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In Andrew NG's Lectures (CS229), the Bayesian Logistic Regression section contained a formula;

$$P(Y|X,S)=\int_\theta P(Y|X,\theta)P(\theta|S)d\theta$$

Here, $\theta$ is treated as a random variable. $S$ is the set of points ${[X^{(i)},Y^{(i)}]_{i=1}^{m}}$.

Using conditional probabilities, it does make intuitive sense although I would really appreciate a rigorous proof of the equation.

From what I got: $$ P(Y|X,S)=\int_\theta P(Y,\theta|X,S)d\theta $$ $$ = \int_\theta P(Y|\theta,X,S)P(\theta|X,S)d\theta$$

Does it assume any sort of independence?

Again, I get the intuition, but I can't seem to arrive at the final answer. A written out proof or required equations to prove the result would be appreciated.

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  • $\begingroup$ What are $X$ and $Y$? Discrete or continuous random variables (or vectors)? Please include as much context as possible directly in the question instead of pointing to some course. $\endgroup$ – paf Jul 7 '18 at 7:21
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    $\begingroup$ @paf In the context of this problem, X(i) and Y(i) are points in the sample space, S. X(i) is an n dimensional vector and Y(i) is the corresponding classification of that point. X and Y themselves are continuous random variables. $\endgroup$ – Utsav Dutta Jul 7 '18 at 7:26
  • $\begingroup$ Treat X as a new point in the hyperspace and P(Y=y|X) as the probability of classifying the new point X as Y=y. $\endgroup$ – Utsav Dutta Jul 7 '18 at 7:27
  • $\begingroup$ related: math.stackexchange.com/questions/1882178/… $\endgroup$ – Henry Jul 7 '18 at 8:17
  • $\begingroup$ What does $\int_\theta \ldots d\theta$ means (where $\theta$ is a r.v.)? $\endgroup$ – d.k.o. Jul 7 '18 at 19:24
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The assumption applied is that of conditional independence. If $Z_1$ and $Z_2$ are independent then $p(Z_1\lvert Z_2) = P(Z_1)$ hence $Z_2$ can be removed from the set of variables on which one is conditioning. Similarly with conditional independence if $Z_1$ and $Z_3$ are conditionally independent given $Z_2$ then $p(Z_1\lvert Z_2,Z_3) = p(Z_1\lvert Z_2)$ so ones $Z_2$ is "controlled for" $Z_1$ does not depend on $Z_3$.

So the author is assuming that

$$p(Y\lvert \theta , X, S) = p(Y\lvert \theta ,X)$$

the dependent variable $Y$ only depends on $S$ through $X$ and $\theta$

$$p(\theta \lvert X,S) = p(\theta \lvert S)$$

so parameters $\theta$ do not depend on $X$ once $S$ is given.

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