Bayesian Inference: Priors, Conjugacy & MCMC

Learning Goal: Construct prior-to-posterior transitions, analyze and derive conjugate priors (such as Beta-Binomial and Normal-Normal models), and design and evaluate Markov Chain Monte Carlo (MCMC) samplers to perform exact and approximate probabilistic modeling.

  • Prerequisites: Basic multi-variable calculus (integration), linear algebra, and introductory Python programming (knowledge of NumPy is helpful).
  • Estimated Study Time: 32 hours

Module 1: Probability Foundations & Bayes' Theorem

This module establishes the foundational probability theory required for Bayesian statistics. You will study conditional probability, visual interpretations of updating beliefs, and the formal mathematical derivation of Bayes' Theorem.

Recommended Videos

Understanding Bayes' Theorem: The Geometry of Changing Beliefs
Understanding Bayes' Theorem: The Geometry of Changing Beliefs
@3blue1brown15:105,535,953
Understanding Bayes' Theorem: The Geometry of Changing Beliefs

Bayes' theorem provides a mathematical framework for updating our beliefs based on new evidence, expressed as P(H|E) = [P(E|H) × P(H)] / P(E), where P(H|E) is the posterior probability of a hypothesis given the evidence, P(E|H) is the likelihood of observing the evidence if the hypothesis is true, P(H) is the prior probability of the hypothesis before seeing the evidence, and P(E) is the total probability of the evidence. The theorem demonstrates that new evidence should not completely determine

@3blue1brown · 15:10

Why this video: This video provides a geometric and visual intuition of Bayes' theorem using area models. It helps build a conceptual foundation before you dive into the algebraic representations of updating beliefs under uncertainty.

Understanding Bayes' Theorem: An Accessible Derivation and Applications in Bayesian Statistics
Understanding Bayes' Theorem: An Accessible Derivation and Applications in Bayesian Statistics
@statquest13:59477,710
Understanding Bayes' Theorem: An Accessible Derivation and Applications in Bayesian Statistics

Bayes' Theorem states that the conditional probability of an event A given event B equals the conditional probability of B given A multiplied by the probability of A, divided by the probability of B (P(A|B) = P(B|A) × P(A)/P(B)). This theorem allows us to update our beliefs about the probability of an event based on new evidence or information, making it fundamental to Bayesian statistics.

@statquest · 13:59

Why this video: StatQuest provides a clear, step-by-step mathematical derivation of Bayes' Theorem. This video bridges the gap between the geometry shown by 3Blue1Brown and the formal notation used in statistical modeling.

Conditional Probability With Venn Diagrams & Contingency Tables
Conditional Probability With Venn Diagrams & Contingency Tables
@TheOrganicChemistryTutor01:281,410,412
Conditional Probability With Venn Diagrams & Contingency Tables

To calculate conditional probability intuitively, ask how much of event A is contained within event B. For example, if rolling a six-sided die where A = {1,3,5} (odd numbers) and B = {3,4,5}, then A ∩ B = {3,5} which contains 2 elements out of B's 3 elements, so P(A|B) = 2/3.

@TheOrganicChemistryTutor · 01:28

Why this video: This short video offers a rapid, highly intuitive guide to calculating conditional probability using visual tools like Venn Diagrams and joint contingency tables.

Module 1 Knowledge Checkpoint

  • Write down the algebraic formula for Bayes' Theorem and label the Prior, Likelihood, Posterior, and Marginal Likelihood (Evidence).
  • Explain how a change in the sample space (conditioning on event BB) affects the denominator when computing conditional probability P(A∣B)P(A|B).
  • Draw a Venn diagram representing two intersecting dependent events and compute P(A∣B)P(A|B) given P(A∩B)P(A \cap B) and P(B)P(B).

Module 2: Probability Distributions & Likelihood

To work with statistical models, you must understand the distinction between discrete and continuous random variables, probability density functions (PDFs), and the key conceptual differences between probability and likelihood.

Recommended Videos

In Statistics, Probability is not Likelihood.
In Statistics, Probability is not Likelihood.
@statquest05:011,516,426
In Statistics, Probability is not Likelihood.

