How to know what production channel Higgs came from? There are four main production channels for SM Higgs: ggF, VBF, Higgstrahlung and associated top.
In multiple occasion I've seen experimental bounds referring to say $\sigma(gg \to H \to \tau^+\tau^-)$.
How can experimental people know which events come from ggF instead of say VBF? I though that when people calculate cross section of say $\sigma(pp\to H \to \tau^+\tau^-)$ they take all the events corresponding to $\tau^+\tau^-$ final state which has the invariant mass of ~125 GeV and then subtract the background.
You don't actually know how the Higgs was produced, you only see the decay products, so how experimentallists put bounds on something like $\sigma(gg \to H \to \tau^+\tau^-)$ specifically.
 A: Have you heard of Monte Carlo calculations? This talk may enlighten you on how it is used in high energy physics experiments.
In this Higgs discovery plot 

Note in the fine detail the various lines beneath the Higgs.  The are Monet Carlo calculations which use the Standard model of particle physics and various assumtions given in the caption, which have in common that no Higgs at 125 is assumed. The bump fits a Higgs at 125GeV in addition.  All these hypothesis enter in the generator of the physics for the plot in the Monte Carlo, thousands of events are produced and the curves fitted to the data. 
The best fit for 125 will have assumed the parameters for different Higgs production, so the fit will be better the closer the guess goes to the production mechanism.
To give reliable numbers for a production mechanism, one can go further. For example eliminate the top events that contribute to the higgs bump above ,from data and Monte Carlo in the same way, and look at a cleaner sample for glue glue  into Higgs and a better fit monte carlo to data ( this done seriously would need new monte carlo events generated, with the main sample of MC events the assumed crossections would be checked). That is why one talks of limits, because this is the only way of "seeing" what is happening at the production, using the statistics and the Monte Carlo simulated events.
