6 ms·

Transformers Are Bayesian Networks

40 points by Anon84 7mo ago

westurner 7mo ago
https://news.ycombinator.com/item?id=45256179 https://news.ycombinator.com/item?id=45256179 : > Which statistical models disclaim that their output is insignificant if used with non-independent features? Naieve Bayes [...] Ironic then, because if transformers are Bayesian networks then we're using Bayesian networks for non-independent features. From "Quantum Bayes' rule and Petz transpose map from the minimum change principle" (2025) https://news.ycombinator.com/item?id=45074143 https://news.ycombinator.com/item?id=45074143 : > Petz recovery map: https://en.wikipedia.org/wiki/Petz_recovery_map https://en.wikipedia.org/wiki/Petz_recovery_map : > In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of as a quantum analog of Bayes' theorem But there aren't yet enough qubits for quantum LLMs: https://news.ycombinator.com/item?id=47203219#47250262 https://news.ycombinator.com/item?id=47203219#47250262 "Transformer is a holographic associative memory" (2025) https://news.ycombinator.com/item?id=43028710#43029899 https://news.ycombinator.com/item?id=43028710#43029899
malcolmgreaves 7mo ago
> Hallucination is not a bug that scaling can fix. It is the structural consequence of operating without concepts. NNs are as close to continuous as we can get with discrete computing. They’re flexible and adaptable and can contain many “concepts.” But their chief strength is also their chief weakness: these “concepts” are implicit. I wonder if we can get a hybrid architecture that has the flexibility of NNs while retaining discrete concepts like a knowledge base does.
- measurablefunc 7mo ago
  There is nothing continuous on the computer, it's all bit strings & boolean arithmetic. The semantics imposed on the bit strings does not exist anywhere in the arithmetic operations, i.e. there is no arithmetic operation corresponding to something as simple as the color red.
  - kelseyfrog 7mo ago
    It sounds like you're saying that if a computer had infinite precision then hallucinations would not occur?
    - measurablefunc 7mo ago
      The way neural networks work is that the base neural network is embedded in a sampling loop, i.e. a query is fed into the network & the driver samples output tokens to append to the query so that it can be re-fed back into the network (q → nn → [a, b, c, ...] → q + sample([a, b, c, ...])). There is no way to avoid hallucinations b/c hallucinations are how the entire network works at the implementation level. The precision makes no difference b/c the arithmetic operations are semantically void & only become meaningful after they are interpreted by someone who knows to associated 1 /w red, 2 w/ blue, 3 w/ clouds, & so on & so forth. The mapping between the numbers & concepts does not exist in the arithmetic.
      - kelseyfrog 7mo ago
        Oh, I thought that the embedding space of the residual stream was precisely that.
        measurablefunc 7mo ago
        The arithmetic is meaningless, it doesn't matter what you call it b/c on the computer it's all bit strings & boolean arithmetic. You can call some sequence of operations residual & others embeddings but that is all imposed top-down. There is nothing in the arithmetic that indicates it is somehow special & corresponds to embeddings or residuals.
        kelseyfrog 7mo ago
        Ah ok, so if we had such a mapping then models wouldn't hallucinate?
- AdieuToLogic 7mo ago
  > NNs are as close to continuous as we can get with discrete computing. This is incorrect. For example, fuzzy logic[0] can model analog ("continuous") truth beyond discrete digital representations, such as 1/0, true/false, etc. 0 - https://en.wikipedia.org/wiki/Fuzzy_logic https://en.wikipedia.org/wiki/Fuzzy_logic
tug2024 7mo ago
[dead]
getnormality 7mo ago
> Transformers are the dominant architecture in AI, yet why they work remains poorly understood. This paper offers a precise answer: a transformer is a Bayesian network. Why would being a Bayesian network explain why transformers work? Bayesian networks existed long before transformers and never achieved their performance.
- Mithriil 7mo ago
  Bayesian network is a really general concept. It applies to all multidimensional probability distribution. It's a graph that encodes independence between variables. Ish. I have not taken the time to review the paper, but if the claim stands, it means we might have another tool to our toolbox to better understand transformers.
warypet 7mo ago
I found this earlier today when looking for research and ended up reporting it for citing fake sources. Please correct me if I'm wrong, but I couldn't find "[9] Jongsuk Jung, Jaekyeom Kim, and Hyunwoo J. Choi. Rethinking attention as belief propagation. In International Conference on Machine Learning (ICML), 2022." anywhere else on the internet
- measurablefunc 7mo ago
  It's "vibe" research. Most of it is basically pure nonsense.
  - kleiba 7mo ago
    Care to elaborate?
    - deleted 7mo ago
      [deleted]
    - jack_pp 7mo ago
      I suspect it means it's LLM generated without it being checked
    - refulgentis 7mo ago
      The headline theorem, "every sigmoid transformer is a Bayesian network," is proved by `rfl` [1]. For non-Lean people: `rfl` means "both sides are the same expression." He defines a transformer forward pass, then defines a BP forward pass with the same operations, wraps the weights in a struct called `implicitGraph`, and Lean confirms they match. They match because he wrote them to match. The repo with a real transformer model (transformer-bp-lean) has 22 axioms and 7 theorems. In Lean, an axiom is something you state without proving. The system takes your word for it. Here the axioms aren't background math, they're the paper's claims: - "The FFN computes the Bayesian update" [2]. Axiom. - "Attention routes neighbors correctly" [3]. Axiom. - "BP converges" [4]. Axiom, with a comment saying it's "not provable in general." - The no-hallucination corollary [5]. Axiom. The paper says "formally verified against standard mathematical axioms" about all of these. They are not verified. They are assumed. The README suggests running `grep -r "sorry"` and finding nothing as proof the code is complete. In Lean, `sorry` means "I haven't proved this" and throws a compiler warning. `axiom` also means "I haven't proved this" but doesn't warn. So the grep returns clean while 22 claims sit unproved. Meanwhile the godel repo has 4 actual sorries [6] anyway, including "logit and sigmoid are inverses," which the paper treats as proven. That same fact appears as an axiom in the other repo [7]. Same hole, two repos, two different ways to leave it open. Final count across all five repos: 65 axioms, 5 sorries, 149 theorems. Claude (credited on page 1) turned it into "Not an approximation of it. Not an analogy to it. The computation is belief propagation." Building to a 2-variable toy experiment on 5K parameters presented as the fulfillment of Leibniz's 350-year-old dream. Ending signed by "Calculemus." [1] https://github.com/gregorycoppola/sigmoid-transformer-lean/blob/28e1918/SigmoidTransformerLean/GeneralTheorem.lean#L157 https://github.com/gregorycoppola/sigmoid-transformer-lean/b... [2] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L309 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [3] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L266 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [4] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L334 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [5] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/TreeCorollary.lean#L27 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [6] https://github.com/gregorycoppola/godel/blob/bc1d138/Godel/ORDecomposition.lean#L60 https://github.com/gregorycoppola/godel/blob/bc1d138/Godel/O... [7] https://github.com/gregorycoppola/sigmoid-transformer-lean/blob/28e1918/SigmoidTransformerLean/Preliminaries.lean#L20 https://github.com/gregorycoppola/sigmoid-transformer-lean/b...
wklm 7mo ago
I like their definition of hallucination

