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Initial Parameter Estimates

Ambient Profile. In high-throughput single-cell assays, a large fraction of recovered barcodes are non-cellular, yet are nonetheless associated with small numbers of molecules originating from the ambient pool. Because the feature counts for these non-cellular barcodes originate solely from ambient contamination, these barcodes' feature expression composition provides an unbiased estimate of the global ambient profile. For these technologies, we estimate the ambient distribution as the normalized average of counts across all barcodes identified as empty:

\[\boldsymbol{a}_g = \frac{\sum_e \boldsymbol{C}_{e,g}}{\sum_e \sum_{g'} \boldsymbol{C}_{e,g'}}, \qquad (8)\]

where \(e\) indexes the non-cellular barcodes.

Under the assumption that non-cellular barcodes sample independently from a shared ambient pool, this estimator corresponds to the maximum likelihood estimate of \(\boldsymbol{a}\) under a multinomial sampling model, and is highly stable due to the large number of non-cellular barcodes typically observed.

Bulk Contamination Profile. Across all technologies, a small fraction of global contamination can arise after pooling during library preparation and amplification. This contamination affects all barcodes approximately uniformly and can be modeled by a shared background profile \(\boldsymbol{m}\). Because this background reflects the aggregate expression composition of the entire experiment, we approximate \(\boldsymbol{m}\) as the normalized mean expression profile across all droplets:

\[\boldsymbol{m}_g = \frac{\sum_i \boldsymbol{C}_{i,g}}{\sum_i \sum_{g'} \boldsymbol{C}_{i,g'}}. \qquad (9)\]

This estimator corresponds to the maximum likelihood estimate of the global background under a multinomial sampling model and provides a stable initialization for subsequent inference. In principle, an even more accurate estimate of \(\boldsymbol{m}\) could be obtained from molecule counts prior to barcode collapsing, as bulk contamination is introduced during library preparation and amplification. However, such information is typically unavailable in standard scRNA-seq outputs, making the collapsed-count estimator a practical and robust approximation.

Cell Typing. The number of cell-types \(K\) is specified or estimated from an initial clustering of the normalized count matrix. The cell-type assignment for each droplet is indicated by the one-hot cell membership vector \(\gamma_i\).

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