Editing Conditional expectation (section)

==== Connections to regression ====

The conditional expectation is often approximated in [[applied mathematics]] and [[statistics]] due to the difficulties in analytically calculating it, and for interpolation.<ref>{{cite book |last1=Hastie |first1=Trevor |title=The elements of statistical learning : data mining, inference, and prediction |date=26 August 2009 |location=New York |isbn=978-0-387-84858-7 |edition=Second, corrected 7th printing |url=https://web.stanford.edu/~hastie/Papers/ESLII.pdf}}</ref>

The Hilbert subspace
:<math> M = \{ g(Y) : \operatorname{E}(g(Y)^2) < \infty \}</math> 
defined above is replaced with subsets thereof by restricting the functional form of {{mvar|g}}, rather than allowing any measurable function. Examples of this are [[Decision tree learning|decision tree regression]] when {{mvar|g}} is required to be a [[simple function]], [[linear regression]] when {{mvar|g}} is required to be [[affine transformation|affine]], etc.

These generalizations of conditional expectation come at the cost of many of [[Conditional expectation#Basic properties|its properties]] no longer holding.
For example, let {{mvar|M}}
be the space  of all linear functions of {{mvar|Y}} and let <math>\mathcal{E}_{M}</math> denote this generalized conditional expectation/<math>L^2</math> projection. If <math>M</math> does not contain the [[constant function]]s, the [[tower property]] 
<math> \operatorname{E}(\mathcal{E}_M(X)) = \operatorname{E}(X) </math>
will not hold.

An important special case is when {{mvar|X}} and {{mvar|Y}} are jointly normally distributed. In this case
it can be shown that the conditional expectation is equivalent to linear regression:
:<math> e_X(Y) = \alpha_0 + \sum_i \alpha_i Y_i</math>
for coefficients <math>\{\alpha_i\}_{i = 0..n}</math> described in [[Multivariate normal distribution#Conditional distributions]].