Improving Predictive Performance in Probabilistic Programs with Stochastic Support by Optimizing Path Weights

Core Concepts

Probabilistic programs with stochastic support can be decomposed into a weighted sum of local posteriors associated with each possible program path. This decomposition reveals that using the full posterior implicitly performs Bayesian model averaging (BMA) over the paths. However, BMA weights can be unstable due to model misspecification or inference approximations, leading to suboptimal predictions. To address this, the authors propose alternative mechanisms for path weighting based on stacking and PAC-Bayes objectives, which can be implemented as a cheap post-processing step on top of existing inference engines.

Abstract

The content discusses the issue of Bayesian model averaging (BMA) in the context of probabilistic programs with stochastic support. Probabilistic programs with stochastic support can be decomposed into a weighted sum of local posteriors associated with each possible program path, and using the full posterior implicitly performs a BMA over these paths. However, the authors argue that BMA weights can be unstable due to model misspecification or inference approximations, leading to suboptimal predictions.
To address this issue, the authors propose two alternative mechanisms for path weighting:

Stacking: This optimizes the path weights to maximize predictive performance on held-out data, using either an explicit validation set or a leave-one-out cross-validation approach. The authors show how stacking can be implemented as a cheap post-processing step on top of existing inference engines.

PAC-Bayes: The authors introduce a regularized stacking objective inspired by PAC-Bayes bounds, which can help prevent overfitting of the path weights.

The authors evaluate the proposed methods on a variety of synthetic and real-world examples, demonstrating that the alternative weighting schemes can lead to more robust weights and better predictive performance compared to the default BMA weights.

Stats

The authors use the following datasets in their experiments:
Subset Regression:

15 covariates, 200 data points for training, 1000 for evaluation
Function Induction:

400 data points for training, 1000 for evaluation
California (regression):

20,640 data points
Diabetes (classification):

442 data points
Stroke (classification):

5,110 data points
Radon Contamination:

Measurements from houses in different US counties

Quotes

"BMA often performs poorly under model misspecification (Gelman and Yao, 2020; Oelrich et al., 2020), wherein it tends to produce overconfident posterior model weights that collapse towards a single model (Huggins and Miller, 2021; Yang and Zhu, 2018)."
"Crucially, BMA implicitly assumes that the data was sampled from exactly one of the constituent models. This is often referred to as the M-closed assumption (Bernardo and Smith, 2009; Clyde and Iversen, 2013; Key et al., 1999)."

Key Insights Distilled From

Beyond Bayesian Model Averaging over Paths in Probabilistic Programs with Stochastic Support

by Tim Reichelt... at arxiv.org 04-15-2024

https://arxiv.org/pdf/2310.14888.pdf

Deeper Inquiries

How can the proposed stacking and PAC-Bayes approaches be extended to handle more complex program structures, such as hierarchical models or models with continuous latent variables

The proposed stacking and PAC-Bayes approaches can be extended to handle more complex program structures by incorporating additional layers of hierarchy and continuous latent variables into the modeling framework.
For hierarchical models, the stacking approach can be adapted to assign weights not only to individual SLPs but also to groups of SLPs that represent different levels of the hierarchy. This can be achieved by introducing additional parameters that capture the hierarchical structure of the model and optimizing the weights accordingly.
In the case of models with continuous latent variables, the stacking and PAC-Bayes approaches can be modified to account for the continuous nature of the variables. This can involve using different types of distributions to model the latent variables and adjusting the weighting schemes to accommodate the continuous space of possible values. Techniques such as variational inference or Gaussian processes can be employed to handle the continuous aspects of the model and optimize the weights effectively.
Overall, by incorporating these modifications, the stacking and PAC-Bayes approaches can be extended to handle a wider range of complex program structures, providing more flexibility and robustness in inference and prediction.

What are the theoretical guarantees, if any, for the robustness and predictive performance of the stacking and PAC-Bayes weighting schemes compared to BMA

Theoretical guarantees for the robustness and predictive performance of the stacking and PAC-Bayes weighting schemes compared to BMA can be analyzed in the context of Bayesian model averaging and model selection.
In terms of robustness, the stacking approach offers a more flexible and adaptive way to assign weights to different models or SLPs, allowing for better handling of model misspecification and overfitting. By optimizing the weights based on predictive performance, stacking can potentially provide more robust and stable results compared to BMA, which tends to be sensitive to model assumptions and can lead to overconfident predictions.
PAC-Bayes objectives, on the other hand, provide a theoretical framework for understanding the generalization properties of Bayesian models and can offer guarantees on the predictive performance of the model. By incorporating PAC-Bayes regularization into the weighting schemes, the approach can control the complexity of the model and prevent overfitting, leading to more reliable predictions.
While BMA weights may have their own advantages in certain scenarios, such as simplicity and interpretability, the stacking and PAC-Bayes approaches offer a more data-driven and adaptive way to assign weights, potentially leading to improved predictive performance and robustness in a wider range of scenarios.

Are there any scenarios where the BMA weights might be preferable to the alternative weighting schemes, and if so, how can these be identified

There may be scenarios where the BMA weights are preferable to the alternative weighting schemes, particularly when the underlying assumptions of the models align well with the true data generating process. In cases where the models are well-specified and the model space is well-defined, BMA weights can provide a principled way to combine the predictions from different models and make inference decisions.
Identifying when BMA weights might be preferable involves assessing the level of model misspecification, the complexity of the model space, and the robustness of the inference algorithms. If the models are well-specified and the assumptions are met, BMA weights can offer a straightforward and interpretable way to combine the models. However, in cases of model misspecification or complex model structures, the alternative weighting schemes like stacking and PAC-Bayes can offer more flexibility and adaptability to handle these challenges and potentially lead to better predictive performance.