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‎src/content/lessons/introduction.mdx‎

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---
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title: 'Introduction to Generative Models'
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description: 'Introduction to Generative Models'
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pubDate: '2025-09-11'
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heroImage: '../../assets/blog-placeholder-3.jpg'
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---
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import T from '../../components/TypstMath.astro'
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This Lecture is an general introduction to generative modelling.
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### Introduction
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This is an introduction to the main settings encountered in generative modelling. The first Lectures will introduce the main algorithms and concept for the vanilla unconditional generative modelling task.
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At the end of the course, we will make excursions to class-conditional and text-conditional generative modelling.
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#### Unconditional Generative Modelling
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**What** In *unconditional* generative modelling, we are given a set of unlabelled data
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<T block v='text("Data: ") underbrace({x_1, x_2, x_3,dots, x_n}, n "observations") in bb(R)^d .' />
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export const catGallery = [
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{
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url: "https://images.pexels.com/photos/57416/cat-sweet-kitty-animals-57416.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x1",
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alt: "Cute cat 1"
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},
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{
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url: "https://images.pexels.com/photos/20787/pexels-photo.jpg?auto=compress&cs=tinysrgb&w=800",
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caption: "x2",
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alt: "Cute cat 2"
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},
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{
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url: "https://images.pexels.com/photos/1183434/pexels-photo-1183434.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x3",
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alt: "Cute cat 3"
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},
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{
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url: "https://images.pexels.com/photos/979247/pexels-photo-979247.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x4",
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alt: "Cute cat 4"
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},
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{
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url: "https://images.pexels.com/photos/2558605/pexels-photo-2558605.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x5",
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alt: "Cute cat 5"
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},
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{
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url: "https://images.pexels.com/photos/1276553/pexels-photo-1276553.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x6",
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alt: "Cute cat 5"
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}
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]
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<figure>
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<div style={{
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display: "grid",
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gridTemplateColumns: "repeat(auto-fit, minmax(200px, 1fr))",
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gap: "1rem",
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alignItems: "start"
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}}>
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{catGallery.map((cat, idx) => (
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<figure key={idx} style={{ margin: 0 }}> {/* remove default margin */}
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<img
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src={cat.url}
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alt={cat.alt}
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style={{ width: "100%", height: "auto", objectFit: "cover", display: "block" }}
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/>
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<figcaption style={{
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textAlign: "center",
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fontSize: "0.85rem",
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color: "#6b7280",
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margin: 0, // remove figcaption margin
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marginTop: "0.25rem" // optional small spacing
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}}>
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{cat.caption}
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</figcaption>
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</figure>
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))}
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</div>
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<figcaption style={{
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textAlign: "center",
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marginTop: "1rem",
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fontStyle: "italic",
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color: "#6b7280"
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}}>
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A dataset of cat photos (source, Pexels.com).
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</figcaption>
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</figure>
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**Assumption** The core underlying assumption of generative modelling is that the data $x_1, \dots, x_n$, is drawn from some *unknown* underlying distribution $p_{data}$: for all $i \in 1, \dots, n$
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<T block v='x_i ~ underbrace(p_"data", "unknown") .' />
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**Goal** Using the empirical data distribution $x_1, \dots, x_n \sim p_{data}$, the goal is to *generate* new samples $x^{\text{new}}$ that look like they were drawn from the same *unknown* distribution $p_{data}$
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<T block v='x^"new" ~ p_"data" .' />
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#### Class-Conditional Generative Modelling
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**What** In *class-conditional* generative modelling, we are given a set of labelled data
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<T block v='text("Data: ") underbrace({(x_1, y_1),dots, (x_n, y_n)}, n "labelled observations") in bb(R)^d times bb(R) .' />
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export const catDogGallery = [
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{
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url: "https://images.pexels.com/photos/57416/cat-sweet-kitty-animals-57416.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x1, y1=cat",
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alt: "Cute cat 1"
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},
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{
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url: "https://images.pexels.com/photos/20787/pexels-photo.jpg?auto=compress&cs=tinysrgb&w=800",
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caption: "x2, y2=cat",
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alt: "Cute cat 2"
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},
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{
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url: "https://images.pexels.com/photos/1183434/pexels-photo-1183434.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x3, y3=cat",
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alt: "Cute cat 3"
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},
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{
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url: "https://images.pexels.com/photos/58997/pexels-photo-58997.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x4, y4=dog",
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alt: "Cute cat 4"
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},
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{
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url: "https://images.pexels.com/photos/731022/pexels-photo-731022.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x5, y5=dog",
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alt: "Cute cat 5"
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},
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{
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url: "https://images.pexels.com/photos/551628/pexels-photo-551628.jpeg?auto=compress&cs=tinysrgb&w=800",
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caption: "x6, y6=dog",
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alt: "Cute cat 5"
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}
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]
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<figure>
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<div style={{
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display: "grid",
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gridTemplateColumns: "repeat(auto-fit, minmax(200px, 1fr))",
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gap: "1rem",
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alignItems: "start"
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}}>
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{catDogGallery.map((cat, idx) => (
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<figure key={idx} style={{ margin: 0 }}> {/* remove default margin */}
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<img
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src={cat.url}
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alt={cat.alt}
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style={{ width: "100%", height: "auto", objectFit: "cover", display: "block" }}
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/>
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<figcaption style={{
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textAlign: "center",
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fontSize: "0.85rem",
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color: "#6b7280",
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margin: 0, // remove figcaption margin
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marginTop: "0.25rem" // optional small spacing
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}}>
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{cat.caption}
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</figcaption>
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</figure>
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))}
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</div>
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<figcaption style={{
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textAlign: "center",
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marginTop: "1rem",
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fontStyle: "italic",
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color: "#6b7280"
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}}>
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A labelled dataset of cat and dog photos (source, Pexels.com).
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</figcaption>
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</figure>
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**Assumption** The core underlying assumption of generative modelling is that the data $x_1, \dots, x_n$, is drawn from some *unknown* underlying distribution $p_{data}( \cdot | y_i)$: for all $i \in 1, \dots, n$
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<T block v='x_i ~ underbrace(p_"data" (dot | y_i), "unknown") .' />
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**Goal** Using the empirical data distribution $x_i \sim p_{data}(\cdot | y_i)$, the goal is to *generate* new samples $x^{\text{new}}$ that look like they were drawn from the same *unknown* distribution $p_{data}$
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<T block v='x^"new cat" ~ p_"data" (dot | y="cat") ,' />
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<T block v='x^"new dog" ~ p_"data" (dot | y="dog") .' />
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#### Text-Conditional Generative Modelling
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### Unconditional Generative Modelling
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#### 1 and 2-Dimensional Examples
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#### Maximum Likelihood
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### References
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- https://people.cs.umass.edu/~domke/courses/sml/10probabilistic.pdf
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- https://people.cs.umass.edu/~domke/courses/sml/11em.pdf

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