reviews: 
Comments: The paper essentially extends the idea of model-based clustering under the Bayesian framework into the clustered federated learning (CFL) problem, where not only the partition of clients but also the model update through federated learning is considered. The paper proposes to tackle the exploded data association incurred by the recursive updates with the $M$-best assignments problem solver.
Strengths: 1. The paper's theoretical contribution that systematically discusses CFL under the Bayesian framework could serve broader interests apart from the proposed algorithms. 

2. The efficiency problem in the recursive update could be tackled with the M-best assignment solver when using the negative log weights as the cost function.
Weaknesses: 1. ***Presentation***

There is a clarity problem before section 4. Specifically, the full procedure of the learning is poorly illustrated. Though Figure 1 is supposed to illustrate the essential steps and communications, the confusing coloring and the lack of clarity on the calculation of key values pose unnecessary challenges for the audience. The pseudo-code for the general learning, clustering procedure, and each variant of association selection, along with a detailed congregate discussion of the communication, would greatly improve the clarity of the paper. Also, it might help to focus on the conceptual discussion in Figure 1 rather than placing notations, which are introduced later in sections 3 and 4. 

2. ***Related work*** 

Section 2 mainly focuses on discussing federated learning works. Yet, the paper needs to pay attention to the existing product partition models and needs more discussion over the progression of existing Bayesian frameworks, which could better position the paper's theoretical contribution.


3. ***Method***

The paper proposes to leverage the cost function to transform the association selection into the M-best assignments problem. Yet, as admitted in the discussion of section 4, the greedy or top $M$ trajectory could be myopic. This problem seems intrinsic in the definition of $L$ and is reflected in the empirical results where the BCFL-MF-6 does not constantly improve upon BCFL-MF-3.

4. ***Experiment***

(1) The datasets tested in section 5 seem to be manually synthesized, raising questions about the motivation of CFL and undermining the significance of the experimental results.

(2) The results only show the downstream performance while lacking evidence about the efficiency, especially for different approximations and corresponding key hyperparameters, including the $M$ discussed in sections 4.3 and 4.4 and $K$ that could potentially impact the communication efficiency and learning convergency.

5. ***Minors***

(1) The paper claims the existing CFL methods do not effectively exploit similarities between different clusters, yet needs more detailed discussion over the statement.

(2) The adopted merge operator in section 4.3 needs to be clarified.

(3) The model warm-up relies on a different partitioning method based on the local model initialization. And it is unclear why it contributes to the performance.
Recommendation: 5: marginally below the acceptance threshold
Reviewer confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness/correctness: 3 good
Clarity: 1 poor