Annealed importance sampling
Algorithms AIMech:AnnealedImportanceSampling DRAFT
Aliases: AIS
A Monte Carlo estimation algorithm that draws from a tractable base distribution, applies Markov transitions invariant for a prescribed sequence of intermediate distributions, and weights each resulting path by successive unnormalized-density ratios to estimate target expectations and normalizing-constant ratios.
Identity Notes & Concept Scope
Compared AIS and tempered transitions against all 442 pinned AIO and 699 native identities; no normalized match. Ignored- and hidden-inclusive three-checkout searches returned 1133,473,482 lines, collapsed into 132 unique content groups. Groups were classified using short full content and long-line paths/identity excerpts; full canonical model-neighbor records were read. Long generated search attributes and ignored dependency linguistic tables were not claimed as complete raw-code readings. Hits concern model-specific AIS uses, historical discovery/review context, pending methods and acronym noise. Semantic assessment covered 103 AIO and306 native definitions, including47 and22 new rows;56 and284 prior rows were verified unchanged. Ten complete AIO contexts and seven full native neighbors assessed. AIS estimates expectations and normalizing-constant ratios using weighted annealing paths. It is distinct from annealing for optimization, parallel-temperature replica exchange, ordinary Metropolis–Hastings, an up/down tempered Metropolis transition, and model-specific uses of AIS already represented in AIMech.
Claims & Grounded Evidence
6 Claims
Mechanism
#path-law-and-weight
For fixed distributions p_j=f_j/Z_j on a common measure space, let p_0 be the target and p_n the base, with finite positive Z_j and p_j positive wherever p_(j−1) is positive. Draw x_(n−1) from p_n, then apply T_(n−1),...,T_1 to obtain x_(n−2),...,x_0. Each T_j must preserve p_j. Return x_0 with weight w=product over j=1,...,n of f_(j−1)(x_(j−1))/f_j(x_(j−1)). Intermediate kernels need not equilibrate, be ergodic, or be reversible. Exact independent base draws and independent subsequent randomness give independent runs. Evaluate the ratios in chronological order j=n down to1. If a numerator is zero, assign that run zero weight and stop; no later ratio or endpoint observable is evaluated, and the run contributes zero to weighted numerator and denominator sums. This explicit zero-weight convention handles valid nested supports without a later 0/0.
Mechanism
#estimand-and-unbiasedness-boundary
With exact base sampling and the stated support and invariance assumptions, E[w]=Z_0/Z_n and E[w*a(x_0)]=(Z_0/Z_n)*E_p0[a] for integrable a. Thus the mean weight is an unbiased estimate of the normalizing-constant ratio. The self-normalized estimate sum(w*a)/sum(w) is generally biased at finite sample size, though consistent under the corresponding law-of-large-numbers assumptions; it is undefined when all sampled weights vanish. Neither averaging log weights nor taking the log of their sample mean generally gives an unbiased log-ratio estimate. Unweighted endpoints need not have distribution p_0. These distinctions follow from the extended-space importance identity, not from a claim of exact equilibrium after annealing.
Scope
#finite-weight-law-checks
A curator three-state example uses f_0=(1,2,7), f_1=(1,1,1), f_2=(3,2,1) and a deterministic forward-cycle T_1. Exact enumeration gives E[w]=5/3 and weighted endpoint masses (1/6,1/3,7/6), while unweighted endpoints have probabilities (1/6,1/2,1/3), distinct from target (1/10,1/5,7/10). For the indicator of state 2, the expected self-normalized estimate is 1/3 with one run and 967/2025 with two, rather than 7/10. A further 192 finite invariant-kernel cases satisfy the weighted-mass identity. Separate reset-kernel and missing-support examples fail, demonstrating why the assumptions matter. These are finite curator arithmetic checks, not general convergence proofs or performance benchmarks. For valid nested supports f_0=f_1=(1,0), f_2=(1,1) and identity T_1, the zero-weight branch stops after its first ratio rather than evaluate a later 0/0. Exact mean weight is1/2, matching Z_0/Z_2.
Scope
#bridge-and-bayesian-scope
A geometric bridge f_j=f_0^beta_j*f_n^(1−beta_j), with beta decreasing from 1 to 0, is an option rather than the algorithm identity. Auxiliary distributions must be proper; merely powering an unnormalized target does not ensure a tractable proper base. When f_n is proportional to a prior and f_0=f_n times the likelihood, Z_0/Z_n is the marginal likelihood. The shared prior constant cancels, but likelihood constants must be retained for this interpretation.
Scope
#terminal-and-correlated-extensions
An optional fixed-length Markov chain preserving p_0 can start from each weighted endpoint; use that run’s weight with its average observable. Error assessment must treat the within-run average as one dependent-data summary rather than count its states as independent AIS runs. The paper also permits correlated base states from an ergodic chain and discusses consistency, but that does not justify finite-sample unbiasedness from an arbitrarily initialized base chain.
Limitation
#diagnostic-and-efficiency-limits
Large or heavy-tailed weights can make the estimator unstable. A small observed weight variance or large empirical effective sample size can miss an important mode that was never sampled. The paper’s temperature-spacing and dimension-scaling analyses depend on idealized distributions and mixing assumptions, and its synthetic and regression examples do not establish universal efficiency or mode discovery. Intermediate distributions, transition effort and evidence for uncertainty remain problem-specific.
⚠️ Curated Limitations & Negative Results
- The maintained identity is the fixed-sequence weighted-path estimator. Resampling particle systems, linked importance sampling, adaptive schedules and full up/down tempered transitions need their own definitions and evidence.
- The intermediate-state paragraph on PDF p. 6 says x_j follows T_(j−1), inconsistent with Equation 4. For 0<=j<=n−2, x_j follows T_(j+1); the top state x_(n−1) is drawn directly from p_n, with no T_n application. The maintained algorithm uses Equations 4–5 and this initial-draw exception.
- Missing support or noninvariant transitions invalidate the weighting identity. Finite precision, underflow and practical transition implementations were not audited.
- All 25 source pages were inspected, using images for plot-heavy pages 14–15; method pages 4–6 were visually checked. Journal bytes, author software, model fits and benchmark replication were not assessed. DRAFT remains.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | AIS uses Markov kernels preserving successive auxiliary distributions to construct weighted paths. The local estimation algorithm is related to the target process/model; it is not asserted to be a neural network, hidden layer or equivalent Markov-chain class. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
neal1998ais
|
Radford M. Neal
|
primary paper
full_text
|
ID: arxiv:physics/9803008v2
sha256:6ae14b83d623…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-14T08:26:48Z
Updated: 2026-10-14T19:46:09Z
Source File:
data/curated/AnnealedImportanceSampling.yaml