Projected Latent Markov Chain Monte Carlo: Conditional Sampling of Normalizing Flows

In this experiment, we infer a missing central quarter (an $8\times 8$ pixel square) of CIFAR-10 (Krizhevsky et al., 2009) images. The CIFAR-10 dataset is composed of $60,000$ full color $32\times 32$ images of $10$ distinct classes of objects, with $6,000$ images provided for each class. The standard training and test set split for the CIFAR-10 dataset is $50,000$ and $10,000$ images, respectively.

Our chosen normalizing flow is a variant of the GLOW (Kingma & Dhariwal, 2018) architecture. We utilized a publicly available, pre-trained model (van Amersfoort, 2019) for this experiment. In the terminology of Kingma & Dhariwal (2018), the model has a depth of flow (K) of 32 and a total of 3 levels (L) and flow layers utilize $512$ hidden channels. The model was reportedly trained for a total of $1,500$ epochs using Adamax with a learning rate of $5\times 10^{-4}$ and a batchsize of $64$. We presume, but cannot guarantee, that the model was trained on the standard $50,000$ example CIFAR-10 training set. Examples of unconditioned samples from this model are provided within Figure 10, as obtained with the standard sampling variance, $\sigma=1.0$ (temperature $T=1.0$, in the terminology of Kingma & Dhariwal (2018)). From these unconditioned samples, it is clear that the model has not collapsed to memorizing the training set.

The particular implementation of the normalizing flow most easily provided access to a coordinate dependent representation of the latent space, which we call absolute coordinates for the latent space. Fundamentally, the Markov Chain within PL-MCMC may utilize any convenient representation of the latent space, so long as a diffeomorphism still maps that representation back to the modeled data. For this CIFAR-10 experiment, we chose to employ a Markov Chain within the architecture’s absolute coordinates. As a general note, the qualifier “absolute” merely refers to the representation favored by the flow’s implementation, while the qualifier “relative” refers to the representation best coinciding with the chosen prior distribution for the flow. The terms only reflect aspects of our practical usage of the representations, as there is no theoretically favored diffeomorphic representation of the latent space.

For inference, we selected images from the standard CIFAR-10 test set. During inference with PL-MCMC, latent space transitions are generated within the above mentioned absolute coordinates of the flow’s latent space by small perturbations from the current latent state. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For this experiment, we select $\sigma_{p}=0.01$ and $\sigma_{a}=1\times 10^{-3}$. PL-MCMC is carried out over 25,000 proposals.

In this experiment, we infer a missing right half (a $64\times 32$ pixel rectangle) of CelebA (Liu et al., 2015) images. The CelebA dataset is composed of $202,599$ full color images of celebrity faces. We utilize the aligned and cropped version of the CelebA dataset, resized to a size of $64\times 64$.

Our chosen normalizing flow is a variant of the GLOW (Kingma & Dhariwal, 2018) architecture. We utilized a publicly available, pre-trained model (Yuki-Chai, 2019) for this experiment. In the terminology of Kingma & Dhariwal (2018), the model has a depth of flow (K) of 32 and a total of 3 levels (L) and flow layers utilize $512$ hidden channels. The model was reportedly trained for a total of $1,500$ epochs using Adamax with a learning rate of $1\times 10^{-3}$ and a batchsize of $12$. We believe that this model was trained on the entirety of the CelebA dataset, with no withheld test or validation set. Examples of unconditioned samples from this model are provided within Figures 11 and 12, as obtained with reduced and standard sampling variance, $\sigma=0.5$ and $\sigma=1.0$ respectively (temperatures $T=0.5$ and $T=1.0$, in the terminology of Kingma & Dhariwal (2018)). From these unconditioned samples, it is clear that the model has not collapsed to memorizing the training set.

