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.

published · difficulty 3/5 · 16 min read

Intuition

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PredictName the object in plain language, then predict what should change.Leave with one reusable mental picture before notation appears.

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.

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Math

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InspectTrack the same object through the notation and check each symbol.Leave with the invariant the equations preserve.

Equation 1Inspect equation objectp_\tau(i) = \frac{e^{z_i/\tau}}{\sum_j e^{z_j/\tau}}.Equation 2Inspect equation objectS_p = \min\{S: \sum_{i\in S} p(i) >= p\}, \qquad p'(i) = (p(i) \mathbf 1[i\in S_p])/(\sum_{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.

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Code

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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))

Section prompt

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Interactive Demo

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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 inquiry checkpoint

Manipulate one control and predict the visible change.

01Choose lensTrace a quantity

02ObserveDemo state pending

03GroundName the equation, invariant, or control that explains it.

04CarryNext: Speculative Decoding: Lossless Multi-Token Generation

Choose what to inspect in Decoding & Sampling: Temperature, Top-p & Inference-Time Control. This shared fallback is an observation guide, not evidence of learning.

Loading interactive demo...

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

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Object

4/4 sections ready

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

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

BeforeMaximum LikelihoodNow4/4 sections readyTryManipulate one control and predict the visible change.NextSpeculative Decoding: Lossless Multi-Token Generation

Object contextLLM Systems

StateLLM Systems

1 source attachedChecking local snapshot

concept:llm-systems/decoding-sampling

Math2 equation objectsp_\tau(i) = \frac{e^{z_i/\tau}}{\sum_j e^{z_j/\tau}}.CodeDecoding & Sampling: Temperature, Top-p & Inference-Time Control code...z = z - z.max()DemoDecoding & Sampling: Temperature, Top-p & Inference-Time Control pred...Manipulate one control and predict the visible change.

ConceptLearner lens

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

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

Mode questionCan I say the mechanism back in one sentence before I reveal anything?

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Take this move

Study modes

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01PredictCommit before reveal.02ExplainName the invariant.03Ground1 source link nearby.04ExperimentChange one knob.05CarrySave the next question.

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Carry the same object through intuition, math, code, and demo.

4/4 sections ready

Carry inMaximum Likelihood

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Work hereDecoding & 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.

Carry outSpeculative Decoding: Lossless Multi-Token Generation

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After The First Pass

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

01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary1 source link attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

Mechanism Storyboard

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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.

Demo notes open01 / Intuition

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

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

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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.

4/4 stages readyDemo notes connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

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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

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Object - ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlQuestion

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

concept:llm-systems/decoding-sampling

Boundary

sources: holtzman-2019-nucleus

Check

Open the closest source note before trusting the local explanation.

Evidence

1 selected-object source shown first; 1 reference total.

Next move

Audit the claim boundary, then ask from the same selected object.

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

Located CF editorial boundary

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.

Used here as

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...

Caveat

Checks temperature softmax and nucleus top-p truncation/renormalization only; not universal decoding-quality settings, safety behavior, task optimality, top-k guarantees, repetition penal...

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.

Object - ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlQuestion

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

concept:llm-systems/decoding-sampling

Boundary

sources: holtzman-2019-nucleus

Check

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Evidence

1 structured claim check on this concept.

Next move

Run the prediction or practice transfer before asking for a grounded review.

1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 1 reference and 3 local witnesses are available for inspection.

Decoding 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.

Registered CF editorial boundary

The Curious Case of Neural Text DegenerationPrimary nucleus sampling source. Sec. 3.1 defines the top-p vocabulary, rescales the truncated distribution, and samples from a dynamically sized n...

Used here as

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....

Local witness

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

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...

Caveat

Checks 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 be...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. 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

Practice notebook

Use the idea, then test it somewhere new

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

AttemptNo learning claim inferred

Object - ConceptDecoding & Sampling: Temperature, Top-p & Inference-Time ControlQuestion

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

concept:llm-systems/decoding-sampling

Boundary

sources: holtzman-2019-nucleus

Check

Use one state from Decoding & Sampling: Temperature, Top-p & Inference-Time Control to explain what changes, why it changes, and which assumption the explanation needs.

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Use one state from Decoding & Sampling: Temperature, Top-p & Inference-Time Control to explain what changes, why it changes, and which assumption the explanation needs.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

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Misconception check

Grounded object roomClose

Selected object routeAsk from this object; carry one invariant back.sources: holtzman-2019-nucleus

ObjectConceptDecoding & Sampling: Temperature, Top-p & Inference-Time Control
PredictBefore revealDecoding & Sampling: Temperature, Top-p & Inference-Time Control pred...
WitnessCompare codeDecoding & Sampling: Temperature, Top-p & Inference-Time Control code...
RoomAsk groundedChecking local snapshot

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 predictionManipulate 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?Checking local snapshot

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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?

Source boundaryInspect source ids: holtzman-2019-nucleusStable content-object key attached

Role lenses for this object

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Learner evidence requestAsk what would make "Decoding & Sampling: Temperature, Top-p & Inference-Time Control" feel predictable rather than familiar.

Assumption

Source ids holtzman-2019-nucleus must support the exact object, not just the surrounding topic.

Source-checking summary

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Proposed experiment

Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo.

Next action

The learner can state the mechanism in their own words

Evidence4 checks

PredictionChecking carried observation

ActionReady for one action

AILearner handoff ready

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01PredictionChecking browser-local route memory

02EvidenceChecking for a carried observation

03BoundaryInspect source ids: holtzman-2019-nucleus

04Next moveSave one next action

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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

Object-attached 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 Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids holtzman-2019-nucleus must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Decoding & Sampling: Temperature, Top-p & Inference-Time Control" feel predictable rather than familiar." | assumption: Source ids holtzman-2019-nucleus must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: holtzman-2019-nucleus" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. 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. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Decoding & Sampling: Temperature, Top-p & Inference-Time Control" feel predictable rather than familiar. - Assumption to keep visible: Source ids holtzman-2019-nucleus must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

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

Learnerpicture first Researchersource check Experimentertest the demo Professorname invariant

01Predict 02Ground 03Code 04Carry

01Intuitionready
02Mathready
03Codeready
04Interactive Demoready

Object Companion

Ask beside the selected object

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

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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.

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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