one way to construct an HMM is by finding all past histories of tokens that condition the future tokens with the same probablity distribution, and make that equivalence class a hidden state in your HMM. Then the conditional distributions determine the arrows coming out of your state and which state you go to next. This is called the “epsilon machine” in Comp Mech, and it is unique. It is one presentation of the data generating process, but in general there are an infinite number of HMM presntations that would generate the same data. The epsilon machine is a particular type of HMM presentation—it is the smallest one where the hidden states are the minimal sufficient statistics for predicting the future based on the past. The epsilon machine is one of the most fundamental things in Comp Mech but I didn’t talk about it in this post. In the future we plan to make a more generic Comp Mech primer that will go through these and other concepts.
The interpretability of these simplexes is an issue that’s in my mind a lot these days. The short answer is I’m still wrestling with it. We have a rough experimental plan to go about studying this issue but for now, here are some related questions I have in my mind:
What is the relationship between the belief states in the simplex and what mech interp people call “features”?
What are the information theoretic aspects of natural language (or coding databases or some other interesting training data) that we can instantiate in toy models and then use our understanding of these toy systems to test if similar findings apply to real systems.
For something like situational awareness, I have the beginnings of a story in my head but it’s too handwavy to share right now. For something slightly more mundane like out-of-distribution generaliztion or transfer learning or abstraction, the idea would be to use our ability to formalize data-generating structure as HMMs, and then do theory and experiments on what it would mean for a transformer to understand that e.g. two HMMs have similar hidden/abstract structure but different vocabs.
Hopefully we’ll have a lot more to say about this kind of thing soon!
The following is text from Claude Opus 3.
I generally find people just dumping answers from LLMs to be kind of cringe, but in this case, as I was using it to try to understand the post and your comments better, I came across some really genuinely insightful-feeling stuff and felt as though Claude’s input might be helpful to you in some way, and that it would be remiss not to share it. I’m sorry if it’s nonsensical, I’m not informed enough on the topic to know.
“Regarding the relationship between belief states and “features”, I think there are a few important connections:
Belief states can be thought of as the “true features” that optimally compress the information needed for prediction. In the simplex visualizations, each point corresponds to a unique belief state—a particular probabilistic weighting of the possible hidden states of the generator at that point in the sequence. These belief states are the most parsimonious representation of the relevant information for predicting the future.
The “features” often discussed in mech interp (e.g. as visualized in attribution maps or extracted by probing classifiers) can be seen as approximations or projections of these true belief states. When we probe an LM’s activations and find that certain neurons or directions correspond to interpretable features like “noun-ness” or “sentiment”, we’re effectively discovering linear projections that approximate some aspect of the true belief state.
The topology and geometry of the belief state simplex can give us intuitions about the structure of these feature spaces. For example, the fact that belief states are constrained to a simplex (rather than an unbounded space) suggests that there may be inherent trade-offs or dependencies between different features. The trajectories of belief states through the simplex as a sequence evolves could also inform us about the dynamics of how features interact and compose.
So in a sense, belief states provide a principled foundation for thinking about what the “ideal features” for a given predictive task would be, and mech interp techniques can be seen as ways to approximate or probe these ideal features in real models.
Regarding toy models and information-theoretic aspects, some elements that could bridge between formal systems and natural language are:
Temporal dependencies and long-range correlations: Natural language exhibits complex temporal structure at multiple scales (e.g. within sentences, across paragraphs, across documents). Toy sequence models with tunable temporal correlations could help us understand how belief states track and compress these dependencies.
Compositionality and recursion: Language exhibits rich compositional structure, where smaller units (morphemes, words, phrases) combine in systematic ways to produce complex meanings. Toy generative grammars or rule systems could help us study how belief states navigate these combinatorial spaces.
Stochasticity and entropy: Language is inherently stochastic and exhibits varying levels of predictability (e.g. some words are much more predictable from context than others). Toy models with tunable entropy could help us understand how belief states adapt to different levels of predictability and how this impacts the geometry of the feature space.
Hierarchical structure: Language operates at multiple levels of abstraction (e.g. phonology, morphology, syntax, semantics, pragmatics). Toy models with explicit hierarchical structure could illuminate how belief states at different levels interact and compose.
The key idea would be to start with minimally complex toy systems that capture some core information-theoretic property of language, fully characterize the optimal belief states in that system, and then test whether the key signatures (e.g. the topology of the belief state space, the trajectory of dynamics) can be found in real language models trained on natural data.
This could provide a roadmap for building up more and more realistic models while maintaining a principled understanding of the underlying information-theoretic structures. The goal would be to eventually bridge the gap between our understanding of toy systems and the much more complex but often qualitatively similar structures found in real language models.
Of course, this is a highly ambitious research program and there are many challenges to contend with. But I believe this kind of cross-pollination between formal methods like Computational Mechanics and more empirical mech interp work could be very fruitful.”
Not at all cringe! This is the age of AI. We either channel its immense power or ignore it at our own peril.
There is no human alive today that is utilizing even last-generation’s LLMs at their full potential. We should all be copying, delegating and cyborging much more from, to and with LLM—not less.
