Is it valid to use latin-hypercube sampling in parallel sampling? I am sampling from a model 10 times in series, doing this in 50 parallel processes. I am using LHS to generate samples each set of ten samples, although each of the 50 parallel runs' samples are generated independently. Will using LHS sampling in this situation create any bias? Is there any reason I should use random sampling instead of LHS?
 A: LHS sampling run 'in parallel' in this way should still lead to unbiased estimates.
In standard LHS sampling, we generate vectors $X_1$, $X_2$,...,$X_n$ (with dimension $d$ equal to the dimension of the sampled parameter space for the model), where $n$ is the desired sample size. We then form the LHS estimate for the function of interest ($f$) using $\hat{\mu}_{LHS}=\frac{1}{n}\sum_{i=1}^n f(X_i)$. Each $X_i$ is distributed uniformly on the unit hypercube $[0,1)^d$ (see Theorem 10.1 in this book chapter), so it follows that $\hat{\mu}_{LHS}$ is an unbiased estimate of $\mu=\int_{[0,1)^d} f(x)dx$ i.e. $E(\hat{\mu}_{LHS})=\mu$.
For 'parallel' LHS sampling, we would generate vectors $X_{jk}$ where $1\leq j\leq N$ and $1\leq k\leq n$, giving an aggregate sample of size $Nn$. Here, each $X_{j1}$,$X_{j2}$,...,$X_{jn}$ is an independent LHS sample. If we define $\hat{\mu}=\frac{1}{Nn}\sum_{j,k}f(X_{jk})$ then, by Theorem 10.1 again, we have $E(f(X_{jk}))=\mu$ for every $j,k$, so $E(\hat{\mu})=\mu$ i.e. $\hat{\mu}$ is unbiased. 
In summary, estimates obtained via 'parallel' LHS sampling are unbiased (same as for standard LHS), so this is not a reason to use simple random sampling rather than LHS. Of course, you could just use standard LHS i.e. generate a single LHS sample of size $Nn$. This should minimise the variance of the estimator. Parallel LHS would be expected to have a higher variance (but lower than for simple random sampling). However, an advantage of splitting into smaller subsamples is that you can increase the sample size simply by appending further subsamples of size $n$ (rather than starting again from nothing, as you would do with standard LHS sampling).
