What is the difference between bootstrap sampling vs multinomial distribution? I feel like bootstrap sampling and multinomial distribution sampling are equivalent. Just want to verify whether my understanding is correct.
Say my data is 1, 1, 2, 2, 2, 2, 3.
Multinomial distribution sampling would be sample 7 observations from a multinomial distribution (2/7, 4/7, 1/7). Isn't bootstrap sampling with replacement essentially the same thing because every time we draw an observation from the data, the chance of getting each value is also (2/7, 4/7, 1/7)? Thanks in advance
 A: Yes, you can think of it as drawing from a multinomial distribution. In fact, when I code bootstrap procedures from scratch, I do exactly that over the indices of my data.
library(MASS)
set.seed(2022)
N <- 100
B <- 1000
X <- MASS::mvrnorm(N, c(0, 0), matrix(c(1, 0.9, 0.9, 1), 2, 2))
for (i in 1:B){
  
  idx <- sample(seq(1, N, 1), N, replace = T) # This is multinomial sampling
                                              # with each index "category"
                                              # having an equal probability
                                                
  X_boot <- X[idx, ] # Select the indices
    
  # Then do something with X_boot, such as calculating the correlation
}

Since you have duplicated values in your 1, 1, 2, 2, 2, 2, 3, drawing uniformly over the indices is in some sense equivalent to doing a multinomial draw with $P(1)=2/7$, $P(2) = 4/7$, and $P(3)=1/7$. There's this issue where the values 1, 2, and 3 are numbers and not categories, so it is debatable if this is multinomial, but this technicality can be resolved by doing a distribution like:
$$
P(\text{Pick 1 and add it to the bootstrap sample}) = 2/7\\
P(\text{Pick 2 and add it to the bootstrap sample}) = 4/7\\
P(\text{Pick 3 and add it to the bootstrap sample}) = 1/7\\
$$
A: The are many variants of the bootstrap and, on the surface, the nonparametric or re-sampling bootstrap has a similarity to drawing from a multinomial distribution. See @Dave's answer.
The bootstrap uses the plug-in principle: we replace an unknown distribution $F$ by an estimate $\widehat{F}$. There are many possible choices for $\widehat{F}$. If $\widehat{F}$ is a parametric distribution, then the bootstrap is parameteric or model-based.
Moreover, as @Tim exaplains in Bootstrapping and ECDF, the bootstrap refers to the entire statistical procedure for estimating uncertainties. That is, once you've sampled $n$ times with replacement from the original dataset of size $n$, the bootstrap also explains how to do inference (estimate standard errors, calculate p-values, compute confidence intervals, etc.)
So drawing an analogy (actually, false equivalence) between bootstrapping and sampling from a multinomial distribution will take your understanding of bootstrapping methods only so far.
B. Efron and T. Hastie. Computer Age Statistical Inference Algorithms, Evidence, and Data Science (2021) It's freely available online.
