Joint probability distribution

As explained in Build a Reasoning Model (From Scratch)

The probability of an entire token sequence under a model, computed by multiplying conditional token probabilities across positions.

How Build a Reasoning Model (From Scratch) covers it

How deep it goes

Defines conditional next-token probabilities and demonstrates their multiplication in code, while deferring later log-probability scoring.

Examples

Compares Berlin and Bridge as candidate completions using per-token probabilities and computed joint scores.

What it assumes

Assumes familiarity with token IDs, logits, and text generation, then revisits their next-token selection mechanism.

What you can do after

Apply it via PyTorch to extract per-token probabilities and multiply them into a sequence score.

How it connects

Connects logits, softmax, tokenization, next-token probabilities, sequence likelihoods, and self-refinement.

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Also explained in

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  Fabulous Adventures in Data Structures and AlgorithmsA joint probability distribution assigns probabilities to combinations of outcomes from multiple random variables.
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  Math and Architectures of Deep LearningThe probability that a particular combination of values for two or more random variables occurs together.
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  Outlier Detection in PythonA joint distribution describes the combined distribution of two or more variables and their dependence across values.
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  Causal AIA full joint distribution is the joint probability distribution over both observed and latent variables in a model.

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Related concepts

• generative modelingused by · Generative modeling is a machine-learning approach that learns patterns or a data distribution to generate new samples resembling the data.
• chain rule of probabilityused by · The chain rule of probability factorizes a joint probability distribution into a product of conditional probabilities.
• observational joint distributionproduces · An observational joint distribution is the true joint probability distribution of variables as observed without externally imposed interventions.
• data-generating processproduced by · A data-generating process is the underlying mechanism, procedure, or probability distribution that produces observed data and their relationships.
• marginal probabilityuses · A marginal probability or distribution describes a variable or event without conditioning on other variables, obtained from a joint distribution by summing or integrating over them.

This explainer was generated by Manning's knowledge graph from the text of Build a Reasoning Model (From Scratch). The section links point to the parts of the book it draws on.