Probability represents the area under a fixed distribution curve for a given range of outcomes, calculated as P(data | distribution), while likelihood represents the y-axis value at a specific data point with distributions that can be moved or modified, calculated as L(distribution | data); in probability, the distribution parameters are fixed and we calculate areas for different data ranges, whereas in likelihood, the observed data is fixed and we evaluate how well different distributions expla

@statquest · 05:01

Why this video: One of the most common confusions in statistical inference is mixing up probability and likelihood. StatQuest clarifies this distinction, explaining how probability deals with finding the area under a fixed distribution, while likelihood varies parameters for a fixed dataset.

Continuous random variable and the probability density function
Continuous random variable and the probability density function
@Jochumzen08:031,584
Continuous random variable and the probability density function

A continuous random variable is defined as a function from the sample space to the real numbers whose range is an uncountable set (typically an interval), and it is characterized by a probability density function (PDF) such that the probability of the variable falling within any interval [a, b] is equal to the integral of the PDF over that interval; unlike discrete random variables, continuous random variables have the property that the probability of taking on any specific exact value is zero,

@Jochumzen · 08:03

Why this video: This lecture covers continuous random variables and how probability density functions behave. It is essential preparation for handling continuous parameters in Bayesian inference.

Likelihood Estimation - THE MATH YOU SHOULD KNOW!
Likelihood Estimation - THE MATH YOU SHOULD KNOW!
@CodeEmporium12:5972,748
Likelihood Estimation - THE MATH YOU SHOULD KNOW!

This section establishes the fundamental distinction between probability and likelihood through visual and mathematical explanations. Probability predicts data from known distributions (e.g., 35% chance house prices fall between $300k-$400k), with total area equaling 1. Likelihood reverses this: given data, it evaluates how well different parameter values explain observations. Ronald Fisher's formal definition states likelihood is proportional to the probability of observing data given specific

@CodeEmporium · 12:59

Why this video: This video provides a rigorous breakdown of likelihood functions. It explains how to construct joint likelihood functions mathematically for independent, identically distributed (i.i.d.) observations.

Module 2 Knowledge Checkpoint

  • Explain why the probability of a single point in a continuous distribution, P(X=x)P(X = x), is always zero, and how we use PDFs to calculate interval probabilities.
  • Given a probability density function f(x∣θ)f(x | \theta), explain what is held constant and what varies when evaluating it as a probability vs. evaluating it as a likelihood.
  • Write the joint likelihood function for nn independent, identically distributed (i.i.d.) normal observations.

Module 3: The Bayesian Framework: Prior to Posterior

This module introduces the Bayesian paradigm: treating parameters as random variables rather than fixed values. You will explore how prior distributions are systematically updated by the likelihood of observed data to yield a posterior distribution.

Recommended Videos

21. Bayesian Statistical Inference I
21. Bayesian Statistical Inference I
@mitocw48:49187,436
21. Bayesian Statistical Inference I

Bayesian statistical inference is a framework for updating beliefs about unknown parameters using observed data through Bayes' rule, where the posterior distribution combines prior beliefs with likelihoods derived from data; when forced to report a single point estimate, the conditional expectation of the unknown parameter given the data is optimal for minimizing mean squared error, though the complete posterior distribution conveys more comprehensive information about uncertainty than any singl

@mitocw · 48:49

Why this video: Part of MIT's open curriculum, this rigorous lecture introduces the formal framework of Bayesian inference, laying out the transition from prior to posterior using solid mathematical foundations.

L14.4 The Bayesian Inference Framework
L14.4 The Bayesian Inference Framework
@mitocw09:4883,719
L14.4 The Bayesian Inference Framework

The Bayesian inference framework treats unknown quantities as random variables with prior distributions, and updates these beliefs using observed data through Bayes' rule to obtain posterior distributions; this complete solution can then be summarized using point estimators like Maximum A Posteriori (MAP) which finds the most probable value, or the conditional expectation estimator which provides the mean of the posterior distribution.

@mitocw · 09:48

Why this video: A concise distillation from MIT that guides you through the components of Bayesian updating, summarizing how prior beliefs are adjusted as new data points arrive.

(ML 7.1) Bayesian inference - A simple example
(ML 7.1) Bayesian inference - A simple example
@mathematicalmonk04:00130,612
(ML 7.1) Bayesian inference - A simple example

Bayesian inference treats all unknown quantities as probability distributions rather than fixed values. Named after Reverend Thomas Bayes, this approach uses probability rules to update beliefs based on data. The framework involves placing distributions on parameters (prior), collecting data, and computing updated distributions (posterior). This differs fundamentally from frequentist approaches by treating uncertainty as a probability distribution over parameters rather than long-run frequencies

@mathematicalmonk · 04:00

Why this video: Mathematical Monk provides a clean, concrete illustration of Bayesian updating, showing how a discrete parameter's distribution shifts after introducing a basic dataset.

