The research is carried out in the problem of identification of causal effects in the frame of probabilistic causality established by Pearl, Spirtes, Glymour and Scheines. The current manuscript expands from current work on unconditional causal effects to conditional causal effects. The main rationale of the solution (alike a previous related one) is to reach the computation of the post-interventional causal effect Q using in the denominator a summatory over the elements of a partition of the causal graph, as long as such partition complies with some condition. As from here, my understanding of the paper I must accept was very limited. The rationale of the idea seems ok, but I’ve been unable to follow the details of the maths (this is sharply different from not having check them! I simply did not fully understood them despite making several attempts at understanding them!). In this sense, I can only apologize to the authors for not having been able to give a more thorough feedback (I have lowered my confidence to the minimum). STRONG POINTS + Given my limited understanding, but as far as I can confirm, the demonstrations of the propositions and lemmas are correct. This is concentrated in the supplementary material.• WEAK POINTS + To help contextualizing, the draft contains a lot of material already available in other papers (likely from the same authors) leaving little room for the new material. In fact, most of the real innovation is what it is given in the supplementary material. Although I found this useful in a really difficult topic, but to an experienced reader (not me!), it may left him with a bit of a deja-vu impression.

Originality; in my humble opinion the paper is moderately original but it addresses and solves a relevant and difficult problem of inference and decision making under partial knowledge and when only observational data are made available. Quality; the technical quality of the paper is excellent, it is well written and structured. However, it is extremely difficult to review a paper like this one in so few days (we have to keep in mind that 5 papers to review in 3 weeks is really an incredible overhead in the case where technical papers like this one have to be judged. Clarity; the paper is well structured and reads well. However, I would stress that is more examples could be inserted to present and discuss the paper this will help the interested reader to grab the many innovations and contributions of the paper. Significance; the paper is highly significant. Indeed, the developed theory and the designed algorithm will help to analyze complex situations, i.e. real world situations where the available knowledge and data are limited and in particular they are of observational nature. I also have few questions for the authors/s; Pag. 2; line five from Structural Causal Models, it is assumed that no U elements can be can be parents of V? Pag. 2; would you please clarify what small pai means in formula (1), is it the specific value of Pai? If so then I do not understand well formula (1), could you please help me understand it?

The paper focuses on an interesting and useful problem (significance), and it provides a novel theoretical contribution (originality) that seems technically correct (high quality). As mentioned in the previous section, it extends the previous work from (Jaber et al. 2019) in terms of conditional causal effects, providing an algorithm that is sound and conjectured to be complete (the lack of this proof is possibly one of the few negative points of the paper). In terms of clarity, the paper seems quite well-written, although I do wonder if a beginner in causal inference would be able to read it. Minor details and typos: L65: not sure about the “semantical” framework, possibly rephrase L121-126: could be improved in terms of clarity, possibly with an example? L272: there existS a potentially causal path