set of sample size $n=1000$.

![image](exp2){width="80.00000%"}

Figure \[fig:latent\_estimation\] reflects the following remarkable results. For the normal case, VAE has little bias, since the true conditional distribution $p_{U|X}(u|x)$ is indeed normal, which is well characterized by the encoder. However, in other two cases, neither the latent distribution $\pi(z)$ nor $p_{U|X}(u|x)$ is normal, and hence VAE gives highly biased estimates for $\pi$. For the three debiasing methods, the refining steps indeed reduce the bias of VAE. However, the debiasing effect of HMC is smaller than that of SAGD, even though theoretically they are similar. HMC also takes more computing time due to the leapfrog steps and the calculation of acceptance probability. For IWAE, it tends to overly truncate the support of the distribution and exaggerate the density of modes. Overall, SAGD provides the most favorable bias reduction results.

[&gt;m[0.11]{}&gt;m[0.01]{}&gt;p[0.15]{}&gt;p[0.15]{}&gt;p[0.15]{}&gt;p[0.15]{}]{} & & VAE & IWAE & HMC & SAGD[\
]{} \[-1.2em\][Normal]{} & \[-0.5em\][$D$]{} & 0.033 (0.0024) & 0.043 (0.0035) & 0.032 (0.0019) & 0.033 (0.0020)[\
]{} & \[-0.5em\][$W$]{} & 0.052 (0.0040) & 0.066 (0.0047) & 0.050 (0.0032) & 0.052 (0.0035)[\
]{} \[-1.2em\][Exp(2)]{} & \[-0.5em\][$D$]{} & 0.095 (0.0018) & 0.059 (0.0039) & 0.085 (0.0016) & 0.064 (0.0029)[\
]{} & \[-0.5em\][$W$]{} & 0.226 (0.0079) & 