Module 3 Knowledge Checkpoint

  • Describe the primary philosophical difference between Bayesian inference and Frequentist inference regarding parameters.
  • Mathematically state why the posterior distribution is proportional to the product of the prior and the likelihood: p(θ∣x)∝p(x∣θ)p(θ)p(\theta | x) \propto p(x | \theta) p(\theta).
  • Explain the role of the marginal likelihood (the denominator in Bayes' theorem) and explain why it is often referred to as the normalizing constant.

Module 4: Conjugate Priors & Analytical Inference

When the posterior distribution belongs to the same probability family as the prior distribution, the prior is called "conjugate" to the likelihood. This allows you to bypass complex integration and compute posteriors analytically.

Note: The feedback pointed out a lack of step-by-step mathematical derivations in general conjugate overviews. We have solved this below by selecting specialized videos that walk through the detailed math of both Beta-Binomial and Normal-Normal conjugate models.

Recommended Videos

17 - Conjugate priors - an introduction
17 - Conjugate priors - an introduction
@oxeduc420905:3046,501
17 - Conjugate priors - an introduction

A conjugate prior is a prior distribution that, when combined with a specific likelihood function, produces a posterior distribution of the same functional form, enabling analytical solutions without complex computations; for example, a normal prior with normal likelihood yields a normal posterior, while a beta prior with Bernoulli likelihood yields a beta posterior, making Bayesian updating straightforward through parameter updates rather than requiring numerical methods.

@oxeduc4209 · 05:30

Why this video: An excellent, mathematically sound introduction to the definition and mechanics of conjugate priors, showing why they are useful for analytical Bayesian calculations.

Bayesian Statistics Crash Course
Bayesian Statistics Crash Course
@svcognac316313:23944
Bayesian Statistics Crash Course

This section demonstrates the mathematical derivation of the Beta-Binomial conjugate prior. Starting with a binomial likelihood and Beta prior, the posterior is derived by multiplying the prior and likelihood and rearranging terms. The posterior parameters are updated by adding successes to alpha and failures to beta. The marginal likelihood (normalizing constant) is calculated by integrating the product of prior and likelihood over all parameter values. This conjugate relationship allows analyt

@svcognac3163 · 13:23

Why this video: Despite its general name, this video directly addresses a key curriculum gap by demonstrating the mathematical derivation of the Beta-Binomial conjugate model. It shows exactly how the prior exponents and likelihood combinatorics combine to form a new Beta distribution.

20 - Beta conjugate prior to Binomial and Bernoulli likelihoods
20 - Beta conjugate prior to Binomial and Bernoulli likelihoods
@oxeduc420905:1149,651
20 - Beta conjugate prior to Binomial and Bernoulli likelihoods

The Beta distribution serves as a conjugate prior to both Binomial and Bernoulli likelihoods because when you multiply the Binomial likelihood (proportional to θ^Z × (1-θ)^(n-Z)) by the Beta prior (proportional to θ^(a-1) × (1-θ)^(b-1)), the resulting posterior distribution maintains the same functional form as the prior, specifically proportional to θ^(a+Z-1) × (1-θ)^(n+b-Z-1), which is itself a Beta distribution with updated parameters a' = a + Z and b' = n + b - Z.

@oxeduc4209 · 05:11

Why this video: This video provides a step-by-step mathematical walkthrough of updating a Beta prior with Binomial and Bernoulli likelihoods, reinforcing the algebraic steps needed to find posterior hyper-parameters.

[Bayesian inference for a mean] Unknown mean posterior derivation
[Bayesian inference for a mean] Unknown mean posterior derivation
@JingchenMonikaHu11:20852
[Bayesian inference for a mean] Unknown mean posterior derivation

This section establishes the mathematical foundations for Bayesian inference in the normal-normal conjugate model. The likelihood function for iid normal observations is expressed as a product of individual densities, with precision τ = 1/σ² used for algebraic simplicity. A normal prior N(μ₀, σ₀²) provides conjugacy, ensuring the posterior remains normal. The posterior is proportional to prior × likelihood, allowing focus on μ-dependent terms while absorbing constants. This conjugate relationshi

@JingchenMonikaHu · 11:20

Why this video: This video addresses our second curriculum gap by providing a step-by-step mathematical derivation of a Normal posterior from a Normal prior and a Normal likelihood. It carefully shows how to complete the square in the exponent.

