How to propose collaboration, when I have an idea for analyzing another group's data? My field is data analysis in which i try to invent/improve algorithms to better analyze existing datasets. During my search i noticed a good medical institute is trying to solve their problem from data analysis point of view, or in other words they are working on designing methods to decipher their own data. 

Based on their publications i noticed that my method might have a better result according to what they look for. So one approach is to ask them if i can have the data and try my method and in case of success i can have my own next publication.

However, i also really like to have collaborations with that institute rather than just using their data, as in my field it'd be a great value to have close works with data-experts like them. 

But i do not know how to approach them? and whom to contact? In fact, i already had a short talk with the post-doc involved in their project. Should i just contact him, or better to put their adviser in the communication loop? 

Also, it would be naive if i say "my approach will work better than what you have devised so let's have collaborations!!", wouldn't be? 

I would be so grateful if you could give me some advice on these situations!
I would contact the postdoc you previously talked with. He could be senior enough to run the whole collaboration without keeping his supervisor in the loop all the time. At the very least he has close access to the supervisor, and would prevent you from "cold-calling" the PI with your idea. Why try that again when you already have a contact? This is of course assuming the talk went well and you think he would be welcoming the collaboration. 

About how to suggest it: Don't tell them that your method is better than theirs. You could tell them you have an alternative idea that might give better results. As far as you could test it, this seems to be the case and you'd like to try it out with more data, preferentially in the form of a collaboration. This avoids telling them you're better than them / they're wrong. Nobody wants to hear that, and in this case they also don't need to hear it.