Thanks!
one way to construct an HMM is by finding all past histories of tokens that condition the future tokens with the same probablity distribution, and make that equivalence class a hidden state in your HMM. Then the conditional distributions determine the arrows coming out of your state and which state you go to next. This is called the “epsilon machine” in Comp Mech, and it is unique. It is one presentation of the data generating process, but in general there are an infinite number of HMM presntations that would generate the same data. The epsilon machine is a particular type of HMM presentation—it is the smallest one where the hidden states are the minimal sufficient statistics for predicting the future based on the past. The epsilon machine is one of the most fundamental things in Comp Mech but I didn’t talk about it in this post. In the future we plan to make a more generic Comp Mech primer that will go through these and other concepts.
The interpretability of these simplexes is an issue that’s in my mind a lot these days. The short answer is I’m still wrestling with it. We have a rough experimental plan to go about studying this issue but for now, here are some related questions I have in my mind:
What is the relationship between the belief states in the simplex and what mech interp people call “features”?
What are the information theoretic aspects of natural language (or coding databases or some other interesting training data) that we can instantiate in toy models and then use our understanding of these toy systems to test if similar findings apply to real systems.
For something like situational awareness, I have the beginnings of a story in my head but it’s too handwavy to share right now. For something slightly more mundane like out-of-distribution generaliztion or transfer learning or abstraction, the idea would be to use our ability to formalize data-generating structure as HMMs, and then do theory and experiments on what it would mean for a transformer to understand that e.g. two HMMs have similar hidden/abstract structure but different vocabs.
Hopefully we’ll have a lot more to say about this kind of thing soon!
The following is text from Claude Opus 3. I generally find people just dumping answers from LLMs to be kind of cringe, but in this case, as I was using it to try to understand the post and your comments better, I came across some really genuinely insightful-feeling stuff and felt as though Claude’s input might be helpful to you in some way, and that it would be remiss not to share it. I’m sorry if it’s nonsensical, I’m not informed enough on the topic to know.
“Regarding the relationship between belief states and “features”, I think there are a few important connections:
Belief states can be thought of as the “true features” that optimally compress the information needed for prediction. In the simplex visualizations, each point corresponds to a unique belief state—a particular probabilistic weighting of the possible hidden states of the generator at that point in the sequence. These belief states are the most parsimonious representation of the relevant information for predicting the future.
The “features” often discussed in mech interp (e.g. as visualized in attribution maps or extracted by probing classifiers) can be seen as approximations or projections of these true belief states. When we probe an LM’s activations and find that certain neurons or directions correspond to interpretable features like “noun-ness” or “sentiment”, we’re effectively discovering linear projections that approximate some aspect of the true belief state.
The topology and geometry of the belief state simplex can give us intuitions about the structure of these feature spaces. For example, the fact that belief states are constrained to a simplex (rather than an unbounded space) suggests that there may be inherent trade-offs or dependencies between different features. The trajectories of belief states through the simplex as a sequence evolves could also inform us about the dynamics of how features interact and compose.
So in a sense, belief states provide a principled foundation for thinking about what the “ideal features” for a given predictive task would be, and mech interp techniques can be seen as ways to approximate or probe these ideal features in real models.
Regarding toy models and information-theoretic aspects, some elements that could bridge between formal systems and natural language are:
Temporal dependencies and long-range correlations: Natural language exhibits complex temporal structure at multiple scales (e.g. within sentences, across paragraphs, across documents). Toy sequence models with tunable temporal correlations could help us understand how belief states track and compress these dependencies.
Compositionality and recursion: Language exhibits rich compositional structure, where smaller units (morphemes, words, phrases) combine in systematic ways to produce complex meanings. Toy generative grammars or rule systems could help us study how belief states navigate these combinatorial spaces.
Stochasticity and entropy: Language is inherently stochastic and exhibits varying levels of predictability (e.g. some words are much more predictable from context than others). Toy models with tunable entropy could help us understand how belief states adapt to different levels of predictability and how this impacts the geometry of the feature space.
Hierarchical structure: Language operates at multiple levels of abstraction (e.g. phonology, morphology, syntax, semantics, pragmatics). Toy models with explicit hierarchical structure could illuminate how belief states at different levels interact and compose.
The key idea would be to start with minimally complex toy systems that capture some core information-theoretic property of language, fully characterize the optimal belief states in that system, and then test whether the key signatures (e.g. the topology of the belief state space, the trajectory of dynamics) can be found in real language models trained on natural data.
This could provide a roadmap for building up more and more realistic models while maintaining a principled understanding of the underlying information-theoretic structures. The goal would be to eventually bridge the gap between our understanding of toy systems and the much more complex but often qualitatively similar structures found in real language models.
Of course, this is a highly ambitious research program and there are many challenges to contend with. But I believe this kind of cross-pollination between formal methods like Computational Mechanics and more empirical mech interp work could be very fruitful.”
Not at all cringe! This is the age of AI. We either channel its immense power or ignore it at our own peril.
There is no human alive today that is utilizing even last-generation’s LLMs at their full potential. We should all be copying, delegating and cyborging much more from, to and with LLM—not less.