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Having

$$\ln p(y) = \ln \frac{p(y, \theta)}{p(\theta, y)}$$

which is a result of the product rule

I do not understand here (slide 17)

$$ \ln p(y) = \int q(\theta) \ln \frac{p(y, \theta)}{p(\theta, y)} d\theta $$

It seems like they are trying to marginalize out the $\theta$ but I am not sure.

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It might be clearer to write it like this:

$$\log p(y) = 1 \times \log \frac{p(y,\theta)}{p(\theta|y)} = \int q(\theta) d\theta \times \log \frac{p(y,\theta)}{p(\theta|y)} $$

And then you can move the thing on the right inside of the integral since it isn't a function of $\theta$

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