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I have to write python code in jupyter due to sampling bivariate normal distribution with 3 sampling methods:

  1. Prior Sampling
  2. Gibbs Sampling
  3. Rejection Sampling

I have done the first two samplings and I also have clear understanding of what accept reject method or so-called rejection sampling is. yet I can not find any proposal distribution for it.


Rejection Sampling and proposal distribution

assume you have a distribution $f(x)$ which is not easy for you to sample. choose another distribution that is easy for you to sample wisely called $g(x)$ so that for some constant $C$ for all $x$ we have $C.g(x) > f(x)$.

we call this $g(x)$ the proposal distribution.

sample from $g(x)$ and name it $X_i$ as you chose $g(x)$ because it was easy to sample. accept this sample with a probability of $p = f(X_i)/C.g(X_i)$.

As I said before I cant find any $g(x)$ easy enough to sample and yet bigger than bivariate normal distribution for some $C$.

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  • $\begingroup$ Can you articulate a specific question? We are a question-and-answer site, so we require you to articulate a concrete question. I see only declarative sentences. Note that coding is off-topic here, but algorithms and methods are on-topic. $\endgroup$
    – D.W.
    May 11 at 18:58
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I came up with this answer which can approximately sample a bivariate normal distribution with accept reject method and I thought it might be useful for others in future.

we choose a 2D uniform distribution called $F_{X,Y}(x,y) = F_X(x).F_Y(y)$ where both $X$ and $Y$ are uniform distributions in range of $(0, +a)$ and they are independent of each other.

We choose a constant $C$. Both $C$ and $a$ are chosen based on our bivariate normal distribution so that:

  1. for all $x, y$ we have $C.F_{X,Y}(x,y) \geq BN(x,y)$, Where $BN$ is our bivariate normal distribution.
  2. $x, y \in (-a, +a)$ supports most of our $BN$ density.

The Accept Reject Method:

  1. Now we sample from our $F_{X,Y}(x,y)$. Suppose it's $X1 = (x1, y1)$.

  2. We uniformly choose from: $$X_{1.1} = (x1, y1),$$ $$X_{1.2}=(-x1, y1), $$ $$X_{1.3}=(x1, -y1), $$ $$X_{1.4} = (-x1, -y1)$$ and name that $\hat{X_1}$ so that we consider all four quadrants of $xOy$.

  3. Now we accept $\hat{X_1}$ as a sample of $BN$ with a probability of $p=BN(x1,y1)/C.f_{X,Y}(x,y)$.

  4. We come back to step 1 and do this cycle to collect as much sample from the bivariate normal distribution as we want.

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