Do Androids Dream of Conditional Probabilities?

This Stanford lecture by Yann Dubois on how Large Language Models (LLMs) are built is a bit dated now (it was delivered in August 2024), but it remains an excellent introduction to the machinery that powers modern LLMs and also highlights some common misconceptions.

Perhaps the most persistent of those is the claim that LLMs are just next-word predictors, usually followed by the assertion that they are not intelligent.

LLMs are artificial neural networks, generally based on the Transformer architecture. During pretraining, an autoregressive language model learns to estimate the conditional probability of the next token (which might be a complete word, part of a word, or another unit of text), given everything that precedes it. 

Now I will bore you with a bit of math, but you can skip it if you are not interested.

Mathematically, this probability is expressed as:

\[P_\theta(x_t \mid x_1,x_2,\ldots,x_{t-1})\]

Here, \(x_t\) represents the next token, the preceding terms represent the context, and \(\theta\) represents the parameters learned by the neural network.

Using the chain rule of probability, the probability assigned to an entire sequence can be written as:

\[P_\theta(x_1,\ldots,x_n)=\prod_{t=1}^{n}P_\theta(x_t\mid x_{\lt t})\]

The model is trained by adjusting its parameters to increase the probabilities assigned to the actual tokens in its training data. To state it more precisely, it maximizes the likelihood (actually minimizes the negative log-likelihood):

\[\mathcal{L}(\theta)=-\sum_{t=1}^{n}\log P_\theta(x_t\mid x_{\lt t})\]

That’s essentially the mathematical foundation of autoregressive pretraining, which you just skipped. Remarkably simple, at least in principle.

Through this process, the model learns statistical regularities involving syntax, semantics, conceptual relationships, logical structures, and how the world works, as expressed through sequences of tokens. The resulting neural network acquires complicated internal representations whose capabilities far exceed what the phrase “predicting the next word” might suggest. It has, in effect, constructed an internalized map of relationships and meaning. Whether this amounts to “understanding”, and exactly how its reasoning processes are implemented internally, are the actual empirical research questions.

The training objective tells us what the system has been optimized to do, but it does not, by itself, establish the limits of what the system can learn to do.

Dubois’s lecture explains why building an effective LLM involves far more than choosing a neural network architecture. He highlights the importance of the availability, quality and composition of training data, the scale of computation, the training procedures themselves, and the evaluation methodology.

These networks not only memorize patterns, they also generalize these learned patterns to unfamiliar inputs. The interesting scientific question therefore concerns the nature, the extent, and thhe limitations of these generalizations.

Much of the underlying knowledge and capability of LLMs is developed during pretraining. Subsequent supervised fine-tuning and reinforcement learning shape how those capabilities are actually expressed. Modern reasoning systems also employ additional post-training techniques. Human input, both in preparing training data and in shaping model behaviour, remains very important.

Despite all of the progress that has been made, these systems still generate convincing nonsense. Such failures (hallucinations) are serious and obviously require better calibration, verification, and evaluation. Today’s strongest systems have made remarkable progress since 2024 and the hallucinations for the best models have decreased in severity, especially in the last few motnhs, although overall reliability remains highly dependent on the task, and all output must still be double checked and verified.

But given the impressive capabilities of contemporary systems, the claim that they are not intelligent is becoming increasingly difficult to defend. Intelligence is neither a binary property nor something that necessarily requires a particular physical substrate. A useful operational definition involves the capacity to acquire and apply knowledge, identify relationships, solve problems, and generalize to unfamiliar situations. Different systems can possess these abilities to different degrees. And I would argue that today we have substantial empirical evidence of such abilities in artificial systems.

Furthermore, I think we already have compelling evidence that these systems demonstrate genuine creativity, at least in the functional sense of producing useful and original solutions.

Here are a few examples from the wider field of machine learning to consider. AlphaGo Zero learned to play Go without training on human games, relying instead on self-play and reinforcement learning. It developed highly unconventional strategies and surpassed the strongest Go players. AlphaFold predicted protein structures before the experimental structures were available to the researchers making the predictions. 

These are specialized AI systems rather than LLMs, but they illustrate how neural networks can discover useful solutions that were never explicitly provided during training.

More directly relevant to LLMs, FunSearch combined a language model with an automated evaluator to discover previously unknown mathematical constructions for the cap set problem, improving established results. AlphaEvolve discovered new algorithms and improved solutions to mathematical and computational problems.

In these  latter examples, the language model generates candidate solutions, which are then tested and refined. This means that the resulting discoveries can be independently verified. The system here produces new knowledge rather than simply retrieving information from its training data.

Another example, and I will end with this, is the 2025 International Mathematical Olympiad. An advanced version of Google’s Gemini Deep Think solved five of the six problems, scoring 35 out of 42 points. This is gold-medal standard, achieved by only a small fraction of human participants. The solutions were evaluated and certified by IMO coordinators. All of the problems were original, previously unseen problems, specifically designed to test sophisticated mathematical problem-solving skills. The models had to construct completely novel solutions. The answers did not exist in their training sets.