Normal prior Normal likelihood Normal posterior distribution
Normal prior Normal likelihood Normal posterior distribution
@deetoher11:0457,199
Normal prior Normal likelihood Normal posterior distribution

When a normal distribution serves as the prior for a normal likelihood function, the resulting posterior distribution is also normal, with its precision equal to the sum of the prior precision and the likelihood precision, and its mean being a weighted average of the prior mean and the sample mean, where weights are proportional to their respective precisions.

@deetoher · 11:04

Why this video: This video reinforces Normal-Normal conjugacy, focusing on how the precision (inverse variance) of the posterior is simply the sum of the prior precision and the data likelihood precision.

Module 4 Knowledge Checkpoint

  • Derive the posterior distribution for a Beta(α,β)\text{Beta}(\alpha, \beta) prior and a Binomial(n,y)\text{Binomial}(n, y) likelihood. Show that the posterior is Beta(α+y,β+n−y)\text{Beta}(\alpha + y, \beta + n - y).
  • Define what a "conjugate prior" is and explain the mathematical advantages of using one.
  • For a Normal-Normal conjugate model, if your prior variance decreases (representing stronger prior beliefs), how does this affect the posterior mean relative to the sample mean?

Module 5: Markov Chain Monte Carlo (MCMC) & Metropolis-Hastings

In complex probabilistic models, calculating the marginal likelihood analytically is often impossible. This module covers Markov Chain Monte Carlo (MCMC) methods, focusing on how the Metropolis-Hastings algorithm allows us to sample from intractable posterior distributions.

Recommended Videos

Why do we need MCMC and how does it work? -- Ben Lambert (Oxford)
Why do we need MCMC and how does it work? -- Ben Lambert (Oxford)
@teachingandlearningmathematics25:026,589
Why do we need MCMC and how does it work? -- Ben Lambert (Oxford)

In Bayesian inference, we cannot directly compute the posterior distribution due to complex multi-dimensional integrals in Bayes' theorem, so we use Markov Chain Monte Carlo (MCMC) sampling to approximate it; unlike independent sampling which converges quickly to characterize distributions, MCMC uses dependent sampling where each draw depends on the current value, making convergence slower and requiring metrics like effective sample size to quantify efficiency, while convergence assessment is ch

@teachingandlearningmathematics · 25:02

Why this video: Dr. Ben Lambert provides a brilliant, clear explanation of why multidimensional integration fails in complex Bayesian models and why we need simulation techniques like MCMC.

Metropolis-Hastings Algorithm Explained: MCMC Data Science
Metropolis-Hastings Algorithm Explained: MCMC Data Science
@ritvikmath18:15138,835
Metropolis-Hastings Algorithm Explained: MCMC Data Science

The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo method that enables sampling from complex probability distributions when only the unnormalized density function f(x) is known. The algorithm works through two key steps: (1) proposing a new candidate state from a symmetric or asymmetric proposal distribution centered at the current state, and (2) accepting or rejecting this candidate based on an acceptance probability calculated as the minimum of 1 and the ratio of target densities

@ritvikmath · 18:15

Why this video: This video breaks down the Metropolis-Hastings algorithm step-by-step. It explains proposal distributions, acceptance ratios, and how the algorithm explores a parameter space without needing to compute the denominator of Bayes' Theorem.

Metropolis-Hastings Algorithm Explained | Markov Chain Monte Carlo
Metropolis-Hastings Algorithm Explained | Markov Chain Monte Carlo
@jaradniemi10:4186,812
Metropolis-Hastings Algorithm Explained | Markov Chain Monte Carlo

The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo method for sampling from a target distribution f(x) when direct sampling is not feasible, requiring only the ability to evaluate f(x) up to a normalizing constant; the algorithm works by iteratively proposing new values from a proposal distribution (which may depend on the current value), calculating an acceptance probability based on the ratio of target densities and proposal densities, and accepting or rejecting the proposal to ge

@jaradniemi · 10:41

Why this video: This video provides a clear mathematical explanation of the Metropolis-Hastings algorithm, making it easy to understand the core formulas behind proposal distributions and acceptance probabilities.

