## Why is this variable in equation 2 of the SQAIR paper a random vector of $n$ ones followed by a zero?

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I've been reading the SQAIR paper lately, and the mathematics involved seems a bit complicated.

Some background, about the paper: SQAIR stands for Sequential Attend, Infer, Repeat - the paper does generative modelling of moving objects. The idea of Attend, Infer, Repeat is to decompose a static scene into constituent objects, where each object is represented by continuous latent variables. The latent variables, $$z^{what}$$,$$z^{where}$$ and $$z^{pres}$$ encode the appearance, position and presence of an object.

Here's a screenshot of the first of many things I'm unable to understand -

Why is $$z^{pres,1:n+1}$$ a random vector of $$n$$ ones followed by a zero? Why do we need the zero? How does it help?

Furthermore, an explanation of equation $$(2)$$ as in the image above, would be great.

P.S. I hope you all find the paper interesting. I'll ask other questions from the paper in separate posts, so as to not crowd one post with too many queries.