Say we have some population, and we obtain a "representative" random sample of that population, $(y_i, x_i)_{i = 1}^n$, where $n$ is very large (millions) and $x_i = (x_{i1}, x_{i2}, ... x_{ip})'$ is a multivariate predictor of the response $y_i$.
This scenario is fairly common when working with large, modern data sets, but still, suppose we wanted to conduct some meaningful inferences about the data using (say) regression.
Assuming we had limited computational power, one approach is to draw a random sample from the larger data. That is, we can draw some sample $(y_j^*, x_j^*)_{j =1}^k$ from $(y_i, x_i)_{i = 1}^n$, where $k << n$.
Assuming this subsample is representative of the larger sample, can we use the subsample to make inferences about the larger, original population?
My thinking is that, yes, this works, our regression coefficients $\beta$ derived from the subsample should reflect the coefficients of the larger sample, which then allows us to make inferences about the population, albeit with slightly higher variance. But if the subsample is large enough (say a million?) then this variance shouldn't be that problematic, since most forms of regression are consistent and we are dealing with a large number of data points. Thoughts?
Edit: On my use of the word "representative", I found this thread:
What exactly does 'representative sample' refer to?
Perhaps as a working definition, we could say that the sample is drawn without bias from the population. Or perhaps: if we could fit the regression on the entire data, we would get unbiased estimates of $\beta$. If the sample were not "representative" then the estimates naturally have some bias, no?
Second question: how would the possible answer to this question change depending on what we mean by representative?
On the word "Random": If our extremely large data set consisted of data points that were obtained via a random sample, then I see no issue with subsampling, but of course, I would like to hear others' thoughts on this. What if the sampling mechanism were not entirely known? Can we still take a simple random subsample?
More generally, under what conditions would a subsample lead to correct inferences about a population?