25x + 0. 2338 . 42Analogous to the law of total probability, the law of total expectation
provides a way of computing an expected value by breaking down a problem into various cases, computing the conditional expected value given each case, and then computing the overall expected value as a probability-weighted average of these case-by-case conditional expected values. Let L (for traffic light) be a discrete random variable taking one value from {Red, Yellow, Green}. 0000 .
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k.
click to investigate The formula for the conditional mean of
given
is a straightforward implementation of the above informal definition: the
weights of the average are given by the
conditional
probability mass function of
.
From the above sections, it should be clear that the conditional expectation
is computed exactly as the expected value, with the only difference that
probabilities and probability densities are replaced by conditional
probabilities and conditional probability densities. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.
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\[\begin{align*}
\text{Discrete $X, Y$ with conditional pmf $p_{Y|X}$:} \textrm{E}(Y|X=x) = \sum_y y p_{Y|X}(y|x)\\
\text{Continuous $X, Y$ with conditional pdf $f_{Y|X}$:} \textrm{E}(Y|X=x) =\int_{-\infty}^\infty y f_{Y|X}(y|x) dy
\end{align*}\]Remember, when conditioning on \(X=x\), \(x\) is treated as a fixed constant. d. e.
\[
\textrm{P}(\textrm{E}(X|Y) \le 5) = \textrm{P}(1. conditional expectation a. 101 P= .
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The conditional pmf of \(X\) given \(Y=4\) places probability 2/7 on each of the values 5, 6, 7, and 1/7 on the value 8. 4778 . f.
In other words,
is a random variable such that its realization equals
when
is the realization of
. Therefore \(\textrm{E}(g(X)Y|X=x) = \textrm{E}(g(x)Y|X=x)= g(x)\textrm{E}(Y|X=x)\), where \(g(x)\) pops out of the expected value since it is just a number.
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3. The conditional pmf of \(Y\) given \(X=5\) places probability 1/2 on the value 4 and 1/2 on the value 3. The “baseline Shapley” and “randomized baseline Shapley” in that paper are identical to SHAP with feature_perturbation=”interventional”. This option is not exactly a causal explanation (as defined in that paper), but it is the best you can do without a background dataset, and importantly it does not suffer from the problem of putting weight on feature not explicitly used by the model.
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To help clarify that SHAP aligns with casual interventional perturbations (as described in that paper) I have actually renamed the feature_dependence=”independent” option to feature_perturbation=”interventional”. 1299 1.
The pink dots in the center of the slices represent the conditional averages, \(\textrm{E}(Y|R = r)\) for the different values of \(r\).
Remember that given \(X=x\), the random variable \(X\) is treated as the fixed constant \(x\). 1 In works of Paul Halmos2 and Joseph L.
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1737 1. 5)/1. Let H be a discrete random variable taking one value from {Hit, NotHit}. I guess that feature_perturbation = ‘tree_path_dependent’ is also useful when your aim is to understand the decision-making process of a model whose training data is unavailable.
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23, displayed in the table below. Shapley values are computed by introducing each feature, one at a time, into a conditional expectation function of the model’s output, f_x (S)=E[f(X)│do(X_S=x_S ) ], and attributing the change produced at each step to the feature that was introduced; then averaging this process over all feature orderings. Scott is saying that you need a background dataset when using the interventional approach to feature perturbation. .