Hi,

What we are able to prove is that the union bound can be tightened using the growth function instead of the number of hyptheses. In order to do that, we considered events involving two samples, because the growth function applies only to finite samples, and the two samples together are still finite (of size

instead of

). However, we need to extend this result to the case we care about, where the events involve a sample (

) and the whole space (

), and the growth function is not directly applicable. This is what the segment is about.

BTW, I added the direct inclusion of a video segment to your post using the [lecture6] tag (see the above sticky thread about including a lecture video segment).