma_{oo}^{-1}(y_{\text{obs}} - \mu_o), \;
     \Sigma_{mm} - \Sigma_{mo}\Sigma_{oo}^{-1}\Sigma_{om}\bigr).

The conditional mean **regresses** the missing entries on the observed ones through the covariance, and
the conditional variance is what a proper imputation must sample from — plugging in only the mean would
be single imputation, discarding exactly the spread this formula provides.

Computation by data augmentation
----------------------------------

Since :math:`\mu` and :math:`\Sigma` are themselves unknown, alternate — the Gibbs pattern of Stage 9,
here with missingness as the latent layer:

.. math::

   \text{(I-step) } y_{\text{mis}} \sim p(y_{\text{mis}} \mid y_{\text{obs}}, \mu, \Sigma), \qquad
   \text{(P-step) } \mu, \Sigma \sim p(\mu, \Sigma \mid y_{\text{obs}}, y_{\text{mis}}).

Impute the missing values given current parameters; update the parameters given the completed data;
repeat. Each cycle produces both a posterior draw of the parameters and a **set of imputations** —
multiple imputation and parameter estimation from one algorithm.

.. code-block:: python

   import pymc as pm
   import numpy as np
   # PyMC imputes automatically when observed data are a masked array
   y_masked = np.ma.masked_invalid(Y)                       # NaNs flagged as missing
   with pm.Model():
       mu = pm.Normal("mu", 0, 5, shape=Y.shape[1])
       chol, _, _ = pm.LKJCholeskyCov("chol", n=Y.shape[1], eta=2,
                                      sd_dist=pm.HalfNormal.dist(1.0), compute_corr=True)
       pm.MvNormal("y", mu=mu, chol=chol, observed=y_masked)   # missing entries become parameters
       idata = pm.sample()                                     # posterior includes the imputations

The t extension, and the limits
---------------------------------

Swap the multivariate normal for a **multivariate** :math:`t` and the imputation becomes **robust**: the
model resists letting an outlier in the observed part of a row distort the imputed values in the rest —
the heavy-tailed lesson applied to filling gaps. Two limits remain. Everything here assumes **MAR**; the
joint model integrates over missing values correctly only when the mechanism is ignorable. And the
approach wants roughly **continuous, jointly-modellable** variables — genuinely mixed data (categorical
with continuous) are handled by chained-equations imputation or by more elaborate joint models, but the
principle is unchanged: **model the variables together, and the conditional distribution of the missing
given the observed is the imputation.**

.. hint::

   **Related lessons:** :doc:`Multiple imputation <119-multiple-imputation>`  ·  :doc:`Notation <118-notation>`  ·  :doc:`Multivariate Normal with Unknown Mean and Variance <025-multivariate-normal-with-unknown-mean-and-variance>`  ·  :doc:`Robust regression using t-distributed errors <117-robust-regression-using-t-distributed-errors>`

.. seealso::

   **Source article** Adapted (context, re-expressed) in our own words from: `https://insightful-data-lab.com/2025/12/07/missing-data-in-the-multivariate-normal-and-t-models/ <https://insightful-data-lab.com/2025/12/07/missing-data-in-the-multivariate-normal-and-t-models/>`__ (insightful-data-lab.com).

.. tags:: purpose: reference, topic: data analysis, domain: bayesian, level: advanced