40 points by Anon84 7mo ago

https://news.ycombinator.com/item?id=45256179 https://news.ycombinator.com/item?id=45256179 : > Which statistical models disclaim that their output is insignificant if used with non-independent features? Naieve Bayes [...] Ironic then, because if transformers are Bayesian networks then we're using Bayesian networks for non-independent features. From "Quantum Bayes' rule and Petz transpose map from the minimum change principle" (2025) https://news.ycombinator.com/item?id=45074143 https://news.ycombinator.com/item?id=45074143 : > Petz recovery map: https://en.wikipedia.org/wiki/Petz_recovery_map https://en.wikipedia.org/wiki/Petz_recovery_map : > In quantum information theory, a mix of quantum mechanics and information theory, the Petz recovery map can be thought of as a quantum analog of Bayes' theorem But there aren't yet enough qubits for quantum LLMs: https://news.ycombinator.com/item?id=47203219#47250262 https://news.ycombinator.com/item?id=47203219#47250262 "Transformer is a holographic associative memory" (2025) https://news.ycombinator.com/item?id=43028710#43029899 https://news.ycombinator.com/item?id=43028710#43029899

malcolmgreaves 7mo ago

> Hallucination is not a bug that scaling can fix. It is the structural consequence of operating without concepts. NNs are as close to continuous as we can get with discrete computing. They’re flexible and adaptable and can contain many “concepts.” But their chief strength is also their chief weakness: these “concepts” are implicit. I wonder if we can get a hybrid architecture that has the flexibility of NNs while retaining discrete concepts like a knowledge base does.

measurablefunc 7mo ago
There is nothing continuous on the computer, it's all bit strings & boolean arithmetic. The semantics imposed on the bit strings does not exist anywhere in the arithmetic operations, i.e. there is no arithmetic operation corresponding to something as simple as the color red.
- kelseyfrog 7mo ago
  It sounds like you're saying that if a computer had infinite precision then hallucinations would not occur?
  - measurablefunc 7mo ago
    The way neural networks work is that the base neural network is embedded in a sampling loop, i.e. a query is fed into the network & the driver samples output tokens to append to the query so that it can be re-fed back into the network (q → nn → [a, b, c, ...] → q + sample([a, b, c, ...])). There is no way to avoid hallucinations b/c hallucinations are how the entire network works at the implementation level. The precision makes no difference b/c the arithmetic operations are semantically void & only become meaningful after they are interpreted by someone who knows to associated 1 /w red, 2 w/ blue, 3 w/ clouds, & so on & so forth. The mapping between the numbers & concepts does not exist in the arithmetic.
    - kelseyfrog 7mo ago
      Oh, I thought that the embedding space of the residual stream was precisely that.
      - measurablefunc 7mo ago
        The arithmetic is meaningless, it doesn't matter what you call it b/c on the computer it's all bit strings & boolean arithmetic. You can call some sequence of operations residual & others embeddings but that is all imposed top-down. There is nothing in the arithmetic that indicates it is somehow special & corresponds to embeddings or residuals.
        kelseyfrog 7mo ago
        Ah ok, so if we had such a mapping then models wouldn't hallucinate?
AdieuToLogic 7mo ago
> NNs are as close to continuous as we can get with discrete computing. This is incorrect. For example, fuzzy logic[0] can model analog ("continuous") truth beyond discrete digital representations, such as 1/0, true/false, etc. 0 - https://en.wikipedia.org/wiki/Fuzzy_logic https://en.wikipedia.org/wiki/Fuzzy_logic

