DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Binary restricted Boltzmann machine, exact likelihood gradient, and finite contrastive divergence

Definition

Assume AC. Let $m, n \geq 1$ , visible states $v \in {0, 1}^{m}$ , hidden states $h \in {0, 1}^{n}$ , biases $a \in R^{m}$ and $b \in R^{n}$ , and weights $W \in R^{m \times n}$ ; a finite bipartite graph is specified by requiring $W_{i j} = 0$ at every absent visible-hidden edge. Define $E (v, h) = - a^{T} v - b^{T} h - v^{T} W h$ , $Z = \sum_{v, h} \exp (- E (v, h))$ , and $p (v, h) = Z^{- 1} \exp (- E (v, h))$ ; $Z$ is finite and positive. Set $p (v) = \sum_{h} p (v, h)$ , and let $V, H$ be the coordinate variables under $p$ . For a finite mass function $q$ , write $E_{q} [g] = \sum_{x} q (x) g (x)$ . With $σ (t) = (1 + \exp (- t))^{- 1}$ , and $Bernoulli (u; r)$ the mass on ${0, 1}$ with probability $r$ at $1$ , the layer conditionals factorize as $p (h ∣ v) = \prod_{j = 1}^{n} Bernoulli ⁣ (h_{j}; σ ⁣ (b_{j} + (W^{T} v)_{j})), p (v ∣ h) = \prod_{i = 1}^{m} Bernoulli ⁣ (v_{i}; σ ⁣ (a_{i} + (W h)_{i})),$ so each layer is conditionally independent given the other. For one observed $v$ , the exact gradients are $\frac{\partial \log p (v)}{\partial W_{i j}} = v_{i} E_{p (h ∣ v)} [h_{j}] - E_{p} [V_{i} H_{j}], \frac{\partial \log p (v)}{\partial a_{i}} = v_{i} - E_{p} [V_{i}], \frac{\partial \log p (v)}{\partial b_{j}} = E_{p (h ∣ v)} [h_{j}] - E_{p} [H_{j}] .$ These are ambient derivatives in all parameters, evaluated at the specified $W$ ; when a graph fixes absent-edge weights to zero, retain only allowed-edge derivatives as free parameters. Define $K (v, v^{'}) = \sum_{h} p (h ∣ v) p (v^{'} ∣ h)$ , a row-stochastic transition matrix. Starting from $q_{0}^{v} = δ_{v}$ , let $q_{k}^{v} = q_{0}^{v} K^{k}$ for a finite integer $k \geq 1$ . For one observed $v$ , define the finite contrastive-divergence components by ${\hat{g}}_{W_{i j}}^{(k)} (v) = v_{i} E_{p (h ∣ v)} [h_{j}] - \sum_{v^{'}} q_{k}^{v} (v^{'}) v_{i}^{'} E_{p (h ∣ v^{'})} [h_{j}], {\hat{g}}_{a_{i}}^{(k)} (v) = v_{i} - \sum_{v^{'}} q_{k}^{v} (v^{'}) v_{i}^{'}, {\hat{g}}_{b_{j}}^{(k)} (v) = E_{p (h ∣ v)} [h_{j}] - \sum_{v^{'}} q_{k}^{v} (v^{'}) E_{p (h ∣ v^{'})} [h_{j}] .$ They replace the exact negative-phase expectations under $p (v^{'})$ by expectations under the finite $k$ -step law $q_{k}^{v}$ and conditional hidden law $p (h ∣ v^{'})$ . This finite-chain statistic is an approximation; no equality to the exact likelihood gradient or chain-convergence claim is asserted.

Definition references

The Elements of Statistical Learning, §17.4.1 and §17.4.4

Hinton, A Practical Guide to Training Restricted Boltzmann Machines, §§2–3

Bipartite architecture diagram

In this instance the left vertices are visible units, the right vertices are hidden units, and each drawn line is an allowed visible-hidden interaction; its weight may be zero for a particular parameter choice. Omitted pairs have weight fixed to zero. The diagram only records the bipartite interaction pattern. The conditional laws and proofs below follow from the displayed finite-state energy formula.

Facts & Assumptions

Given: The finite binary state spaces, parameters and energy in the Definition, with AC for the library's conditional-independence convention.

[F1]

A random element is a measurable map on a probability space, so the visible and hidden coordinate maps are random elements of finite spaces. Random elements and real random variables.

