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the ABC algorithm is given as

  1. Draw $\theta \sim \pi(\theta)$
  2. Simulate data $X \sim \pi(x | \theta)$
  3. Accept $\theta$ if $\rho(X, D) < \varepsilon$

where $\pi(\theta)$ is the prior, $\pi(x | \theta)$ is the likelihood, $\rho(\cdot | \cdot)$ is some distance measure, $D$ is the observed data and $\varepsilon$ is the tolerance that represents a trade off between accuracy and computability.

Generally, in papers that I have seen on this, a proof is given where it states we actually sample from $\pi_{\varepsilon} = \pi(\theta | \rho(X, D) < \varepsilon)$ and then if $\varepsilon \to 0$, this converges to the true posterior $\pi(\theta | D)$.

If in Step 3, we had

3*. Accept $\theta$ if $X = D$

I was wondering if anyone knew how to prove that in this new algorithm, we sample from the true posterior? So there is no $\varepsilon \to 0$ argument?

Thanks in advance to anyone that can offer some help!

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This case is the original version of the algorithm, as in Rubin (1984) and Tavaré et al. (1997). Assuming that $$\mathbb{P}_\theta(X=D)>0$$ the values of $\theta$ that come out of the algorithm are distributed from a distribution with density proportional to $$\pi(\theta) \times \mathbb{P}_\theta(X=D)$$ since the algorithm generates the pair $(\theta,\mathbb{I}_{X=D})$ with joint distribution $$\pi(\theta) \times \mathbb{P}_\theta(X=D)^{\mathbb{I}_{X=D}} \times \mathbb{P}_\theta(X\ne D)^{\mathbb{I}_{X\ne D}}$$ Conditioning on $\mathbb{I}_{X=D}=1$ leads to $$\theta|\mathbb{I}_{X=D}=1 \sim \pi(\theta) \times \mathbb{P}_\theta(X=D)\Big/\int \pi(\theta) \times \mathbb{P}_\theta(X=D) \,\text{d}\theta$$ which is the posterior distribution.

On the side, I gave this very proof in class a few hours ago!

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    $\begingroup$ Thanks Xi'an! If this was extended outside of the Bayesian context, so for arbitrary functions $f_{1}(x)$ and $f_{2}(x)$, does this work to find samples from $f \propto f_{1}f_{2}$? $\endgroup$ – user-2147482565 Dec 3 '18 at 16:04
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    $\begingroup$ Yes, indeed!, this is actually the most rudimentary form of acceptance-rejection: simulate $X$ from $f_1$ and accept if the realisation of $Y\sim f_2$ is equal to the realisation of $X$. $\endgroup$ – Xi'an Dec 3 '18 at 18:28

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