As in the CIFAR-10 experiment, the particular implementation of the normalizing flow most easily provided access to a coordinate dependent representation of the latent space, which we call absolute coordinates for the latent space. For this CelebA experiment, we chose to employ a Markov Chain within what we call relative coordinates of the latent space. For this architecture, we have convenient access to three subsets of latent variables, $\bm{\xi}_{1},\bm{\xi}_{2},$ and $\bm{\xi}_{3}$ within absolute coordinates. Fixing $\bm{\xi}_{1}$ (resp. fixing $\bm{\xi}_{1}$ and $\bm{\xi}_{2}$) there is an invertible transformation $h_{2}(\bm{\xi}_{2};\bm{\xi}_{1})$ (resp. $h_{3}(\bm{\xi}_{3};\bm{\xi}_{1},\bm{\xi}_{2})$) such that $h_{2}(\bm{\xi}_{2};\bm{\xi}_{1})\sim\mathcal{N}(0,I)$ (resp. $h_{3}(\bm{\xi}_{3};\bm{\xi}_{1},\bm{\xi}_{2})\sim\mathcal{N}(0,I)$) under our flow’s prior. The Markov Chain in relative coordinates simply follows and proposes transitions for the triplet $(\bm{\xi}_{1},h_{2}(\bm{\xi}_{2};\bm{\xi}_{1}),h_{3}(\bm{\xi}_{3};\bm{\xi}_{1},\bm{\xi}_{2}))$.

As it appears that no test set had been withheld during training of the model, we selected images at random from the full dataset for our experiment. During inference with PL-MCMC, latent space transitions are generated within relative coordinates of the flow’s latent space by small perturbations from the current latent state. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\xi^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For this experiment, we select $\sigma_{p}=0.01$ and $\sigma_{a}=1\times 10^{-3}$. PL-MCMC is carried out over 25,000 proposals.

In this experiment, we infer missing portions of MNIST (LeCun et al., 1998) digits. The MNIST dataset is composed of $70,000$ monochrome $28\times 28$ images of handwritten digits. The standard training and test set split for the MNIST dataset is $60,000$ and $10,000$ images, respectively. Our data missingness mechanisms are independent missingness, where pixels are lost uniformly at random, patch missingness, where randomly located contiguous rectangular blocks are missing, and square observation, where only a randomly located contiguous square is observed.

Our chosen normalizing flow is a variant of the NICE (Dinh et al., 2014) architecture. Our implementation is a modification of that by Mu (2019). In the terminology of Kingma & Dhariwal (2018), the model has a depth of flow (K) of 5 and a total of 4 levels (L) and intermediate flow layers have a dimension of $1000$. The flow utilizes an independent logistic prior distribution. Rather than splitting even and odd pixels within coupling layers, we split between two randomly selected partitions that are chosen at the time of the flow’s initialization and remain fixed for all layers of the flow. Of course, better performance would be expected by selecting a flow architecture that best suits the spatial organization of image data. However, random partitioning ensures that the expected performance of the flow remains independent of the spatial structuring of the data.

The normalizing flow is trained for $1000$ epochs over the standard $60,000$ element MNIST training set using RMSprop with a learning rate of $1\times 10^{-5}$ and a momentum of $0.9$ and a batch size of $200$. The data is pre-processed by performing pixel-wise whitening of the dataset (subtracting pixel-wise observed means and dividing by pixel-wise observed standard deviation). The training procedure is intended to follow that used later for training from incomplete MNIST digits to serve as a baseline for comparison and is not intended to produce the best possible generative model of MNIST digits. Examples of unconditioned samples from this model are provided within Figure 13, as obtained with the standard sampling variance (temperature $T=1.0$, in the terminology of Kingma & Dhariwal (2018))

During inference with PL-MCMC, latent space transitions are generated within the absolute coordinates of the flow’s latent space, half of the time at random by small perturbations from the current latent state and half of the time entirely resampled at a reduced standard deviation. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. When completely resampling the latent state, proposals are generated following a resampling kernel, $g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\mathbf{0},\sigma_{r}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For this experiment, we select $\sigma_{p}=0.05$, $\sigma_{r}=0.5$, and $\sigma_{a}=1\times 10^{-3}$. To simplify calculation of Metropolis-Hastings acceptance probabilities, we employ the assumption that small displacements in the latent space result from $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$ while large displacements result from $g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$, which is valid in the limit of $\sigma_{p}<<\sigma_{r}$. Therefore, rather than utilize the true transition kernel $g(\bm{\xi}^{\prime}|\bm{\xi})$, we simply assume that $g(\bm{\xi}^{\prime}|\bm{\xi})\propto g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$ following a perturbation and $g(\bm{\xi}^{\prime}|\bm{\xi})\propto g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$ following a resample. PL-MCMC is carried out over 2,000 proposals. To determine conditional expectations, the results of $20$ independent PL-MCMC chains are averaged together.

