LLM Systems

Decoding & Sampling: Temperature, Top-p & Inference-Time Control

How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

status: publishedimportance: importantdifficulty 3/5math: undergraduateread: 16mlive demo

Back to LLM Systems Next: Speculative Decoding: Lossless Multi-Token Generation

Editorial LLM-systems illustration of logit distributions, top-p filtering, temperature shaping, and sampled token paths.

Concept Structure

Decoding & Sampling: Temperature, Top-p & Inference-Time Control

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

2prerequisites

1next concepts

1related links

Learner Contract

What this page should let you do.

You are here becauseHow inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

This LLM Systems concept is the current object: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

Before thisMaximum Likelihood

2 prerequisites listed; refresh them before leaning on the math or code.

Maximum Likelihood Scaled Dot-Product Attention & Transformer Layers

By the end4/4 sections ready | code witness expected | live demo

Explain the mechanism, trace the main notation, and test one prediction in the live demo.

Do this firstIntuition

Read the intuition before the notation; the math should name a mechanism you already felt.

Then go nextSpeculative Decoding: Lossless Multi-Token Generation

Follow this edge after making one prediction here; the next page should reuse the result, not restart the route.

Test the linkManipulate one control and predict the visible change.Then continue to Speculative Decoding: Lossless Multi-Token Generation

Claim/source review status

Substantive review recorded

1/1 claims have bounded review metadata; still check caveats and source scope.Metadata-derived; review may be AI-assisted. Not a human certification.

Claims1/1 reviewed

Sources1 cited

Codeattached

Demolive

Reviewed2026-05-07

Updatedpage 2026-05-07

First Route Readiness

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01Homenext action 02Paper Mappernext action 03Knowledge Graphnext action 04Searchnext action 05Attention To Servingnext action +1 more

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Object flow

4/4 sections readyAsk about this Research room

ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlLLM Systems EquationDecoding & Sampling: Temperature, Top-p & Inference-Time Control equa...Exact equation object CodeDecoding & Sampling: Temperature, Top-p & Inference-Time Control code...Exact code witness DemoDecoding & Sampling: Temperature, Top-p & Inference-Time Control inte...Visualization object ClaimDecoding settings reshape the next-token distribution at inference ti...Exact claim check SourceThe Curious Case of Neural Text DegenerationExact source object

ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlLLM Systems

1 source attachedLocal snapshot ready

concept:llm-systems/decoding-sampling

Codewitness nearby Predictbefore reveal Roomobject handoff

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

Training gives you a next-token distribution. Decoding is what turns that distribution into actual behavior.

Two products can ship the "same model" and feel completely different because decoding choices differ:

greedy vs sampling,
temperature (how sharp the distribution is),
top-p/top-k truncation (how much tail you delete),
penalties and constraints (repetition, min length, format forcing).

So decoding is not a cosmetic detail. It's an inference-time control surface that trades off determinism, diversity, and failure modes like repetition loops.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1 $p_\tau(i) = \frac{e^{z_i/\tau}}{\sum_j e^{z_j/\tau}}.$ Equation 2 $S_p = \min\left\{S: \sum_{i\in S} p(i) \ge p\right\}, \qquad p'(i) = \frac{p(i)\,\mathbf 1[i\in S_p]}{\sum_{j\in S_p} p(j)}.$

Let $z_i$ be the logits for token $i \in \mathcal V$ at some step.

Temperature rescales logits

Temperature $\tau > 0$ produces:

p_\tau(i) = \frac{e^{z_i/\tau}}{\sum_j e^{z_j/\tau}}.

Smaller $\tau$ sharpens (more deterministic).
Larger $\tau$ flattens (more exploratory).

Nucleus (top-p) truncation deletes the tail, then renormalizes

Let $S_p$ be the smallest set of tokens whose probability mass is at least $p$ (after sorting by probability):

S_p = \min\left\{S: \sum_{i\in S} p(i) \ge p\right\}, \qquad p'(i) = \frac{p(i)\,\mathbf 1[i\in S_p]}{\sum_{j\in S_p} p(j)}.

Then you sample from $p'$ , the truncated and renormalized distribution.

This is why top-p changes behavior: it literally changes the distribution you sample from.

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Code witness 1import numpy as np def softmax(z): z = z - z.max() ez = np.exp(z) return ez / ez.sum() def to...python

import numpy as np

def softmax(z):
    z = z - z.max()
    ez = np.exp(z)
    return ez / ez.sum()

def top_p_filter(p, probs):
    idx = np.argsort(-probs)
    keep = np.zeros_like(probs, dtype=bool)
    cum = 0.0
    for i in idx:
        keep[i] = True
        cum += probs[i]
        if cum >= p:
            break
    q = probs * keep
    return q / q.sum()

rng = np.random.RandomState(0)
logits = np.array([3.0, 2.0, 1.0, 0.2, -0.5])  # 5 tokens

for tau, p in [(1.0, 1.0), (0.7, 0.9), (1.4, 0.9)]:
    probs = softmax(logits / tau)
    probs = top_p_filter(p, probs)
    samples = rng.choice(len(probs), size=20000, p=probs)
    counts = np.bincount(samples, minlength=len(probs)) / len(samples)
    print("tau=", tau, "top_p=", p, "freqs=", np.round(counts, 3))

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Use the demo to see how temperature/top-p reshape the distribution step by step, and how small changes alter which toy tokens survive.

Live Concept Demo

Explore Decoding & Sampling: Temperature, Top-p & Inference-Time Control

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 3/5undergraduatecode-aligned

Demo Prediction Checkpoint

Manipulate one control and predict the visible change.

