I am trying to understand the benefits of joint genotyping and would be grateful if someone could provide an argument (ideally mathematically) that would clearly demonstrate the benefit of joint vs. single-sample genotyping.This is what I've gathered from other resources (Biostars, GATK forums, etc.)Joint-genotyping helps control FDR because errors from individually genotyped samples are added up, and amplified when merging call-sets (by Heng Li on https://www.biostars.org/p/10926/)If someone understands this, can you please clarify what is the difference on the overall FDR rate between the two scenarios (again, with an example ideally)Greater sensitivity for low-frequency variants - By sharing information across all samples, joint calling makes it possible to rescue genotype calls at sites where a carrier has low coverage but other samples within the call set have a confident variant at that location. (from https://software.broadinstitute.org/gatk/documentation/article.php?id=4150)I don't understand how the presence of a confidently called variant at the same locus in another individual can affect the genotyping of an individual with low coverage. Is there some valid argument that allows one to consider reads from another person as evidence of a particular variant in a third person? What are the assumptions for such an argument? What if that person is from a different population with entirely different allele frequencies for that variant?Having read several of the papers (or method descriptions) that describe the latest haplotype-aware SNP calling methods (HaplotypeCaller, freebayes, Platypus) the overall framework seems to be:Establish a prior on the allele frequency distribution at a site of interest using one (or combination) of: non-informative prior, population genetics model-based prior like Wright Fisher, prior based on established variation patterns like dbSNP, ExAC, or gnomAD.Build a list of plausible haplotypes in a region around the locus of interest using local assembly.Select haplotype with highest likelihood based on prior and reads data and infer the locus genotype accordingly.At which point(s) in the above procedure can information between samples be shared or pooled? Should one not trust the AFS from a large-scale resource like gnomAD much more than the distribution obtained from other samples that are nominally party of the same cohort but may have little to do with each other because of different ancestry, for example?I really want to understand the justifications and benefits offered by multi-sample genotyping and would appreciate your insights.    
Single-sample vs. joint genotyping
genotyping;gatk;freebayes;platypus;multi sample
Say you are sequencing to 2X coverage. Suppose at a site, sample S has one reference base and one alternate base. It is hard to tell if this is a sequencing error or a heterozygote. Now suppose you have 1000 other samples, all at 2X read depth. One of them has two ALT bases; 10 of them have one REF and one ALT. It is usually improbable that all these samples have the same sequencing error. Then you can assert sample S has a het. Multi-sample calling helps to increase the sensitivity of not so rare SNPs. Note that what matters here is the assumption of error independency. Ancestry only has a tiny indirect effect.Multi-sample calling penalizes very rare SNPs, in particular singletons. When you care about variants only, this is for good. Naively combining single-sample calls yields a higher error rate. Multi-sample calling also helps variant filtering at a later stage. For example, for a sample sequenced to 30X coverage, you would not know if a site at 45X depth is caused by a potential CNV/mismapping or by statistical fluctuation. When you see 1000 30X samples at 45X depth, you can easily know you are looking at a CNV/systematic mismapping. Multiple samples enhance most statistical signals.Older methods pool all BAMs when calling variants. This is necessary because a single low-coverage sample does not have enough data to recover hidden INDELs. However, this strategy is not that easy to massively parallelized; adding a new sample triggers re-calling, which is very expensive as well. As we are mostly doing high-coverage sequencing these days, the old problem with INDEL calling does not matter now. GATK has this new single-sample calling pipeline where you combine per-sample gVCFs at a later stage. Such sample combining strategy is perhaps the only sensible solution when you are dealing with 100k samples.The so-called haplotype based variant calling is a separate question. This type of approach helps to call INDELs, but is not of much relevance to multi-sample calling. Also, of the three variant callers in your question, only GATK (and Scalpel which you have not mentioned) use assembly at large. Freebayes does not. Platypus does but only to a limited extent and does not work well in practice.I guess what you really want to talk about is imputation based calling. This approach further improves sensitivity with LD. With enough samples, you can measure the LD between two positions. Suppose at position 1000, you see one REF read and no ALT reads; at position 1500, you see one REF read and two ALT reads. You would not call any SNPs at position 1000 even given multiple samples. However, when you know the two positions are strongly linked and the dominant haplotypes are REF-REF and ALT-ALT, you know the sample under investigation is likely to have a missing ALT allele. LD transfers signals across sites and enhances the power to make correct genotyping calls. Nonetheless, as we are mostly doing high-coverage sequencing nowadays, imputation based methods only have a minor effect and are rarely applied.