In this experiment, we train normalizing flows to model the distribution of MNIST digits from incomplete training data. Our data missingness mechanisms are independent missingness, where pixels are lost uniformly at random, patch missingness, where randomly located contiguous rectangular blocks are missing, and square observation, where only a randomly located contiguous square is observed. For each missingness mechanism, we consider missingness rates of $0.3,0.6,$ and $0.9$. For training, we apply the missingness mechanism to the standard MNIST training set, resulting in a training set of $60,000$ incomplete digits. For testing, we apply the missingness mechanism to the standard MNIST test set, resulting in a test set of $10,000$ incomplete digits.

Our chosen normalizing flow is a variant of the NICE (Dinh et al., 2014) architecture. In the terminology of Kingma & Dhariwal (2018), the model has a depth of flow (K) of 5 and a total of 4 levels (L) and intermediate flow layers have a dimension of $1000$. The flow utilizes an independent logistic prior distribution. Rather than splitting even and odd pixels within coupling layers, we split between two randomly selected partitions that are chosen at the time of the flow’s initialization and remain fixed for all layers of the flow. The flow architecture and implementation is the same as used above for training from complete MNIST digits. Better performance could be obtained by selecting an architecture that best suits the spatial organization of image data or by scaling model complexity along with the missingness rate (at a missingness rate of $0.9$, we have a tenth as much observed data available for training as in the complete data case).

In all cases, the normalizing flow is trained for $1000$ epochs using RMSprop with a learning rate of $1\times 10^{-5}$ and a momentum of $0.9$ and a batch size of $200$. The data is pre-processed by performing pixel-wise whitening of the dataset (subtracting pixel-wise observed means and dividing by pixel-wise observed standard deviation). At each of the first 50 epochs of training, missing pixels are resampled following an independent normal distribution, such that $\mathbf{x}_{M}\sim\mathcal{N}(\mathbf{0},I)$ (in whitened coordinates). Every 50 epochs thereafter, missing values are resampled using PL-MCMC as applied to the flow being trained. During inference with PL-MCMC, latent space transitions are generated within the absolute coordinates of the flow’s latent space, half of the time at random by small perturbations from the current latent state and half of the time entirely resampled at a reduced standard deviation. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. When completely resampling the latent state, proposals are generated following a resampling kernel, $g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\mathbf{0},\sigma_{r}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For computation of Metropolis-Hastings probabilities, we employ the same approximation as used when inferring MNIST digits within the qualitative experiments. Throughout training, we use $\sigma_{p}=0.05$ and $\sigma_{a}=1\times 10^{-3}$. Within the first $500$ epochs, we utilize $\sigma_{p}=1.814$. After the first $500$ epochs, we resample at reduced variance with $\sigma_{p}=0.5$. These parameters and this training procedure were chosen because they provided acceptable performance when applied to training data with the moderate missingness rate of $0.6$. It would certainly be beneficial to determine a more principled approach to their selection. After PL-MCMC sampling, we clamp values between pixel-wise observed minimal and maximal values to produce a newly imputed training set for use in the next $50$ epochs of training. Figure 14 below illustrates the progression of how the training set is imputed over training.

