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| = Conditional Expectations = | | == Summary == |
| The '''conditional expectation''' of a random variable <math>Y</math> with respect to an event <math>\mathcal{E}</math> is defined by
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| :<math>
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| \mathbf{E}[Y\mid \mathcal{E}]=\sum_{y}y\Pr[Y=y\mid\mathcal{E}].
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| </math>
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| In particular, if the event <math>\mathcal{E}</math> is <math>X=a</math>, the conditional expectation
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| :<math>
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| \mathbf{E}[Y\mid X=a]
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| </math>
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| defines a function
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| :<math>
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| f(a)=\mathbf{E}[Y\mid X=a].
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| </math>
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| Thus, <math>\mathbf{E}[Y\mid X]</math> can be regarded as a random variable <math>f(X)</math>.
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| ;Example
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| :Suppose that we uniformly sample a human from all human beings. Let <math>Y</math> be his/her height, and let <math>X</math> be the country where he/she is from. For any country <math>a</math>, <math>\mathbf{E}[Y\mid X=a]</math> gives the average height of that country. And <math>\mathbf{E}[Y\mid X]</math> is the random variable which can be defined in either ways:
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| :* We choose a human uniformly at random from all human beings, and <math>\mathbf{E}[Y\mid X]</math> is the average height of the country where he/she comes from.
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| :* We choose a country at random with a probability ''proportional to its population'', and <math>\mathbf{E}[Y\mid X]</math> is the average height of the chosen country.
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| The following proposition states some fundamental facts about conditional expectation.
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| {{Theorem
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| |Proposition (fundamental facts about conditional expectation)|
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| :Let <math>X,Y</math> and <math>Z</math> be arbitrary random variables. Let <math>f</math> and <math>g</math> be arbitrary functions. Then
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| :# <math>\mathbf{E}[X]=\mathbf{E}[\mathbf{E}[X\mid Y]]</math>.
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| :# <math>\mathbf{E}[X\mid Z]=\mathbf{E}[\mathbf{E}[X\mid Y,Z]\mid Z]</math>.
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| :# <math>\mathbf{E}[\mathbf{E}[f(X)g(X,Y)\mid X]]=\mathbf{E}[f(X)\cdot \mathbf{E}[g(X,Y)\mid X]]</math>.
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| }}
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| The proposition can be formally verified by computing these expectations. Although these equations look formal, the intuitive interpretations to them are very clear.
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| The first equation:
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| :<math>\mathbf{E}[X]=\mathbf{E}[\mathbf{E}[X\mid Y]]</math>
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| says that there are two ways to compute an average. Suppose again that <math>X</math> is the height of a uniform random human and <math>Y</math> is the country where he/she is from. There are two ways to compute the average human height: one is to directly average over the heights of all humans; the other is that first compute the average height for each country, and then average over these heights weighted by the populations of the countries.
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| The second equation:
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| :<math>\mathbf{E}[X\mid Z]=\mathbf{E}[\mathbf{E}[X\mid Y,Z]\mid Z]</math>
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| is the same as the first one, restricted to a particular subspace. As the previous example, inaddition to the height <math>X</math> and the country <math>Y</math>, let <math>Z</math> be the gender of the individual. Thus, <math>\mathbf{E}[X\mid Z]</math> is the average height of a human being of a given sex. Again, this can be computed either directly or on a country-by-country basis.
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| The third equation:
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| :<math>\mathbf{E}[\mathbf{E}[f(X)g(X,Y)\mid X]]=\mathbf{E}[f(X)\cdot \mathbf{E}[g(X,Y)\mid X]]</math>.
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| looks obscure at the first glance, especially when considering that <math>X</math> and <math>Y</math> are not necessarily independent. Nevertheless, the equation follows the simple fact that conditioning on any <math>X=a</math>, the function value <math>f(X)=f(a)</math> becomes a constant, thus can be safely taken outside the expectation due to the linearity of expectation. For any value <math>X=a</math>,
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| :<math>
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| \mathbf{E}[f(X)g(X,Y)\mid X=a]=\mathbf{E}[f(a)g(X,Y)\mid X=a]=f(a)\cdot \mathbf{E}[g(X,Y)\mid X=a].
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| </math>
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| The proposition holds in more general cases when <math>X, Y</math> and <math>Z</math> are a sequence of random variables.
