**Conditional Probability**

The conditional probability P ( *Y* ≤ 1/3 | *X* ) may be defined as the best predictor of the indicator

given *X*. That is, it minimizes the mean square error E ( *I* - *g*(*X*) )2 on the class of all random variables of the form *g* (*X*).

In the case *f* = *f*_{1} the corresponding function *g* = *g*_{1} may be calculated explicitly,

Alternatively, the limiting procedure may be used,

giving the same result.

Thus, P ( *Y* ≤ 1/3 | *X* ) = *g*_{1} (*X*). The expectation of this random variable is equal to the (unconditional) probability, E ( P ( *Y* ≤ 1/3 | *X* ) ) = P ( *Y* ≤ 1/3 ), namely,

which is an instance of the law of total probability E ( P ( *A* | *X* ) ) = P ( *A* ).

In the case *f* = *f*_{2} the corresponding function *g* = *g*_{2} probably cannot be calculated explicitly. Nevertheless it exists, and can be computed numerically. Indeed, the space L_{2} (Ω) of all square integrable random variables is a Hilbert space; the indicator *I* is a vector of this space; and random variables of the form *g* (*X*) are a (closed, linear) subspace. The orthogonal projection of this vector to this subspace is well-defined. It can be computed numerically, using finite-dimensional approximations to the infinite-dimensional Hilbert space.

Once again, the expectation of the random variable P ( *Y* ≤ 1/3 | *X* ) = *g*_{2} (*X*) is equal to the (unconditional) probability, E ( P ( *Y* ≤ 1/3 | *X* ) ) = P ( *Y* ≤ 1/3 ), namely,

However, the Hilbert space approach treats *g*_{2} as an equivalence class of functions rather than an individual function. Measurability of *g*_{2} is ensured, but continuity (or even Riemann integrability) is not. The value *g*_{2} (0.5) is determined uniquely, since the point 0.5 is an atom of the distribution of *X*. Other values *x* are not atoms, thus, corresponding values *g*_{2} (*x*) are not determined uniquely. Once again, "*the concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is inadmissible.*" (Kolmogorov; quoted in ).

Alternatively, the same function *g* (be it *g*_{1} or *g*_{2}) may be defined as the Radon–Nikodym derivative

where measures μ, ν are defined by

for all Borel sets That is, μ is the (unconditional) distribution of *X*, while ν is one third of its conditional distribution,

Both approaches (via the Hilbert space, and via the Radon–Nikodym derivative) treat *g* as an equivalence class of functions; two functions *g* and *g′* are treated as equivalent, if *g* (*X*) = *g′* (*X*) almost surely. Accordingly, the conditional probability P ( *Y* ≤ 1/3 | *X* ) is treated as an equivalence class of random variables; as usual, two random variables are treated as equivalent if they are equal almost surely.

Read more about this topic: Conditioning (probability), Conditioning On The Level of Measure Theory

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