Probability often changes when we learn new information.
Conditional probability and Bayes’ theorem give a structured way to update beliefs using evidence.
Conditional probability updates probabilities after observing an event.
Bayes’ theorem lets you estimate a hidden cause from observed evidence.
Naïve Bayes turns Bayes’ theorem into a practical classifier by assuming conditional independence of features given the class.
flowchart TD
A[Conditional<br/>probability] -->|foundation| B[Bayes<br/>theorem]
D[Independent<br/>events] -->|implies| C[Independence]
C -->|simplifies| A
E[Prior] -->|with likelihood| B
F[Likelihood] -->|updates| H[Posterior]
G[Evidence] -->|normalises| B
B -->|yields| H
I[Naïve<br/>Bayes] -->|uses| B
J[Naïve<br/>assumption] -->|assumes| C
K[Features] -->|given class| J
L[Class] -->|conditions| J
I -->|predicts| M[Classification]
M -->|selects| L
style A fill:#90CAF9,stroke:#1E88E5,color:#000
style B fill:#90CAF9,stroke:#1E88E5,color:#000
style C fill:#90CAF9,stroke:#1E88E5,color:#000
style D fill:#CE93D8,stroke:#8E24AA,color:#000
style E fill:#CE93D8,stroke:#8E24AA,color:#000
style F fill:#CE93D8,stroke:#8E24AA,color:#000
style G fill:#CE93D8,stroke:#8E24AA,color:#000
style J fill:#CE93D8,stroke:#8E24AA,color:#000
style K fill:#CE93D8,stroke:#8E24AA,color:#000
style L fill:#CE93D8,stroke:#8E24AA,color:#000
style H fill:#C8E6C9,stroke:#2E7D32,color:#000
style I fill:#C8E6C9,stroke:#2E7D32,color:#000
style M fill:#C8E6C9,stroke:#2E7D32,color:#000
Probability distributions are the bridge between:
real-world randomness and mathematical modelling.
A random experiment produces outcomes.
A random variable turns those outcomes into numbers.
A probability distribution tells you how likely each number (or range of numbers) is.
Key takeaway:
A distribution is a complete “story” about uncertainty:
what values are possible, how likely they are, and how we summarise them (mean, variance).
Many ML models are probabilistic:
they assume data (or errors) follow a distribution.
Loss functions often come from distribution assumptions:
squared loss aligns with Gaussian noise.
Naïve Bayes (from the previous module) becomes practical once you can model:
\( P(X\mid Y) \)
using suitable distributions.
In practice:
choosing a distribution is a modelling decision.
It affects:
prediction, uncertainty estimates, and what “rare” or “typical” means in your data.
A random variable is a way to attach numbers to outcomes of a random experiment.
It lets us move from:
“what happened?”
to:
“what number should we analyse?”
Key takeaway:
A random variable is a function from the sample space to real numbers.
Once you define the random variable clearly, the rest (pmf/pdf/cdf, mean, variance) becomes systematic.
flowchart TD
PD["Probability<br/>distributions"] --> RV["Random<br/>variables"]
RV --> T["Types"]
T --> RV1["Discrete<br/>RVs"]
T --> RV2["Continuous<br/>RVs"]
RV --> F["PMF / PDF / CDF"]
RV --> S["Mean / Variance<br/>Covariance"]
RV --> J["Joint & Marginal<br/>distributions"]
RV --> X["Transformations"]
style PD fill:#90CAF9,stroke:#1E88E5,color:#000
style RV fill:#90CAF9,stroke:#1E88E5,color:#000
style T fill:#CE93D8,stroke:#8E24AA,color:#000
style F fill:#CE93D8,stroke:#8E24AA,color:#000
style S fill:#CE93D8,stroke:#8E24AA,color:#000
style J fill:#CE93D8,stroke:#8E24AA,color:#000
style X fill:#CE93D8,stroke:#8E24AA,color:#000
style RV1 fill:#CE93D8,stroke:#8E24AA,color:#000
style RV2 fill:#CE93D8,stroke:#8E24AA,color:#000
Once you can describe a random variable using a pmf or pdf, the next step is to use
named distributions that appear repeatedly in real data and in ML models.
Key takeaway:
Named distributions give you ready-made probability models for common patterns:
binary outcomes, counts, and measurement noise.
Machine Learning is built on mathematical principles that allow models to:
represent data
learn patterns
optimise performance
flowchart LR
DATA[Data]
MATH[Math Models]
OPT[Optimisation]
MODEL[Trained Model]
DATA --> MATH
MATH --> OPT
OPT --> MODEL
ML requires core mathematical tools to understand how ML algorithms work internally. Algebra deals with relationships between variables and quantities, while Calculus focuses on change and optimization.
March 12, 2026