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| = Martingales =
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| A '''martingale''' is a random sequence <math>X_0,X_1,\ldots</math> satisfying the following so-called ''martingale property''.
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| {{Theorem
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| |Definition (martingale)|
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| :A sequence of random variables <math>X_0,X_1,\ldots</math> is a '''martingale''' if for all <math>i> 0</math>,
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| :: <math>\begin{align}
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| \mathbf{E}[X_{i}\mid X_0,\ldots,X_{i-1}]=X_{i-1}.
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| \end{align}</math>
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| }}
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| ==Examples ==
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| ;coin flips
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| :A fair coin is flipped for a number of times. Let <math>Z_j\in\{-1,1\}</math> denote the outcome of the <math>j</math>th flip. Let
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| ::<math>X_0=0\quad \mbox{ and } \quad X_i=\sum_{j\le i}Z_j</math>.
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| :The random variables <math>X_0,X_1,\ldots</math> defines a martingale.
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| {{Proof| We first observe that <math>\mathbf{E}[X_i\mid X_0,\ldots,X_{i-1}] = \mathbf{E}[X_i\mid X_{i-1}]</math>, which intuitively says that the next number of HEADs depends only on the current number of HEADs. This property is also called the '''Markov property''' in statistic processes.
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| ::<math>
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| \begin{align}
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| \mathbf{E}[X_i\mid X_0,\ldots,X_{i-1}]
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| &= \mathbf{E}[X_i\mid X_{i-1}]\\
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| &= \mathbf{E}[X_{i-1}+Z_{i}\mid X_{i-1}]\\
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| &= \mathbf{E}[X_{i-1}\mid X_{i-1}]+\mathbf{E}[Z_{i}\mid X_{i-1}]\\
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| &= X_{i-1}+\mathbf{E}[Z_{i}\mid X_{i-1}]\\
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| &= X_{i-1}+\mathbf{E}[Z_{i}] &\quad (\mbox{independence of coin flips})\\
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| &= X_{i-1}
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| \end{align}
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| </math>
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| }}
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| ;edge exposure in a random graph
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| :Consider a '''random graph''' <math>G</math> generated as follows. Let <math>[n]</math> be the set of vertices, and let <math>[m]={[n]\choose 2}</math> be the set of all possible edges. For convenience, we enumerate these potential edges by <math>e_1,\ldots, e_m</math>. For each potential edge <math>e_j</math>, we independently flip a fair coin to decide whether the edge <math>e_j</math> appears in <math>G</math>. Let <math>I_j</math> be the random variable that indicates whether <math>e_j\in G</math>. We are interested in some graph-theoretical parameter, say [http://mathworld.wolfram.com/ChromaticNumber.html chromatic number], of the random graph <math>G</math>. Let <math>\chi(G)</math> be the chromatic number of <math>G</math>. Let <math>X_0=\mathbf{E}[\chi(G)]</math>, and for each <math>i\ge 1</math>, let <math>X_i=\mathbf{E}[\chi(G)\mid I_1,\ldots,I_{i}]</math>, namely, the expected chromatic number of the random graph after fixing the first <math>i</math> edges. This process is called edges exposure of a random graph, as we "exposing" the edges one by one in a random graph.
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| It is nontrivial to formally verify that the edge exposure sequence for a random graph is a martingale. However, we will later see that this construction can be put into a more general context.
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| ==Generalization ==
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| The martingale can be generalized to be with respect to another sequence of random variables.
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| {{Theorem
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| |Definition (martingale, general version)|
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| :A sequence of random variables <math>Y_0,Y_1,\ldots</math> is a martingale with respect to the sequence <math>X_0,X_1,\ldots</math> if, for all <math>i\ge 0</math>, the following conditions hold:
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| :* <math>Y_i</math> is a function of <math>X_0,X_1,\ldots,X_i</math>;
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| :* <math>\begin{align}
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| \mathbf{E}[Y_{i+1}\mid X_0,\ldots,X_{i}]=Y_{i}.
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| \end{align}</math>
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| }}
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| Therefore, a sequence <math>X_0,X_1,\ldots</math> is a martingale if it is a martingale with respect to itself.
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| The purpose of this generalization is that we are usually more interested in a function of a sequence of random variables, rather than the sequence itself.
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| The following definition describes a very general approach for constructing an important type of martingales.
