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# Foundations of Probability and Statistics

(Stanford University - Hank Ping Han Hsieh)

### - Overview

Artificial intelligence (AI) is nearly always associated with computer science and engineering. Although obviously dependent on massive computational resources, AI has nevertheless required substantial statistical input along its entire developmental path.

Both probability and statistics are important in AI and machine learning (ML). Probability is a fundamental concept in ML. It allows you to model uncertainty, variability, and noise in your data and algorithms. Probability is about predicting the likelihood of future events.

ML algorithms are designed to learn from data and make predictions or decisions based on that data. However, this process is inherently uncertain, as the data may contain noise or be incomplete. Probability and statistics are best applied in ML by using them to model uncertainty, characterize predictor behavior, and make predictions about future events.

Statistics is used for making predictions and drawing insights from data. Statistics involves the analysis of the frequency of past events. Statistical foundation is crucial to finding insights and drawing conclusions from the data.

ML heavily utilizes statistics, and statistics is built upon probability theory. However, there are fundamental differences in the mindsets. Probability enables us to reason about uncertainty; statistics quantifies and explains it. ML makes predictions from data.

### - Important Topics in Probability and Statistics

In the context of ML and data science, some important topics in probability and statistics include:

• Probability theory: Including concepts such as random variables, probability distributions, and conditional probability.
• Statistical inference: Covering topics like estimation, hypothesis testing, and confidence intervals.
• Regression analysis: Understanding linear regression, logistic regression, and other regression techniques for modeling relationships between variables.
• Classification: Exploring techniques such as decision trees, support vector machines, and k-nearest neighbors for classifying data.
• Clustering: Studying methods like k-means clustering and hierarchical clustering for grouping similar data points.
• Bayesian statistics: Understanding the principles of Bayesian inference and its applications in machine learning.
• Resampling methods: Including bootstrapping and cross-validation for evaluating the performance of predictive models.
• Dimensionality reduction: Exploring techniques like principal component analysis (PCA) and t-distributed stochastic neighbor embedding (t-SNE) for reducing the dimensionality of data.
• Time series analysis: Covering methods for analyzing and forecasting time-dependent data, such as autoregressive models and moving averages.
• Anomaly detection: Understanding techniques for identifying abnormal patterns or outliers in data, such as isolation forests and one-class SVM.

These topics form a foundational understanding of probability and statistics in the context of ML and data science.

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### - TensorFlow Probability (TFP)

TensorFlow Probability (TFP) is a Python library built on TensorFlow that makes it easy to combine probabilistic models and deep learning on modern hardware (TPU, GPU). It's for data scientists, statisticians, ML researchers, and practitioners who want to encode domain knowledge to understand data and make predictions.

TFP includes:

• A wide selection of probability distributions and bijectors.
• Tools to build deep probabilistic models, including probabilistic layers and a `JointDistribution` abstraction.
• Variational inference and Markov chain Monte Carlo.
• Optimizers such as Nelder-Mead, BFGS, and SGLD.

Since TFP inherits the benefits of TensorFlow, you can build, fit, and deploy a model using a single language throughout the lifecycle of model exploration and production. TFP is open source and available on GitHub. To get started, see the TensorFlow Probability Guide.

### - TensorFlow Statistics (TFS)

TensorFlow Statistics (TFS) is a library that provides statistical tools and functions for TensorFlow. It includes a variety of features for data analysis, such as:

• Descriptive statistics: TFS provides functions for calculating basic descriptive statistics, such as mean, median, mode, standard deviation, and variance.
• Hypothesis testing: TFS provides functions for performing hypothesis tests, such as t-tests, chi-squared tests, and ANOVA.
• Regression analysis: TFS provides functions for performing regression analysis, such as linear regression, logistic regression, and polynomial regression.
• Time series analysis: TFS provides functions for performing time series analysis, such as autoregressive integrated moving average (ARIMA) models and exponential smoothing models.
• Machine learning: TFS provides functions for building and training machine learning models, such as support vector machines (SVMs), decision trees, and random forests.

TFS is a powerful tool for data analysis and ML. It can be used to analyze a wide variety of data sets, and it provides a variety of features for building and training ML models.

[More to come ...]