Could observing unknowingly cause the wave function to collapse? I've recently been reading (at a basic level) about the double slit experiment and how the mere act of observing can cause the "wave function to collapse", as they say. I find this mind-blowingly fascinating, and it naturally sparks some questions.
In particular, consider the following scenario: I perform the double slit experiment and detect which slit was traversed, sending the data to some computer. This would cause the wave function to collapse, as I understand. Now, imagine that I delete the data as it is being read by the computer. If I am not mistaken, the wave function would not collapse in this case, since there would be no way for me to determine which slit was traversed. Now, imagine that I am under the belief that the data is being deleted when in reality it is being saved to some disk. What would happen in this case? Assuming I never come to know about the existence of the saved data, would the wave function collapse?
 A: 
In particular, consider the following scenario: I perform the double
  slit experiment and detect which slit was traversed, sending the data
  to some computer. This would cause the wave function to collapse, as I
  understand. Now, imagine that I delete the data as it is being read by
  the computer.

I don't think this quite the correct way to be thinking about this.
If the particle is detected as being located in one slit or the other, it is in one slit or the other, i.e, the state of the particle has "collapsed" to the state of being in one slit or the other.  This is a postulate of QM.  From Wiki:

A measurement results in the system being in the eigenstate
  corresponding to the eigenvalue result of the measurement.

A: Consider the following case which shows the effect of destroying information on wavefunction collapse:
Suppose the experiment consists of Bob at hole 2 in the double slit experiment being able to open and close the hole instantly. Let the intensity be so low that on average only one particle at a time is in the apparatus. By closing the hole he ensures the particle must go by route 1 if it is to hit the screen. Now what happens if he manages to re-open hole 2 just before the particle is detected at the screen? By repetition of the experiment the pattern built up at the screen is the interference pattern (case a). If the hole were closed at the time the particle is irreversibly detected at the screen then the pattern built up would not show interference at all (case b). The state of the apparatus at the exact moment of the irreversible detection of the particle at the screen determines if the particle is contributing to case a or case b pattern. Note that in this experiment Bob hasn't detected any particles himself but keeps a record of the time at which he opens or closes the hole which can be correlated with the particle arrival times at the screen. we can thus group the screen observations into two groups- those that occurred with the hole open and those with it closed. The first ones show the interference pattern the second do not. What happens if Bobs records are destroyed before the screen results are analyzed into these two groups? We see a mixed pattern of both a and b so interference fringes on top of high background which tends to wash out the fringes. The point is destroying Bobs information doesn't change the results which were observed at the screen. But those results were determined by the now lost information, they don't suddenly change.