Module 5 Knowledge Checkpoint

  • Explain why computing the marginal likelihood p(x)=∫p(x∣θ)p(θ)dθp(x) = \int p(x|\theta)p(\theta)d\theta becomes mathematically intractable when θ\theta has many dimensions.
  • Write the mathematical expression for the Metropolis-Hastings acceptance probability α=min⁡(1,p(θ∗)q(θ(t)∣θ∗)p(θ(t))q(θ∗∣θ(t)))\alpha = \min\left(1, \frac{p(\theta^*) q(\theta^{(t)} | \theta^*)}{p(\theta^{(t)}) q(\theta^* | \theta^{(t)})}\right).
  • Describe the difference between a symmetric proposal distribution (Metropolis algorithm) and an asymmetric proposal distribution (Metropolis-Hastings algorithm).

Module 6: Gibbs Sampling & Probabilistic Programming

This module explores Gibbs Sampling, a special case of the Metropolis-Hastings algorithm where parameters are updated one by one using their full conditional distributions. You will also learn how to build Bayesian models using Python and R libraries like PyMC and Stan.

Recommended MCMC Diagnostics Guide (Addressing the Video Gap)

The video pool lacks coverage of MCMC convergence diagnostics. To bridge this gap, study the following key diagnostic metrics, which are essential for verifying your samplers before drawing conclusions:

  • Traceplots: Time-series plots of your parameter values across sampling steps. A healthy, converged chain looks like a stationary "fuzzy caterpillar." Flat lines or drifting trends suggest poor mixing.
  • Gelman-Rubin Diagnostic (R^\hat{R} / R-hat): Compares the variance between multiple independent chains to the variance within the chains. A converged model should have R^<1.05\hat{R} < 1.05 (ideally <1.01<1.01). If R^≫1.05\hat{R} \gg 1.05, your chains have not converged to the same distribution.
  • Autocorrelation: Measures how correlated successive samples are. High autocorrelation means your sampler is exploring slowly and needs more steps to yield independent samples.
  • Effective Sample Size (ESS): Estimates the number of independent samples your highly correlated chain is actually equivalent to.

Recommended Videos

Gibbs sampling
Gibbs sampling
@jaradniemi08:38112,159
Gibbs sampling

Gibbs sampling is a Markov chain Monte Carlo technique for drawing samples from a multivariate joint distribution by iteratively sampling from conditional distributions of each variable given the current values of all other variables, making it particularly useful for high-dimensional problems where direct simulation or other methods like accept-reject or Metropolis-Hastings are impractical.

@jaradniemi · 08:38

Why this video: This video introduces Gibbs sampling, explaining how to sample from joint multivariate distributions by iteratively drawing from conditional univariate distributions.

Gibbs Sampling Tutorial: Discrete Bivariate Example & Formal Definition
Gibbs Sampling Tutorial: Discrete Bivariate Example & Formal Definition
@SpartacanUsuals18:5886,072
Gibbs Sampling Tutorial: Discrete Bivariate Example & Formal Definition

Gibbs sampling is a Markov chain Monte Carlo method for sampling from multi-dimensional probability distributions by iteratively sampling each variable from its conditional distribution given the current values of all other variables, accepting all proposals without rejection; unlike the Metropolis algorithm, it requires knowing the conditional distributions and being able to independently sample from them, making it particularly useful for Bayesian inference but potentially less efficient when

@SpartacanUsuals · 18:58

Why this video: A comprehensive and mathematically detailed tutorial on the mechanics of Gibbs sampling. It walks you through a complete bivariate discrete example, showing how the math works step-by-step.

[41] Intro to Probabilistic Programming with PyMC (Austin Rochford)
[41] Intro to Probabilistic Programming with PyMC (Austin Rochford)
@DataUmbrella05:2713,072
[41] Intro to Probabilistic Programming with PyMC (Austin Rochford)

Probabilistic programming reverses the traditional data science workflow from data → story to story → inference. It encodes narratives about how data is generated and performs Bayesian inference to learn unknown quantities. Mathematically, this connects to Monte Carlo methods for approximating integrals that arise in Bayes' theorem. The core challenge is that Bayesian inference requires computing intractable integrals for complex models, motivating the use of probabilistic programming with Monte

@DataUmbrella · 05:27

Why this video: This video introduces probabilistic programming in Python using PyMC. It shows how to translate theoretical math into computer code that automates Bayesian updating.