[F2]

Conditional independence given a sigma-algebra means factorization of conditional expectations for every pair of bounded measurable real functions. Conditional independence given a sigma-algebra.

[F3]

Expectation of an integrable random variable is its Lebesgue integral; on a finite probability space it is the finite weighted sum used in the Definition. Expectation of a nonnegative or integrable random variable.

[A1]

AC is the explicit hypothesis of the conditional-independence convention, and here means that every family of nonempty sets has a choice function. Conditional independence given a sigma-algebra The Axiom of Choice.

Verification

technique · direct

1.1givenconstructalgebra

The state space ${0, 1}^{m} \times {0, 1}^{n}$ has $2^{m + n}$ elements. Every term $\exp (- E (v, h))$ is finite and positive, so their finite sum $Z$ is finite and positive; therefore $p$ is a probability mass function.

2.1givenstep 1.1algebra

For fixed $v$ , put $c_{j} = b_{j} + (W^{T} v)_{j}$ . Then $\exp (- E (v, h)) = \exp (a^{T} v) \prod_{j} \exp (h_{j} c_{j})$ , and summing over binary $h$ gives $\exp (a^{T} v) \prod_{j} (1 + \exp (c_{j}))$ . Thus $p (h ∣ v)$ is the product of Bernoulli masses with success probabilities $\exp (c_{j}) / (1 + \exp (c_{j})) = σ (c_{j})$ .

2.2givenstep 1.1algebra

For fixed $h$ , put $d_{i} = a_{i} + (W h)_{i}$ . Then $\exp (- E (v, h)) = \exp (b^{T} h) \prod_{i} \exp (v_{i} d_{i})$ , and summing over binary $v$ gives $\exp (b^{T} h) \prod_{i} (1 + \exp (d_{i}))$ . Hence $p (v ∣ h)$ is the product of Bernoulli masses with success probabilities $\exp (d_{i}) / (1 + \exp (d_{i})) = σ (d_{i})$ .

2.3F3step 1.1algebra

For fixed observed $v$ , $p (v) = Z^{- 1} \exp (a^{T} v) \sum_{h} \exp (b^{T} h + v^{T} W h)$ . Differentiating these finite sums gives $\partial_{W_{i j}} \log p (v) = v_{i} E_{p (h ∣ v)} [h_{j}] - \partial_{W_{i j}} \log Z$ ; the same calculation gives $\partial_{a_{i}} \log p (v) = v_{i} - \partial_{a_{i}} \log Z$ and $\partial_{b_{j}} \log p (v) = E_{p (h ∣ v)} [h_{j}] - \partial_{b_{j}} \log Z$ .

3.1F1F2A1step 2.1step 2.2

The product conditional mass functions in steps 2.1–2.2 make the conditional expectations of any bounded measurable functions of disjoint coordinates factor into the product of their conditional expectations. By [F1]–[F2], each layer is therefore conditionally independent given the other in the stated sigma-algebra sense. This use of the conditional-expectation convention is the only proof point needing [A1].

3.2F3step 2.3algebra

Differentiating the finite partition sum yields $\partial_{W_{i j}} \log Z = \sum_{v^{'}, h^{'}} v_{i}^{'} h_{j}^{'} p (v^{'}, h^{'}) = E_{p} [V_{i} H_{j}]$ , $\partial_{a_{i}} \log Z = E_{p} [V_{i}]$ , and $\partial_{b_{j}} \log Z = E_{p} [H_{j}]$ . Substitution in step 2.3 gives the exact gradients in the Definition.

4.1step 2.1step 2.2step 3.2algebra∎

For each visible state $v$ , $\sum_{v^{'}} K (v, v^{'}) = \sum_{h} p (h ∣ v) \sum_{v^{'}} p (v^{'} ∣ h) = \sum_{h} p (h ∣ v) = 1$ . Thus K is a stochastic matrix and each $q_{k}^{v}$ is a well-defined finite probability distribution. Replacing the exact negative-phase marginal $p (v^{'})$ in the gradient formulas by $q_{k}^{v} (v^{'})$ gives the finite contrastive-divergence statistic defined in the Statement.

Depends on

Used by

Visible RBM units are conditionally independent but dependent marginally Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Elements of Statistical Learning, second edition (standard reference, not scraped)
Geoffrey Hinton, A Practical Guide to Training Restricted Boltzmann Machines (standard reference, not scraped)