Commit to what Decoding & Sampling: Temperature, Top-p & Inference-Time Control should make visible before reading the result.

After The First Pass

Turn the concept into an inspected object.

Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

See the idea move before the page explains it

How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

Prediction open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Decoding & Sampling: Temperature, Top-p & Inference-Time Control should make visible.

Visual Inquiry

Make the image answer a mathematical question

How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

4/4 stages readyLive demo connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Decoding & Sampling: Temperature, Top-p & Inference-Time Control easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

paper · 2019The Curious Case of Neural Text DegenerationHoltzman et al.

Primary nucleus sampling source. Sec. 3.1 defines the top-p vocabulary, rescales the truncated distribution, and samples from a dynamically sized nucleus; Sec. 3.3 defines temperature softmax.

Open source

Claim Review

How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

Status1 substantive review recorded

Claims without a substantive review badge still need exact source-support review.

Sources1 reference

holtzman-2019-nucleus

Witnesses4 local objects

Use equation, code, and demo objects to check whether the source support is operational.

Substantively reviewedDecoding settings reshape the next-token distribution at inference time: temperature rescales probabilities, while nucleus sampling truncates the unreliable tail and renormalizes the remaining dynamic token set.Claim metadata: source checked

Holtzman et al. define nucleus sampling as a top-p vocabulary whose cumulative probability exceeds p, then rescale that truncated distribution for sampling; the candidate set changes with distribution shape. Sec. 3.3 defines temperature softmax over logits. Local math/code/demo are toy distribution-shaping witnesses.

Sources: The Curious Case of Neural Text DegenerationChecks temperature softmax and nucleus top-p truncation/renormalization only; not universal decoding-quality settings, safety behavior, task optimality, top-k guarantees, repetition penalties, beam-search behavior, or production sampler implementations. Demo is toy/local only.A bounded review summary is present; still check caveats and exact source scope.

Checked Holtzman Sec. 3.1 and 3.3: temperature is softmax re-estimation over logits divided by temperature, while nucleus/top-p selects a smallest cumulative-probability set, rescales the truncated distribution, samples from a dynamically sized nucleus, and targets the unreliable tail. The first two equations, first code witness, and updated demo are toy/local distribution-shaping witnesses only.

Reviewer: codex+oracle; reviewed 2026-05-07

source-span-holtzman-2019-nucleus math-object-1 math-object-2 code-witness-1 interactive-demo

Source support candidates

paper 2019The Curious Case of Neural Text Degeneration

Primary nucleus sampling source. Sec. 3.1 defines the top-p vocabulary, rescales the truncated distribution, and samples from a dynamically sized nucleus; Sec. 3.3 defines temperature softmax.

Mechanism witnesses

Equation 1

p_\tau(i) = \frac{e^{z_i/\tau}}{\sum_j e^{z_j/\tau}}.

Equation 2

S_p = \min\left\{S: \sum_{i\in S} p(i) \ge p\right\}, \qquad p'(i) = \frac{p(i)\,\mathbf 1[i\in S_p]}{\sum_{j\in S_p} p(j)}.

Code witness 1import numpy as np def softmax(z): z = z - z.max() ez = np.exp(z) return ez / ez.sum() def to...Demo stateLive mechanism probe

Practice Loop

Try the idea before it explains itself

How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Decoding & Sampling: Temperature, Top-p & Inference-Time Control.

Hint 1

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Hint 2

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Hint 3

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Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Object research drawerClose

ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlLLM Systems

Code witness comparisonDecoding & Sampling: Temperature, Top-p & Inference-Time Control code witness 1z = z - z.max()Prediction before revealDecoding & Sampling: Temperature, Top-p & Inference-Time Control interactive demoManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Decoding & Sampling: Temperature, Top-p & Inference-Time Control click without losing the math?Local snapshot ready

Research Room

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conceptLLM Systems

Decoding & Sampling: Temperature, Top-p & Inference-Time Control

Anchored question

What is the smallest example that makes Decoding & Sampling: Temperature, Top-p & Inference-Time Control click without losing the math?

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Evidence to inspect

Source ids to inspect: holtzman-2019-nucleus
Definition, prerequisite, and contrast concept links
The equation or code witness that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Decoding & Sampling: Temperature, Top-p & Inference-Time Control Object key: concept:llm-systems/decoding-sampling Context: LLM Systems Anchor id: concept/concept-notebook/llm-systems/decoding-sampling Open question: What is the smallest example that makes Decoding & Sampling: Temperature, Top-p & Inference-Time Control click without losing the math? Evidence to inspect: - Source ids to inspect: holtzman-2019-nucleus - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

Open source object

concept/concept-notebook/llm-systems/decoding-sampling
concept:llm-systems/decoding-sampling

Learning Map

Before / Now / Try / Next

BeforeMaximum Likelihood

NowIntuition → Math → Code → Demo

TryManipulate one control and predict the visible change.

NextSpeculative Decoding: Lossless Multi-Token Generation

Intuitionready
Mathready
Codeready
Interactive Demoready

Object Companion

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How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: LLM Systems concept. Learning surface: Decoding & Sampling: Temperature, Top-p & Inference-Time Control. What this page says: How inference settings reshape the next-token distribution into actual model behavior: temperature, nucleus sampling, and why decoding is a control knob. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

LLM Systems

llm-systemsdecodingsamplingtemperaturetop-p

Prerequisites

Maximum Likelihood Scaled Dot-Product Attention & Transformer Layers

Leads To

Speculative Decoding: Lossless Multi-Token Generation

Diffusion, Score-Based Models & Flow Matching

Within this domain