For testing, we utilize the normalizing flows to reconstruct the $10,000$ element incomplete test sets. For this reconstruction, we utilize PL-MCMC chains over $2,000$ proposals following the same procedure as in training. During testing, we use $\sigma_{p}=0.05$, $\sigma_{p}=0.5$, and $\sigma_{a}=1\times 10^{-3}$. To collect statistics for performance measures, we train three normalizing flows on distinctly prepared (i.e. resampled missingness patterns) training sets, each of which is tested on five distinctly prepared test sets. We consider two methods of employing PL-MCMC to produce reconstructions of missing data. The first method is imputation with a single sample from a PL-MCMC chain (imputation with a sample from the conditional distribution) and the second method is imputation with the average across multiple samples from independent PL-MCMC chains (imputation with the conditional mean). As we average across the samples from $10$ independent PL-MCMC chains for the second method, our reported statistics for the first method encompass these additional $10$ replications for individual sample imputation performance.

As imputation performance measures, we consider per-pixel reconstruction RMSE and Fréchet Inception Distance (Heusel et al., 2017). To compute Fréchet Inception Distance, we use the implementation provided along with that of MisGAN (Li, 2019). Reconstruction RMSE is recorded within the original, unwhitened representation of pixel data. When imputing $T$ test examples with $\mathcal{M}_{i}$ denoting the set of pixel indices that are missing for the $i$-th example and $\hat{x}_{i,j}$ denoting our imputed estimate for the $j$-th pixel of the $i$-th example (having ground truth value $x_{i,j}$), our reconstruction RMSE is calculated as:

For comparison, we consider imputation using pixel-wise observed means and imputation using the convolutional variant of MisGAN (Li et al., 2018). We use the implementation of MisGAN provided by Li (2019). The MisGAN models are trained following the provided default parameters (500 epochs with a batch size of 64 with $\tau=0,\alpha=0.1,\beta=0.1,\gamma=0$, $\mathtt{maskgen}=\mathtt{fusion}$, $\mathtt{gp\_lambda}=10$, $\mathtt{n\_critic}=5$, and $\mathtt{n\_latent}=128$, with a three layer fully connected imputer network with 784 units in each layer).

In this experiment, we train normalizing flows to model the distributions of various continuous UCI datasets (Bache & Lichman, 2013) from incomplete training data. In all cases, our data missingness mechanism is independent missingness with a missingness rate of $0.5$. A summary of the UCI datasets used in this experiment is provided below in Table 3. For training, we apply the missingness mechanism to the entirety of a single copy of the dataset. For testing, we attempt to reconstruct the missing portions of the training set.

Our chosen normalizing flows are variants of the NICE (Dinh et al., 2014) architecture. In the terminology of Kingma & Dhariwal (2018), all models have a depth of flow (K) of 5 and a total of 4 levels (L) and intermediate flow layers have a dimension of $120$. The flows utilize an independent normal prior distribution. Rather than splitting even and odd pixels within coupling layers, we split between two randomly selected partitions that are chosen at the time of the flow’s initialization and remain fixed for all layers of the flow. As our implementation works most easily with an even number of attributes, we copy the concrete attributes and data missingness to double the number of attributes. By copying data missingness patterns, we ensure that the doubling does not introduce additional information for training. A summary of input attribute dimensions and training batch sizes is provided within Table 4.

In all cases, the normalizing flow is trained for $1000$ epochs using Adamax with a learning rate of $0.002$, $\beta_{1}=0.9$, and $\beta_{2}=0.999$. We duplicate the training sets (data missingness patterns included) $10$ times to form a larger training set without introducing additional information beyond that present in the original incomplete training set. The batch sizes used are listed in Table 4. The data is pre-processed by performing attribute-wise whitening of the dataset (subtracting attribute-wise observed means and dividing by attribute-wise observed standard deviation). At each of the first 50 epochs of training, missing attribute are resampled following an independent normal distribution, such that $\mathbf{x}_{M}\sim\mathcal{N}(\mathbf{0},I)$ (in whitened coordinates). Every 50 epochs thereafter, missing values are resampled using PL-MCMC as applied to the flow being trained. During inference with PL-MCMC, latent space transitions are generated within the absolute coordinates of the flow’s latent space, half of the time at random by small perturbations from the current latent state and half of the time entirely resampled at a reduced standard deviation. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\xi^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. When completely resampling the latent state, proposals are generated following a resampling kernel, $g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\mathbf{0},\sigma_{r}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For computation of Metropolis-Hastings probabilities, we employ the same approximation as used when inferring MNIST digits within the qualitative experiments. Throughout training, we use $\sigma_{p}=0.01$, $\sigma_{r}=1.0$, $\sigma_{a}=1\times 10^{-3}$. These parameters and this training procedure were chosen because they provided acceptable performance across the UCI datasets considered. It would certainly be beneficial to determine a more principled approach to their selection. After PL-MCMC sampling, we clamp values between attribute-wise observed minimal and maximal values to produce a newly imputed training set for use in the next $50$ epochs of training.

