Explain why Polonius thinks Hamlet is or is not insane.

In Shakespeare's Hamlet, Polonius
thinks Hamlet is insane or "mad," because Hamlet wants him to think he is mad--it's part
of his plan.

In Act 1.5.171, after Hamlet has been told by
the Ghost that Claudius killed King Hamlet, Hamlet tells Horatio that he may put an
"antic disposition on," or fake madness, and that Horatio should not let on that Hamlet
is just faking.  The Ghost gives his blessing, so to speak, to the plan by calling out
for Horatio to "Swear" he will do as Hamlet asks. 

Ophelia
incorporates herself into Hamlet's plan by obeying her father and returning his letters
to him and denying him access to see her.  She later tries to spy on Hamlet while
engaging him in a conversation as Polonius and Claudius listen.  Thus, Hamlet uses her
to reveal his "madness," knowing, presumably, that his antics will get back to the king,
which they eventually do, of course.    

When Hamlet grabs
Ophelia's arm and studies her face as if he were an artist, sighs as if he were dying,
then walks away without looking where he's going, he is pretending to be mad.  He wants
the king to be preoccupied with Hamlet's madness, rather than with the fact that Hamlet
is going to revenge his father.  He acts mad so Polonius will think he's mad.  And it
works.