Talk Down

It is very common to say, "the police are trying to talk him down" or "the police are talking him down." However, the other options you listed (yell, shout, scream) are not used in this manner.

So, when will we use those options(yell, shout, scream)?

When it is said in a loud, excited voice.

How about this:

"The police are trying to talk him down from the rooftop."
"The police are trying to yell him down from the rooftop."
"The police are trying to shout him down from the rooftop."
"The police are trying to scream him down from the rooftop."

I want the last three to mean that the police are yelling/shouting/screaming at the guy to get down to the grou

No, those do not work. 'Talk' here is idiomatic: 'talk' = 'verbally coax'.

The specific phrase "talk him down" would be used to say that the police are engaged in the act of negotiating with him to convince him not to jump. I would never use "yell/shout/scream him down." I'm not sure what the exact grammatical explanation for this is, other than to say that "talk him down" is a unique phrase.

If you wanted to express that the police are yelling at someone on

I found an interesting use of "talk down":

"For eight years of the Bush administration, the MSM talked down the US economy relentlessly."

So, figuratively speaking, the economy was about to "commit suicide on a rooftop" and MSM somehow talked the economy down to the ground?

the MSM talked down the US economy relentlessly = The MSM disparaged the US economy continuously