1
$\begingroup$

Suppose $Q\ll P$ so that $\frac{dQ}{dP}$ is well defined. Obviously it is true that $\int\frac{dQ}{dP}dP = Q(\Omega) =1$. Is it true that $$\int\left|\log\left(\frac{dQ}{dP}\right)\right|dP < \infty \quad ?$$

Is is true that $\int \frac{dQ}{dP} dQ = \int (\frac{dQ}{dP})^2 dP < \infty$? What about $\int \frac{dQ}{dP} |\log(\frac{dQ}{dP})|dP$ ? I'm trying to read about these things and explore their properties but such things aren't discussed in the books I have.

$\endgroup$
1
$\begingroup$

One can write $$\int|\log(\frac{dQ}{dP})|dP=\int_{1>{dQ}/{dP}>0}\log(\frac{dP}{dQ})dP+\int_{\infty>{dQ}/{dP}>1}\log(\frac{dQ}{dP})dP\\=D(P,Q|R_1)-D(P,Q|R_2)$$ where $D$ stands for the KL-divergence, and $R_1$ and $R_2$ are the related domains where the integral is taken. For the final result to be finite one needs Q>>P, else it is always possible to find a counterexample, where Q>>P is violated on some mesurable set and over this set the integral is infinite.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.