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So I was looking at factors that control the relative strengths of acids and bases.

We know that $\ce{HI}$ is a stronger acid that $\ce{HBr}$ because $\ce{I}$ is a much larger atom.

But then, I read somewhere that $\ce{HBrO4}$ is stronger than $\ce{HIO4}$, because in this case it's electronegativity that plays the important role.

How do you know whether to use size or EN as a guideline for comparing strengths of acids/bases?

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    $\begingroup$ In HI and HBr, the hydrogen is directly bonded to the halogen, but in the perhalic acids it isn't. $\endgroup$ – f'' Aug 14 '16 at 16:25
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"We know that HI is a stronger acid that HBr because I is a much larger atom"

That is only half the story. It is about the strength of the H-Br versus H-I bond. Since I is larger than Br, the H-I bond is longer, weaker, and therefore more prone to dissociation. See here for more in-depth discussion.

$\ce{HBrO4}$ and $\ce{HIO4}$

both have the acidic proton bound to oxygen:

enter image description here

The H-O bond is approximately of equal strength in both acids and therefore you turn to electronegativity arguments.

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