For testing, we utilize the normalizing flows to reconstruct the missing values from their training sets. For this reconstruction, we utilize PL-MCMC chains over $2,000$ proposals following the same procedure as in training. During testing, we use $\sigma_{p}=0.01$, $\sigma_{p}=1.0$, and $\sigma_{a}=1\times 10^{-3}$. To collect statistics for performance measures, we train five normalizing flows on distinctly prepared (i.e. resampled missingness patterns) training sets. We consider two methods of employing PL-MCMC to produce reconstructions of missing data. The first method is imputation with a single sample from a PL-MCMC chain (imputation with a sample from the conditional distribution) and the second method is imputation with the average across multiple samples from independent PL-MCMC chains (imputation with the conditional mean). As we average across the samples from $25$ independent PL-MCMC chains for the second method, our reported statistics for the first method encompass these additional $25$ replications for individual sample imputation performance. In the case of the concrete dataset, we consider the copied attribute values as an additional single sample from the conditional distribution that is also incorporated into averaging.

As an imputation performance measure, we consider per-attribute normalized MSE. When imputing $T$ test examples with $\mathcal{M}_{i}$ denoting the set of attribute indices that are missing for the $i$-th example and $\hat{x}_{i,j}$ denoting our imputed estimate for the $j$-th attribute of the $i$-th example (having ground truth value $x_{i,j}$), our normalized MSE is calculated as:

where $\sigma_{j}$ denotes the ground truth standard deviation of the $j$-th attribute.

For comparison, we consider imputation using attribute-wise observed means, imputation using the missForest (Stekhoven & Bühlmann, 2012) R package with default parameters, and imputation with VAEs using MIWAE (Mattei & Frellsen, 2019). For imputation with missForrest, no data preprocessing is employed. We use the implementation of MIWAE provided by Mattei (2019). In all cases, the VAE architecture employed has an intrinsic dimension $d$ of $10$, an encoder and decoder comprised of $3$ layers each with $128$ hidden units with ReLU activation functions, an independent normal prior, and a Student’s $t$ distribution observation model. In all cases, we utilize zero imputation as the MIWAE imputation function. For training, we use $20$ importance weights while for inference we use $10,000$ importance weights. We train the models using the provided default parameters ($2,000$ epochs using Adam with a learning rate of $0.001$ and a batch size of $64$). In all cases, the data is pre-processed by performing attribute-wise whitening of the dataset (subtracting attribute-wise observed means and dividing by attribute-wise observed standard deviation).

In these experiments, we use MCMC to estimate the conditional expectation for the missing portion of a single MNIST digit. In this case, the data missingness mechanism is a checkerboard pattern masking half of the digit, as shown in Figure 15.

To perform this inference, we use the same normalizing flow as used in the qualitative experiments in Section 5.3, detailed within Appendix D.3. During inference with PL-MCMC, latent space transitions are generated within the absolute coordinates of the flow’s latent space, half of the time at random by small perturbations from the current latent state and half of the time entirely resampled at a reduced standard deviation. When perturbing the latent state, proposals are generated following a perturbation kernel, $g_{p}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\xi^{\prime}\sim\mathcal{N}(\bm{\xi},\sigma_{p}^{2}I)$. When completely resampling the latent state, proposals are generated following a resampling kernel, $g_{r}(\bm{\xi}^{\prime}|\bm{\xi})$, such that $\bm{\xi}^{\prime}\sim\mathcal{N}(\mathbf{0},\sigma_{r}^{2}I)$. The auxiliary distribution, $q$, is chosen to target $\mathbf{y}_{O}\sim\mathcal{N}(\mathbf{x}_{O},\sigma_{a}^{2}I)$. For this experiment, unless otherwise specified, we select $\sigma_{p}=0.01$, $\sigma_{r}=1.0$, and $\sigma_{a}=1\times 10^{-3}$. For computation of Metropolis-Hastings probabilities, we employ the same approximation as used when inferring MNIST digits within the qualitative experiments.

