Probability and Statistics for AI

Math for AI: Probability and Statistics with a Computer Science Perspective (Using Python)

Introduction

After exploring Linear Algebra for AI, the next logical step in building a strong foundation for Artificial Intelligence is understanding Probability and Statistics. These two branches of mathematics are essential for reasoning under uncertainty, which is at the heart of most AI applications.

In real-world problem solving, we often start with qualitative ideas, but when we express these in numbers — quantitative data — we begin to apply statistics. In short, statistics is the science of making decisions based on data.

Why Probability & Statistics Matter in Computer Science

In pure math, we usually deal with small-scale values (e.g., 10–20 values), which is manageable. But in computer science, especially in AI, we work with thousands to millions of data points. This creates scalability challenges, making statistics and probability crucial for:

Data Science (large-scale statistics)
Machine Learning (glorified probability)
Algorithm analysis, cryptography, and system modeling

They help us model and analyze uncertainty, a core concern in most computing problems.

Getting Started with Python for Stats & Probability

We’ll use Python as it's general-purpose and well-supported for scientific computing.

We’ll also use a virtual environment (via Conda) to isolate packages and ensure compatibility across systems. This environment can be easily ported to another machine, similar to a virtual machine.

Before setting up the environment, it's helpful to explore an open-source GitHub repository that provides foundational code and material:

git clone https://github.com/recluze/stats-prob-cs

This repository includes course notebooks and examples for deeper understanding. Browsing the code will help you get comfortable with the basics.

Note: I'm using Arch Linux. Some commands may vary slightly on other distros. For best results, use Linux or macOS over Windows.

Install Python:

sudo pacman -Syu python

Set up a virtual environment:

wget https://repo.anaconda.com/miniconda/Miniconda3-latest-Linux-x86_64.sh
bash Miniconda3-latest-Linux-x86_64.sh
conda create -n stat-env python=3.11
conda activate stat-env

Install required libraries:

conda install jupyterlab pandas matplotlib

jupyterlab: Web-based IDE for notebooks.
pandas: For structured data manipulation.
matplotlib: For plotting graphs and visualizations.

Understanding Data Types

In this course, we deal with two types of data:

Artificial Data – Collected through surveys (manual input).
Organic Data – Generated by systems and logs (e.g., website user activity).

Organic data is especially relevant in computing as it reflects real-world system behavior.

In machine learning, data must often be i.i.d. (independent and identically distributed) for modeling.

One-Hot Encoding

When comparing categorical variables (e.g., male vs. female), one-hot vectors are used to represent them numerically without implying order or magnitude:

Example:

[male, female] = [1, 0], [0, 1]

This avoids incorrect assumptions like "greater = male" and "smaller = female" which may happen if scalar values are used.

If we represent unordered categorical data as scalars instead of arrays, we distort the structure and meaning of that data.

Foundations of Probability

Probability is axiomatic — based on agreed-upon rules that help us model uncertainty. It’s not about proving what's true, but about agreeing on how to handle unknowns.

Topics and Their Computer Science, Hardware & Software Perspectives

Probability and statistics touch almost every aspect of computing. Below is a brief overview of key topics in this domain, described from three perspectives: computer science use cases, hardware impact, and software support.

These concepts help build intelligent systems that make decisions, process uncertainty, and handle large-scale data.

Topic	Description	Computer Science Perspective	Hardware Perspective	Software Perspective
Data Types	Different forms of data: structured, unstructured	Organize and clean datasets	Efficient memory access	pandas, NumPy
One-Hot Vectors	Binary representation for categories	Preprocessing for ML models	Stored efficiently in RAM	Used in embeddings, TensorFlow
Histograms & Visualizations	Graphical summary of data distribution	Detect patterns and anomalies	GPU acceleration for big data	matplotlib, seaborn
Central Tendency	Mean, Median, Mode	Summarize large datasets	Requires optimized storage	numpy.mean(), pandas functions
Variance & Standard Deviation	Measure of data spread	Feature scaling and normalization	Computed using parallel float ops	scikit-learn, NumPy
Entropy & Information	Measure of uncertainty	Decision trees, NLP models	GPU-based entropy calculation	sklearn, scipy.stats
Probability & Events	Likelihood of an event	Foundation of model prediction	RNG hardware/simulation	Python’s random, numpy.random
Conditional Probability	Probability given a condition	Bayesian models, filters	Efficient table lookups	scikit-learn, PyMC3
Bayes’ Rule	Update belief with evidence	Spam filters, medical diagnosis	Uses logarithmic transformations	Naive Bayes classifiers
Random Variables	Variables with probabilistic outcomes	Modeling random processes	Supported by dedicated chips	NumPy, PyMC3, TensorFlow
Distributions (PMF/PDF)	Probability models for data	Classification, noise estimation	SIMD instructions for speed	scipy.stats, torch.distributions
Joint & Marginal Probability	Modeling multivariate scenarios	NLP, image classification	Parallel computation across cores	pandas, statsmodels
Expected Values	Long-run average outcomes	Policy evaluation in RL	Summation using vector units	NumPy, torch.mean
KL Divergence	Measure of distribution difference	Regularization, GANs	Log-sum-exp ops on GPU	PyTorch, TensorFlow
Bayesian Inference	Probabilistic belief update	Uncertainty-aware learning	Monte Carlo simulations	PyMC3, NumPyro, Edward

Summary

Probability and statistics are crucial for making decisions under uncertainty, which is common in all computing systems.
They form the mathematical backbone of machine learning and data science.
From a hardware view, modern CPUs/GPUs accelerate statistical computation.
On the software side, Python libraries like NumPy, Pandas, SciPy, and PyTorch simplify implementation.

Whether you're analyzing logs, training models, or building systems like Jetson Nano for robotics — understanding probability and statistics is essential to designing intelligent, efficient, and scalable solutions.

P.S:if you spot any mistakes, feel free to point them out — we’re all here to learn together! 😊

Haris
FAST-NUCES
BS Computer Science | Class of 2027

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