Why are there changes in the histogram with a convert to profile?

When you convert an image to a different profile the color numbers change, in order for the colors to represented the same (as closely as possible given the gamut differences) in the new color space. The histogram also changes, quite noticeably. Why? Is it representing the raw numbers and not their interpretation in the new space? And if so -- why is that? Is it of some use?
Thanks!

I understand that for an export operation, but still puzzled why the histograms are almost identical in LR for images that had been brought into PS from the same RAW file and converted to different spaces (with slight differences undoubtedly due to the different gamuts). In that case, LR must be "honoring" the different color spaces, with the same appearance and histograms... But when I soft-proof the same RAW file I get very different histograms -- the sRGB more stretched out than the ProPhoto one. Those soft-proof histograms are "the same" (realizing they are all quick calculations) as when I do the corresponding convert to profile in PS.
Maybe I'm not getting my head around "the histogram of the proof target." That would be the more "dramatic" (changed) histogram I'm seeing in LR's soft proof and PS's convert to profile. But is there a reason to show it that way? Doesn't a histogram need to be interpreted in a color space to be meaningful?
If I go ahead with something like a Save for Web or Export from LR in that target color space, I'll then see the histogram return to more or less what it was before (given that rendering intent has done its thing).
Is it that the more dramatic histogram is simply representing the image in the wrong color space?? I can see that as a pragmatic decision.

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