We are grateful to Jakob Runge for pointing out the additonal work that has been done since the original publication on MIT and look forward to a fruitful discussion on this forum.
In response to the above comment we would like to stress three important points, however:
(1) In the example given in our study, MIT does not recover the correct delay. We do give a quantitative explanation of this failure by detailing how MIT measures the time interval between the time point when the information in question is first seen in the source (and potentially stored there for a while, invisible to the target, before being transfered), and when it's seen in the target. Hence, MIT measures the sum of information storage time in the source and the actual transfer delay. We do believe that this summed time is a useful quantity in itself, but it is not exactly the information transfer delay, as was claimed incorrectly. The incorrectness of this claim was demonstrated by example in our study. The formulation of MIT that is based on scalar observables makes it impossible for variables to store information in the same way systems states do this, which is possibly why this subtlety was overlooked.
(2) Information transfer as defined by Schreiber is based on (Markov) states of systems and explicitely NOT on scalar observations (see definition of transfer entropy on page 462 in Schreiber, Phys Rev Lett (85)2, 2000). As a consequence, a failure to reconstruct the systems' states properly (i.e. by using scalar observations) may yield spurious results -- a simple example for this phenomenon was already given in Vicente et al., J Comp Neurosci, 2011.
(3) The Pompe and Frenzel (2007) partial mutual information estimator that is mentioned in the above comment is not a transfer entropy estimator. Therefore it does not measure information TRANSFER as pointed out in Schreiber 2000. Schreiber clearly separates estimators for information transfer from lagged mutual information estimators in all generality. The Pompe and Frenzel estimator is therefore quite different from our estimator, both in formulation and in purpose. As far as the actual entropy estimation technique employed by Pompe and Frenzel is concerned, we cite the source of the idea to use neighbour statistics as: Kraskov et al., Phys. Rev. E 69, 066138 (2004). This is the same source that Pompe and Frenzel cite for their estimator. We also note that the possibility to estimate transfer entropy functionals using this technique was also already demonstrated in Kraskov's PhD thesis from 2004.

In his comment Jakob Runge claims that MIT fails to recover the correct delay in our example because "...Rather, their example model seems to not fulfill the assumptions implicitly used in their own proof. ".

We have carefully checked this and conclude:
(a) We are not aware of any assumptions that are implict in our proof. The relevant assumptions are explicitely given. The confusion about this point (i.e. implicit assumptions) may arise from the fact that both our proof, and the graph supporting it are based on a state space notation (Bold typeface) that seems to have been misunderstood as a representsation of a scalar process.
(b) The (explicit) assumptions in our proof are simply that the two processes in question do have a state space representation, i.e. are Markovian. It is easy to see for our test case I that p(x(t) | x(t-1), x(t-2)) = p(x(t) | x(t-1)), i.e. X is a Markov chain of order 1, in this case even in scalar representation. If it were not it could of course be transformed into a Markov chain of order 1 in a state space representation. The same holds for y(t) which is a noisy copy of x(t). We note that both processes are also stationary, although this is not necessary for our proof. Hence the causal graph of the processes in our proof (which is explicitely given in a state space representation in figure 2, so that all Markov processes are of order one) does apply to our example.
Hence, we concluce that the failure of MIT to reconstruct the correct delay is not due to our example not meeting "implicit assumptions" of our proof.

Jakob Runge states that our proof does also hold for MIT, and give a formal argument in in arxiv 1304.7930.
We note, that for their argument they use the causal graph in our figure 2. However, this causal graph is only valid if the quanatities included as nodes in the grasph are states in a state space. If the quantities in the graph represent any other (smaller) collections of variables the graph looses it is generality and does not represent generic Markov processes, and d-separation can not be shown using this graph. Since MIT is not based on a state space representation, our graph may not be used to proof properties of MIT. Hence, the derivation given in arxiv 1304.7930 so far lacks the necessary prerequisites to arrive at their formula 2.

Jakob Runge states that the failure of MIT to recover the correct interaction delay seems to be because "Rather, their example model seems to not fulfill the assumptions implicitly used in their own proof. "

We are not aware of what these implicit assumptions would be and note that the example processes in test case 1 have a state space representation and are Markovian of order 1, and hence, do fullfill the assumptions necessary for our proof to hold.
We also note that our transfer entropy estimator does indeed recover the correct delay for this process, as predicted by pour proof.

Dear Jakob,

after critical evaluation we found that the proof of interaction delay reconstruction via MIT given in your above reference on arXiv contains an error in equation (4):

On both sides of the equation the conditioning is performed for the Parents of x(t-delta-xi); for a proof of delay reconstruction using the momentary information transfer, however, the conditioning on the right hand side of equation (4) should be on the parents of x(t-delta) not on those of x(t-delta-xi). Otherwise the the conditioning ...

(a) ... does not represent MIT for x(t-delta), as it should for a proof of maximality at the correct delay.
(b) ... is potentially on a future sample of x as xi is allowed to be negative in our original proof.

In addition, we have carefully checked the example given in our PLOS one paper, and still find that MIT does not reconstruct the delay correctly, even though this is a very simple case with one source and one target, and the number of Parents of the source x, and the target y is very small.

Michael Wibral