When data is missing some values for some features, the missing data cannot be ignored. Example 10.13 gives an example where ignoring missing data leads to wrong conclusions. Making inference from missing data is a causality problem, as it cannot be solved by observation, but requires a causal model.

A missingness graph, or m-graph for short, is used to model data where some values might be missing. Start with a belief network model of the domain. In the m-graph, all variables in the original graph exist with the same parents. For each variable $V$ that could be observed with some values missing, the m-graph contains two extra variables:

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  $M\mathrm{\_}V$, a Boolean variable that is true when $V$’s value is missing. The parents of this node can be whatever variables the missingness is assumed to depend on.

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  A variable $V^{\ast }$, with domain $dom(V)\cup \{missing\}$, where $missing$ is a new value (not in the domain of $V$). The only parents of $V^{\ast }$ are $V$ and $M\mathrm{\_}V$. The conditional probability table contains only 0 and 1, with the 1s being

  	$P(V^{\ast }=missing\mid M\mathrm{\_}V=true)=1$
  	$P(V^{\ast }=v\mid M\mathrm{\_}V=false\wedge V=v)=1.$

If the value of $V$ is observed to be $v$, then $V^{\ast }=v$ is conditioned on. If the value for $V$ is missing, $V^{\ast }=missing$ is conditioned on. Note that $V^{\ast }$ is always observed and conditioned on, and $V$ is never conditioned on, in this augmented model. When modeling a domain, the parents of $M\mathrm{\_}V$ specify what the missingness depends on.

A distribution is recoverable or identifiable from missing data if the distribution can be accurately measured from the data, even with parts of the data missing. Whether a distribution is recoverable is a property of the underlying graph. A distribution that is not recoverable cannot be reconstructed from observational data, no matter how large the dataset. The distribution in Example 11.5 is not recoverable.

Data is missing completely at random (MCAR) if $V$ and $M\mathrm{\_}V$ are independent. If the data is missing completely at random, the examples with missing values can be ignored. This is a strong assumption that rarely occurs in practice, but is often implicitly assumed when missingness is ignored.

A weaker assumption is that a variable $Y$ is missing at random (MAR), which occurs when $Y$ is independent of $M\mathrm{\_}Y$ given the observed variables $V_o$. That is, when $P(Y\mid V_o,M\mathrm{\_}Y)=P(Y\mid V_o)$. This occurs when the reason the data is missing can be observed. The distribution over $Y$ and the observed variables is recoverable by $P(Y,V_o)=P(Y\mid V_o,M\mathrm{\_}Y=false)P(V_o)$. Thus, the non-missing data is used to estimate $P(Y\mid V_o)$ and all of the data is used to estimate $P(V_o)$.

The main points to remember are:

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  You cannot learn from missing data without making modeling assumptions.

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  Some distributions are not recoverable from missing data, and some are. It depends on the independence structure of the underlying graph.

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  If the distribution is not recoverable, a learning algorithm still may be able to learn parameters, but the resulting distributions should not be trusted.