Probabilistic-Programming

I recently read the paper Carbon-Aware Computing for Data Centers with Probabilistic Performance Guarantees which demonstrated the benefits and power of a Distributionally Robust Optimization method and I wanted to dive deeper into the method. This blog post is a result of the deep dive. Introduction In Stochastic Optimization (SO), we are interested in solutions that account for different possible outcomes of uncertain data. Accounting for this uncertainty requires us to define a probability distribution over the uncertain quantities—a forecast error, a demand signal, a compute load—and then optimize an objective defined with respect to that distribution. The choice of how to use the distribution depends on the application: one might minimize expected cost, guard against a high-percentile loss using Conditional Value-at-Risk (CVaR), or enforce a chance constraint that a capacity limit is violated with probability no greater than some threshold. What all of these approaches share is the assumption that we know, or can accurately estimate, the distribution itself. Distributionally Robust Optimization is the framework we reach for when that assumption breaks down. ...

Many applications of optimization and control are performed in a deterministic setting. That is, the quantities of the problem such as the state variables, control variables, and or parameters of a model are treated as fixed known values. This assumption may be reasonable in applications such as robotics, but in others such as resource allocation, we may need to incorporate the uncertainty of various quantities in order to compute the expectation of the objective. ...

Hierarchical regression, also known as multilevel modeling, is a powerful modeling technique that allows one to analyze data with a nested structure. This approach is particularly useful when dealing with data that has natural groupings, such as students within schools, patients within hospitals, or in the example below, product configurations within manufacturing processes. One of the key advantages of hierarchical regression lies in its ability to handle missing data in groups, i.e., when one group may not share the same covariates as another group or some groups may contain missong observations. ...

Alternative sampling backends This blog post is a copy of the alternative samplers documentation I wrote for Bambi. The original post can be found here. In Bambi, the sampler used is automatically selected given the type of variables used in the model. For inference, Bambi supports both MCMC and variational inference. By default, Bambi uses PyMC’s implementation of the adaptive Hamiltonian Monte Carlo (HMC) algorithm for sampling. Also known as the No-U-Turn Sampler (NUTS). This sampler is a good choice for many models. However, it is not the only sampling method, nor is PyMC the only library implementing NUTS. ...

Interpret Advanced Usage The interpret module is inspired by the R package marginaleffects and ports the core functionality of {marginaleffects} to Bambi. To close the gap of non-supported functionality in Bambi, interpret now provides a set of helper functions to aid the user in more advanced and complex analysis not covered within the comparisons, predictions, and slopes functions. These helper functions are data_grid and select_draws. The data_grid can be used to create a pairwise grid of data points for the user to pass to model.predict. Subsequently, select_draws is used to select the draws from the posterior (or posterior predictive) group of the InferenceData object returned by the predict method that correspond to the data points that “produced” that draw. ...

#| code-fold: true import matplotlib.pyplot as plt import torch import numpy as np from botorch.acquisition import qLogExpectedImprovement from botorch.fit import fit_gpytorch_model from botorch.models import SingleTaskGP from botorch.optim import optimize_acqf from gpytorch.mlls import ExactMarginalLogLikelihood from torch.distributions import Normal plt.style.use("https://raw.githubusercontent.com/GStechschulte/filterjax/main/docs/styles.mplstyle") Outcome constraints In optimization, it is often the goal that we need to optimize an objective function while satisfying some constraints. For example, we may want to minimize the scrap rate by finding the optimal process parameters of an manufacturing machine. However, we know the scrap rate cannot be below 0. In another setting, we may want to maximize the throughput of a machine, but we know that the throughput cannot exceed the maximum belt speed of the machine. Thus, we need to find regions in the search space that both yield high objective values and satisfy these constraints. In this blog, we will focus on inequality outcome constraints. That is, the domain of the objective function is ...

Survival Models Survival models, also known as time-to-event models, are specialized statistical methods designed to analyze the time until the occurrence of an event of interest. In this notebook, a review of survival analysis (using non-parametric and parametric methods) and censored data is provided, followed by a survival model implementation in Bambi. This blog post is a copy of the survival models documentation I wrote for Bambi. The original post can be found here. ...

Predict New Groups In Bambi, it is possible to perform predictions on new, unseen, groups of data that were not in the observed data used to fit the model with the argument sample_new_groups in the model.predict() method. This is useful in the context of hierarchical modeling, where groups are assumed to be a sample from a larger group. This blog post is a copy of the zero inflated models documentation I wrote for Bambi. The original post can be found here. ...

#| code-fold: true import arviz as az import matplotlib.pyplot as plt from matplotlib.lines import Line2D import numpy as np import pandas as pd import warnings import bambi as bmb warnings.filterwarnings("ignore", category=FutureWarning) WARNING (pytensor.tensor.blas): Using NumPy C-API based implementation for BLAS functions. Ordinal Regression This blog post is a copy of the ordinal models documentation I wrote for Bambi. The original post can be found here. In some scenarios, the response variable is discrete, like a count, and ordered. Common examples of such data come from questionnaires where the respondent is asked to rate a product, service, or experience on a scale. This scale is often referred to as a Likert scale. For example, a five-level Likert scale could be: ...

#| code-fold: true import arviz as az import matplotlib.pyplot as plt from matplotlib.lines import Line2D import numpy as np import pandas as pd import scipy.stats as stats import seaborn as sns import warnings import bambi as bmb warnings.simplefilter(action='ignore', category=FutureWarning) WARNING (pytensor.tensor.blas): Using NumPy C-API based implementation for BLAS functions. Zero inflated models This blog post is a copy of the zero inflated models documentation I wrote for Bambi. The original post can be found here. ...