There's a direct answer to this excellent question in Act 4, Scene 5. Ophelia has gone mad, and Claudius tells Gertrude that the people are
The people are confused ('muddied'), and spreading negative rumours about Polonius' death: Claduius worries that he has been 'green' (naive) to bury ('inter') him in secret ('hugger-mugger').
Later in the same speech, Claudius reveals why he has buried Polonius in secret: no-one knows that Hamlet did the deed, he tells Gertrude, and therefore
Because they have no other information (they are 'beggared' of 'matter') nothing will stop the people from blaming ('arraigning') Claudius (who speaks here with the royal we), and whispering their suspicions in ear after ear.
Claudius buried Polonius in secret because he didn't want anyone to know about the death: but word has got out, and now that Claudius himself is rumoured to be to blame for the death, he wonders whether he has made the wrong decision.
In Act 4, sc. 5, Laertes asks Claudius why his father's funeral was so limited - why Polonius wasn't given any military rites or formality. Claudius merely responds that Laertes has a right to question how Polonius died and that he hopes the guilty party will be punished by death. Then in sc. 7 of that act, Laertes asks Claudius why Hamlet wasn't dealt with like a criminal since Hamlet killed Polonius. The question regarding the limited funeral isn't addressed directly. Claudius explains though that he couldn't deal with Hamlet like any other criminal because Hamlet is so loved by the people of Denmark. He says that any faults Hamlet has are turned into virtues by the people so if he, Claudius, tried to punish Hamlet, the people would have turned against Claudius. Claudius goes on, later in the scene, to tell Laertes that he has a plan that may allow Laertes to kill Hamlet.
polonius was from the court and was an important person to the kingdom, he didn't want anyone to know who killed polonius, so he had the funeral private because denmark loved hamlet and it was an accident that hamlet killed polonius, he thought it was claudius behind the curtain.