We study the linear contextual bandit problem where an agent has to select one candidate from a pool and each candidate belongs to a sensitive group. In this setting,candidates' rewardsmaynotbedirectly comparable between groups,for example when the agent is an employer hiring candidates from different ethnic groups and some groups have a lower reward due to discriminatory bias and/or socialinjustice.

This can be written succinctly as Y, b Y | ( X) (18) A.2 Calibration in Classification ECE used for break ties. For each model and dataset, the best performing model is then re-run with 50 random seeds to gather information about standard errors and statistical significance. Kernel Bandwidth We select the RBF kernel bandwidth for training on each dataset using the aforementioned hyperparameter optimization. For each county, we track the weather sequence of each year into a few summary statistics for each month (average/maximum/minimum temperatures, precipitation, cooling/heating degree days). All other hyperparameters are held constant, including the number of training steps.

Probabilistic Answer Set Programming under the credal semantics (PASP) extends Answer Set Programming with probabilistic facts that represent uncertain information. The probabilistic facts are discrete with Bernoulli distributions. However, several real-world scenarios require a combination of both discrete and continuous random variables. In this paper, we extend the PASP framework to support continuous random variables and propose Hybrid Probabilistic Answer Set Programming (HPASP). Moreover, we discuss, implement, and assess the performance of two exact algorithms based on projected answer set enumeration and knowledge compilation and two approximate algorithms based on sampling. Empirical results, also in line with known theoretical results, show that exact inference is feasible only for small instances, but knowledge compilation has a huge positive impact on the performance. Sampling allows handling larger instances, but sometimes requires an increasing amount of memory.

We analyze the estimation of information theoretic measures of continuous random variables such as: differential entropy, mutual information or Kullback-Leibler divergence. The objective of this paper is two-fold. First, we prove that the information theoretic measure estimates using the k-nearest-neighbor density estimation with fixed k converge almost surely, even though the k-nearest-neighbor density estimation with fixed k does not converge to its true measure. Second, we show that the information theoretic measure estimates do not converge for k growing linearly with the number of samples. Nevertheless, these nonconvergent estimates can be used for solving the two-sample problem and assessing if two random variables are independent.

In this post, we will walk through the building blocks of probability theory and use these learnings to motivate fundamental ideas in machine learning. In the first section, we will talk about random variables and how they help quantify real world experiments. The final section will talk about how these mathematical concepts are used together to solve machine learning problems. Let's begin our journey with a fun experiment. Take a pen and paper; go outside to the main street in front of your house. Look at every person that walks passed you and take note their hair color; some approximation of their height in centimeters; and any other detail you find interesting. Do this for about 10 minutes. You conducted your first experiment! With this experiment, you can now answer some questions: How many people walked passed you?

The power and expressivity of Probabilistic Logic Programming (PLP) [8, 18] have been utilized to represent many real world situations [2, 9, 14]. Usually, probabilistic logic programs involve only discrete random variables with Bernoulli or Categorical distributions. Numerous solutions emerged to also handle continuous distributions [10, 12, 25], increasing the expressiveness of PLP and giving birth to hybrid probabilistic logic programs, that is, programs that include discrete and continuous random variables. Inference in this type of programs is hard since it combines the complexity of the grounding computation with the intractability of a distribution defined by a mixture of random variables. Usually, inference in general hybrid probabilistic logic programs (i.e., without imposing restrictions on the type of distributions allowed) is done by leveraging knowledge compilation and using external solvers [25] or by sampling [4, 16].

Data science is an interdisciplinary field. One of the building blocks of data science is statistics. Without a decent level of statistics knowledge, it would be highly difficult to understand or interpret the data. Statistics helps us explain the data. We use statistics to infer results about a population based on a sample drawn from that population.

We analyze the estimation of information theoretic measures of continuous random variables such as: differential entropy, mutual information or Kullback-Leibler divergence. The objective of this paper is two-fold. First, we prove that the information theoretic measure estimates using the k-nearest-neighbor density estimation with fixed k converge almost surely, even though the k-nearest-neighbor density estimation with fixed k does not converge to its true measure. Second, we show that the information theoretic measure estimates do not converge for k growing linearly with the number of samples. Nevertheless, these nonconvergent estimates can be used for solving the two-sample problem and assessing if two random variables are independent.