in debugging this kind of error, all you need to do is see where you are losing the value. use print or echos to show you the value of the radio buttons at various points. Also check that the name of the radio buttons is coming through appropriately (view source is your friend).
If you think that the problem is with those lines, then try this instead:
Yes<input name="<?php echo $_POST['visible']; ?>" type="radio" value="yes">
No<input name="<?php echo $_POST['visible']; ?>" type="radio" value="no">