tug2024 7mo ago

[dead]

getnormality 7mo ago

> Transformers are the dominant architecture in AI, yet why they work remains poorly understood. This paper offers a precise answer: a transformer is a Bayesian network. Why would being a Bayesian network explain why transformers work? Bayesian networks existed long before transformers and never achieved their performance.

Mithriil 7mo ago
Bayesian network is a really general concept. It applies to all multidimensional probability distribution. It's a graph that encodes independence between variables. Ish. I have not taken the time to review the paper, but if the claim stands, it means we might have another tool to our toolbox to better understand transformers.

warypet 7mo ago

I found this earlier today when looking for research and ended up reporting it for citing fake sources. Please correct me if I'm wrong, but I couldn't find "[9] Jongsuk Jung, Jaekyeom Kim, and Hyunwoo J. Choi. Rethinking attention as belief propagation. In International Conference on Machine Learning (ICML), 2022." anywhere else on the internet

measurablefunc 7mo ago
It's "vibe" research. Most of it is basically pure nonsense.
- kleiba 7mo ago
  Care to elaborate?
  - deleted 7mo ago
    [deleted]
  - jack_pp 7mo ago
    I suspect it means it's LLM generated without it being checked
  - refulgentis 7mo ago
    The headline theorem, "every sigmoid transformer is a Bayesian network," is proved by `rfl` [1]. For non-Lean people: `rfl` means "both sides are the same expression." He defines a transformer forward pass, then defines a BP forward pass with the same operations, wraps the weights in a struct called `implicitGraph`, and Lean confirms they match. They match because he wrote them to match. The repo with a real transformer model (transformer-bp-lean) has 22 axioms and 7 theorems. In Lean, an axiom is something you state without proving. The system takes your word for it. Here the axioms aren't background math, they're the paper's claims: - "The FFN computes the Bayesian update" [2]. Axiom. - "Attention routes neighbors correctly" [3]. Axiom. - "BP converges" [4]. Axiom, with a comment saying it's "not provable in general." - The no-hallucination corollary [5]. Axiom. The paper says "formally verified against standard mathematical axioms" about all of these. They are not verified. They are assumed. The README suggests running `grep -r "sorry"` and finding nothing as proof the code is complete. In Lean, `sorry` means "I haven't proved this" and throws a compiler warning. `axiom` also means "I haven't proved this" but doesn't warn. So the grep returns clean while 22 claims sit unproved. Meanwhile the godel repo has 4 actual sorries [6] anyway, including "logit and sigmoid are inverses," which the paper treats as proven. That same fact appears as an axiom in the other repo [7]. Same hole, two repos, two different ways to leave it open. Final count across all five repos: 65 axioms, 5 sorries, 149 theorems. Claude (credited on page 1) turned it into "Not an approximation of it. Not an analogy to it. The computation is belief propagation." Building to a 2-variable toy experiment on 5K parameters presented as the fulfillment of Leibniz's 350-year-old dream. Ending signed by "Calculemus." [1] https://github.com/gregorycoppola/sigmoid-transformer-lean/blob/28e1918/SigmoidTransformerLean/GeneralTheorem.lean#L157 https://github.com/gregorycoppola/sigmoid-transformer-lean/b... [2] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L309 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [3] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L266 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [4] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/Preliminaries.lean#L334 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [5] https://github.com/gregorycoppola/transformer-bp-lean/blob/767a498/TransformerBPLean/TreeCorollary.lean#L27 https://github.com/gregorycoppola/transformer-bp-lean/blob/7... [6] https://github.com/gregorycoppola/godel/blob/bc1d138/Godel/ORDecomposition.lean#L60 https://github.com/gregorycoppola/godel/blob/bc1d138/Godel/O... [7] https://github.com/gregorycoppola/sigmoid-transformer-lean/blob/28e1918/SigmoidTransformerLean/Preliminaries.lean#L20 https://github.com/gregorycoppola/sigmoid-transformer-lean/b...

wklm 7mo ago

I like their definition of hallucination