This paper proposes a new tighter lower bound for variational inference by using auxiliary variable methods. R1 and R3 found the work significant and interesting. R4 had some concerns, some of which were resolved by authors' response. R4 decided to increase their score, however they do still have some concerns. I recommend the authors to take R4's comments into account and either remove the erroneous statements mentioned by them, or add explanation/proofs to show that the statements are correct, or at the very least clarify them so that such confusion does not arise to the reader in the future. There are many other small comments by the reviewers that need to be taken into account. In addition, the authors should do all the additional things they promised in the rebuttal. Given that this is done, this paper can be accepted.