Ideally, for comparison with PL-MCMC, we would consider employing a Metropolis-Hastings MCMC through the modeled data space proposing $\mathbf{x}_{M}^{\prime}\sim\mathcal{N}(\mathbf{x}_{M},\sigma_{MH}^{2}I)$, for some appropriately chosen $\sigma_{MH}$. However, we found that naive Metropolis-Hastings through the modeled data space required such a small $\sigma_{MH}$ that the Markov Chain made no discernible progress. We therefore resorted to per-missing-pixel Gibbs sampling, with missing pixel proposals generated following $x_{M,i}^{\prime}\sim\mathcal{N}(\mu_{i},\sigma_{i}^{2})$, with $\mu_{i}$ and $\sigma_{i}$ denoting the $i$-th missing pixel’s observed mean and standard deviation throughout the training set.

For both techniques, the initial state of Markov Chain (latent state for PL-MCMC and missing pixel values for Gibbs sampling) is determined by unconditional sampling from the model at standard variance (temperature $T=1.0$, in the terminology of Kingma & Dhariwal (2018)). In each replication, the conditional mean is estimated from the average of 100 independent Markov Chains. To gather statistics regarding the variance of estimated conditional mean RMSE, we perform 10 separate replications of the experiment.

As the evaluation of a PL-MCMC Metropolis-Hastings proposal involved two transformations through the normalizing flow while the evaluation of a Gibbs sampling proposal involves only one transformation, it can be argued that each PL-MCMC proposal was twice as costly as each Gibbs sample. For this reason, Figures 16 and 17 perform the same comparisons as Figures 7 and 8, but with respect to the number of flow transformations utilized in the Markov Chains. We can see that, even accounting for the relative computational costs of the two methods, PL-MCMC offers significantly improved sampling performance.

Figures 18 and 19 compare the effect of altering proposal distribution scale when the transition proposals are generated only by the perturbation kernel. For reference, the results from our default $\sigma_{p}=0.01$, $\sigma_{r}=1.0$, and $\sigma_{a}=1\times 10^{-3}$ PL-MCMC implementation are also included in red.

Figures 20 and 21 compare the effect of altering the scale of the transition proposal’s resampling kernel.

Given that our results from auxiliary distributions with standard deviations of $\sigma_{a}=10^{-3}$ and $\sigma_{a}=10^{-6}$ closely overlap, we may be concerned that the auxiliary distribution might dominate PL-MCMC’s behavior and reduce the procedure to a simple search in the latent space to best rebuild the observed data. To determine whether an auxiliary distribution with $\sigma_{a}=10^{-3}$ overwhelmingly dominates the conditional sampling process, we follow a PL-MCMC implementation with $\sigma_{a}=10^{-3}$ and determine how often the its decisions regarding proposal acceptance would be contradicted by an implementation with $\sigma_{a}=\infty$. These results are provided within Figure 22.

Fundamentally, we see that, at least with $\sigma_{a}=10^{-3}$ in this particular task, the two implementations are likely to agree in their decisions to accept or reject particular proposal transitions. Choosing an auxiliary distribution with $\sigma_{a}=10^{-3}$ does not overwhelmingly “bully” the Markov Chain into mindlessly reconstructing observed data. Still, there is a substantial probability that this choice of auxiliary distribution will alter the decision, which is the mechanism by which the auxiliary distribution helps to guide the Markov Chain and improve conditional sampling performance. We suspect that these decision change probability computations could be useful for tuning the choice of auxiliary distribution.