I think the simple answer is probably that the statistical inference assumes that your markers are independent. Linkage disequilibrium means that the markers are non-independent, i.e. if you take two SNP loci, A and B, and you observe that one of your samples has the A1 allele .. if the loci have significant LD, the probability of the allele at locus B being B1 or B2 will not necessarily be equal to their respective allele frequencies in the population.

Founder populations might have undergone expansion and contraction leading to the different magnitude of selection and genetic bottleneck that results in ancestry based LD in the contemporary population. Therefore, background LD analysis of markers are important if you are dealing with an admixed population with different ancestry.

Most methods will assume independence among sites as a part of the underlying model. Building on Jamie's example, the likelihood of assignment for a two-site multilocus genotype of A1/A1 , B1/B2 to either population "X" or "Y" could be calculated like so:

If "X" has frequencies:

A1: 90%

A2: 10%

B1: 60%

B2: 40%

And "Y" has frequencies:

A1: 30%

A2: 70%

B1: 20%

B2: 80%

P(genotype| ind is from X) = P(A1|X) * P(A1|X) * P(B1|X) * P(B2|X) = 0.9*0.9*0.6*0.4 = 0.1944

P(genotype| ind is from Y) = P(A1|Y) * P(A1|Y) * P(B1|Y) * P(B2|Y) = 0.3*0.3*0.2*0.8 = 0.0144

So, the probability of e.g. genotype A1/A2 being sampled from population X is the probability of sampling an A1 allele multiplied by the probability of sampling an A2 allele. We multiply by the same for locus B, but this assumes that the probability of sampling an allele at locus B doesn't depend on the genotype of locus A. If the occurence of alleles at any two sites is biased e.g. by their physical linkage, then the assumption inherent to this calculation is broken. Note that another assumption we are making is that the probability of sampling a given allele just depends on its frequency in the population. (Also, this is a simplified version for the sake of demonstration)

Some methods consider LD among sites, such as the more recent version of Structure (Falush et al 2003), although as Binod mentioned, genetic drift can drive spurious results (Kaueffer et al 2007). As a result, most studies of this kind will sub-sample a single SNP per locus, and test for LD pairwise among loci. Some methods derive information from among-site LD though (Malinsky et al 2018). Final takeaway is that you'll need to think about the assumptions being made by a given analysis.

thanks for your help, Jamie McCann and Tyler Chafin