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| {{Theorem
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| |Definition (The Doob sequence)|
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| : The Doob sequence of a function <math>f</math> with respect to a sequence of random variables <math>X_1,\ldots,X_n</math> is defined by
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| ::<math>
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| Y_i=\mathbf{E}[f(X_1,\ldots,X_n)\mid X_1,\ldots,X_{i}], \quad 0\le i\le n.
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| </math>
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| :In particular, <math>Y_0=\mathbf{E}[f(X_1,\ldots,X_n)]</math> and <math>Y_n=f(X_1,\ldots,X_n)</math>.
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| }}
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| The Doob sequence of a function defines a martingale. That is
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| ::<math>
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| \mathbf{E}[Y_i\mid X_1,\ldots,X_{i-1}]=Y_{i-1},
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| </math>
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| for any <math>0\le i\le n</math>.
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| To prove this claim, we recall the definition that <math>Y_i=\mathbf{E}[f(X_1,\ldots,X_n)\mid X_1,\ldots,X_{i}]</math>, thus,
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| :<math>
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| \begin{align}
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| \mathbf{E}[Y_i\mid X_1,\ldots,X_{i-1}]
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| &=\mathbf{E}[\mathbf{E}[f(X_1,\ldots,X_n)\mid X_1,\ldots,X_{i}]\mid X_1,\ldots,X_{i-1}]\\
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| &=\mathbf{E}[f(X_1,\ldots,X_n)\mid X_1,\ldots,X_{i-1}]\\
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| &=Y_{i-1},
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| \end{align}
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| </math>
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| where the second equation is due to the fundamental fact about conditional expectation introduced in the first section.
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| The Doob martingale describes a very natural procedure to determine a function value of a sequence of random variables. Suppose that we want to predict the value of a function <math>f(X_1,\ldots,X_n)</math> of random variables <math>X_1,\ldots,X_n</math>. The Doob sequence <math>Y_0,Y_1,\ldots,Y_n</math> represents a sequence of refined estimates of the value of <math>f(X_1,\ldots,X_n)</math>, gradually using more information on the values of the random variables <math>X_1,\ldots,X_n</math>. The first element <math>Y_0</math> is just the expectation of <math>f(X_1,\ldots,X_n)</math>. Element <math>Y_i</math> is the expected value of <math>f(X_1,\ldots,X_n)</math> when the values of <math>X_1,\ldots,X_{i}</math> are known, and <math>Y_n=f(X_1,\ldots,X_n)</math> when <math>f(X_1,\ldots,X_n)</math> is fully determined by <math>X_1,\ldots,X_n</math>.
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| The following two Doob martingales arise in evaluating the parameters of random graphs.
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| ===edge exposure martingale===
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| :Let <math>G</math> be a random graph on <math>n</math> vertices. Let <math>f</math> be a real-valued function of graphs, such as, chromatic number, number of triangles, the size of the largest clique or independent set, etc. Denote that <math>m={n\choose 2}</math>. Fix an arbitrary numbering of potential edges between the <math>n</math> vertices, and denote the edges as <math>e_1,\ldots,e_m</math>. Let
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| ::<math>
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| X_i=\begin{cases}
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| 1& \mbox{if }e_i\in G,\\
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| 0& \mbox{otherwise}.
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| \end{cases}
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| </math>
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| :Let <math>Y_0=\mathbf{E}[f(G)]</math> and for <math>i=1,\ldots,m</math>, let <math>Y_i=\mathbf{E}[f(G)\mid X_1,\ldots,X_i]</math>.
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| :The sequence <math>Y_0,Y_1,\ldots,Y_n</math> gives a Doob martingale that is commonly called the '''edge exposure martingale'''.
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| ===vertex exposure martingale===
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| : Instead of revealing edges one at a time, we could reveal the set of edges connected to a given vertex, one vertex at a time. Suppose that the vertex set is <math>[n]</math>. Let <math>X_i</math> be the subgraph of <math>G</math> induced by the vertex set <math>[i]</math>, i.e. the first <math>i</math> vertices.
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| :Let <math>Y_0=\mathbf{E}[f(G)]</math> and for <math>i=1,\ldots,n</math>, let <math>Y_i=\mathbf{E}[f(G)\mid X_1,\ldots,X_i]</math>.
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| :The sequence <math>Y_0,Y_1,\ldots,Y_n</math> gives a Doob martingale that is commonly called the '''vertex exposure martingale'''.
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