Bayesian Modeling with R and Stan (Reupload)
Bayesian Modeling with R and Stan (Reupload)
@slcrug52:4742,538
Bayesian Modeling with R and Stan (Reupload)

Bayesian inference reverses conditional probability using Bayes' theorem to update beliefs about parameters from data, treating parameters as random variables with probability distributions rather than fixed unknowns; Stan is a probabilistic programming language that implements Markov Chain Monte Carlo methods to sample from complex posterior distributions, enabling practical Bayesian analysis through hierarchical modeling and flexible prior specification.

@slcrug · 52:47

Why this video: A deeper look at Bayesian programming using Stan. This presentation explains how to define priors, likelihoods, and parameter constraints within a production-grade probabilistic programming library.

Module 6 Knowledge Checkpoint

  • Explain how Gibbs sampling can be considered a special case of the Metropolis-Hastings algorithm where the acceptance probability is always 1.
  • Given a bivariate distribution p(x,y)p(x, y), outline the exact pseudocode steps for running a Gibbs sampler.
  • Identify a "fuzzy caterpillar" traceplot vs. a poorly mixing traceplot, and state what an R^\hat{R} value of 1.151.15 indicates about your model's convergence.
  • Write a simple script using a probabilistic programming library (like PyMC or Stan) to estimate the mean and standard deviation of a normal distribution.

Course Map

This flowchart maps the structure of the course, showing module dependencies and the recommended learning path.

Key People Index

  • Thomas Bayes (1701–1761): An English statistician and Presbyterian minister who formulated the foundational theorem of conditional probability that bears his name.
  • Michael Betancourt: A leading applied statistician and core developer of Stan, known for his work on the geometry and mathematical foundations of Hamiltonian Monte Carlo (HMC).
  • Osvaldo Martin: A prominent Bayesian researcher and core developer of PyMC and ArviZ, widely known for his books and educational efforts in probabilistic programming.
  • Austin Rochford: An applied mathematician and PyMC contributor who writes extensively on leveraging probabilistic programming for real-world data science.

Final Self-Assessment

Test your understanding of the concepts covered in this curriculum:

  • I can derive Bayes' Theorem from the definition of conditional probability: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}.
  • I can explain the mathematical and conceptual differences between probability (area under a curve) and likelihood (height of a curve).
  • I can write out the prior, likelihood, and posterior setup for a coin-flipping experiment (Beta-Binomial) and solve for the posterior analytically.
  • I can state the conjugate prior families for Poisson, Binomial, and Normal likelihoods.
  • I can explain why the marginal likelihood (the denominator of Bayes' Theorem) is computationally intractable in high dimensions, creating the need for simulation.
  • I can write the pseudocode for the Metropolis-Hastings algorithm, including the candidate generation and acceptance/rejection steps.
  • I can describe how a Gibbs sampler works by iteratively sampling from full conditional distributions.
  • I can list and explain the four main MCMC convergence diagnostics: Traceplots, Gelman-Rubin (R^\hat{R}), Autocorrelation, and Effective Sample Size (ESS).
  • I can write a working model in PyMC or Stan to estimate the posterior of parameters from a real dataset and run MCMC sampling on it.
Explore Further

Related Mathematics Roadmaps

View All→
mathematics

Non-Euclidean Geometry: Models & Geodesics

01Euclid's Fifth Postulate and the Birth of New Geometries
02Hyperbolic Space and the Poincaré Disk Model
03Geodesics: Defining Straight Lines in Curved Spaces
+ 2 more learning modules
15h5 modules
Explore Roadmap
mathematics

Knot Theory — Moves, Coloring & Polynomials

01Introduction to Topology and Knots
02Reidemeister Moves & Knot Equivalence
03Knot Invariants and Tricolorability
+ 2 more learning modules
12h5 modules
Explore Roadmap
mathematics

Category Theory: Functors, Monads & FP

01Mathematical Foundations: Functions, Composition & Types
02Introduction to Category Theory: Objects & Morphisms
03Functors: Mapping Between Categories
+ 2 more learning modules
12h5 modules
Explore Roadmap
mathematics

Complex Analysis: Residues & Contour Integrals

01Foundations of Complex Numbers
02Complex Functions and Holomorphic Properties
03Complex Integration and Cauchy's Theorem
+ 2 more learning modules
18h5 modules
Explore Roadmap