Probability & Statistics for AI: The Complete Guide

Core Concepts:

• Random Variables: Discrete vs continuous
• Distributions: Gaussian, Bernoulli, Poisson
• Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B)
• Expectation/Variance: E[X], Var(X)

AI Applications:

• Naive Bayes classifiers (NLP)
• Gaussian processes (surrogate models)
• Bernoulli distributions (binary outcomes)

Python Implementation:

# Working with distributions in Python
import numpy as np
from scipy.stats import norm, bernoulli

# Gaussian PDF
x = np.linspace(-3, 3, 100)
pdf = norm.pdf(x, loc=0, scale=1)  # μ=0, σ=1

# Bernoulli sampling
samples = bernoulli.rvs(p=0.3, size=1000)  # 30% probability of 1
        

Key Principles:

• Bias-Variance Tradeoff: Underfitting vs overfitting
• VC Dimension: Model complexity measure
• PAC Learning: Probably Approximately Correct
• Empirical Risk: R̂(h) = (1/n)ΣL(h(x_i),y_i)

Learning Curves:

Scenario	Training Error	Validation Error	Diagnosis
High Bias	High	High	Underfitting
High Variance	Low	High	Overfitting

Theoretical Insight:

Modern deep learning models often violate traditional statistical learning assumptions due to their overparameterization

Advanced Techniques:

• Markov Chains: Memoryless processes
• Hidden Markov Models: Sequential data
• Bayesian Networks: Directed acyclic graphs
• Variational Inference: Approximate Bayesian methods

Pyro Implementation (VAE):

# Simplified Variational Autoencoder in Pyro
import pyro
import pyro.distributions as dist

def model(x):
    z = pyro.sample("z", dist.Normal(0, 1))
    return pyro.sample("x", dist.Bernoulli(logits=decoder(z)), obs=x)

def guide(x):
    loc = pyro.param("loc", torch.zeros(100))
    scale = pyro.param("scale", torch.ones(100), constraint=dist.constraints.positive)
    pyro.sample("z", dist.Normal(loc, scale))
        

Modern Applications:

Diffusion models use Markov chains to gradually denoise data, while transformers employ self-attention over probabilistic embeddings