Now I consider all of this strong evidence of intelligence and creativity in the strict  functional sense. Whether these systems experience creativity, possess subjective awareness, or could develop consciousness are separate open philosophical and scientific questions. But none of them need to be resolved before we can establish what these systems can actually do.

Despite all of this, modern systems still fail frustratingly in apparently very simple tasks. They can become confused when unfamiliar constraints are introduced, they are easy to misdirect, and they produce and stand behind errors far too confidently. They are still too uncomforatble with uncertainty and are pushed, by their training and the demands of their users, to provide answers. A 2025 Apple study argued there are serious limitations in controlled reasoning tasks as problem complexity increased. 

So we need to be cautious about their capabilities and need rigorous, reproducible tests of generalization, reasoning, reliability, and the ability to identify and correct errors. 

But all of this suggests that artificial intelligence may be developing differently from human intelligence. 

These systems are highly capable and can already solve unfamiliar problems, construct complex mathematical proofs, write and debug software, recognize relationships, and transfer knowledge between domains. In some specialized, objectively testable tasks they already reach or exceed expert human performance. These are capabilities that any reasonable scientific account of intelligence must explain.

It goes without saying that we should not play down the risks associated with rapid AI development. It is not just about reliability and mistakes these models make, or about the problem of energy usage. It  is also about concentration of corporate power, rising inequality, and the political and institutional challenges of regulating increasingly capable systems. We need to take all of this very seriously and push our elected representatives to better understand the capabilities and limitations of these systems, to develop and enforce proportionate regulations, and to ensure that innovation serves the wider public interest.

Now, if after reading all of this, you insist that no matter what a machine accomplishes, its performance can never count as intelligence, you have constructed a claim that no conceivable experiment could falsify. That’s bad science.

The more productive question is to ask exactly what kinds of intelligence these systems exhibit, how those capabilities arise, how understandable their reasoning is, how reliably they generalize, and where their limitations lie. 

And those questions we can actually investigate.

Rubio in Athens

Rubio's speech at the Pnyx in Athens contained some interesting observations about the ancient Greek intellectual inheritance, along with the expected nods to his audience. Massaging national egos pays off everywhere. What is more interesting, however, is where he chose to place his emphasis. He spoke of ambition, honour, duty, glory, military strength, loyalty to the state and national power. He spoke of civilisational danger and external enemies. Toward the end of the speech, among his examples of the Western spirit he specifically named Alexander, Caesar, Charlemagne and Cortés. But what exactly is distinctively Western about conquest and military glory? Countless other civilisations have their own great conquerors, imperial ambitions and sanitised national myths.

What *is* distinctive about Western civilisation is the development of principles that challenged the arbitrary exercise of power: individual liberty, freedom of conscience, equality before the law, limits on political authority and, perhaps most importantly, institutionalised self-criticism. The ability of a society to question its own assumptions, traditions and institutions, and recursively improve upon them.

John Stuart Mill famously argued in *On Liberty* that even if all humanity agreed on something except one individual, the majority would have no right to silence that person. The Scientific Revolution and Enlightenment further established that no authority, however prestigious, should be beyond scrutiny. This principle underpins both scientific progress and liberal democracy.

Of course none of these ideas emerged fully formed, nor were they exclusively Western or consistently applied. Western history is full of conquest, slavery, persecution and imperial domination, but it also produced powerful intellectual and institutional mechanisms for challenging these injustices. And that particular intellectual and institutional development is a distinctive Western achievement, shaped in part by centuries of intellectual exchange with other cultures. Its successful adoption in countries as diverse as Japan and South Korea, demonstrates that the principles themselves are universal, not some exclusive Western inheritance.

The obvious irony here is that Rubio stood at the birthplace of Athenian democracy, praising Western civilisation on behalf of an administration seeking to concentrate executive power, challenging judicial constraints, pressuring independent media, imposing political conditions on universities, disparaging democratic allies and proposing drastic cuts to scientific research. What exactly is he asking us to defend here?

He spoke of the need for a strong military. Fair enough. Military strength matters for deterrence. A society must be able to defend itself and its values. But military strength is a means to an end, not a measure of civilisational greatness. And we should be particularly wary when military expansion goes hand in hand with the erosion of democratic checks and balances. The greater the concentration of power, the stronger the institutions needed to constrain it. Not weaker. Otherwise, what distinguishes Rubio's vision of Western greatness from that of any other empire throughout history?

So yeah, absolutely, Western civilisation's finest achievements are worth defending. But we should first be very clear about which achievements we are actually talking about. And we should recognise that some of the most serious threats to these achievements today come from within, from the very people who most loudly proclaim themselves their guardians.

Berlin elections 2026

Die Linke won 25.7% in Berlin, roughly doubling its 2023 vote. The SPD fell to 12.1%, the Greens to 14.3% and the CDU to 18.8%, while the AfD rose to 16.3%. So the story here is that both Die Linke and the AfD gained substantially while the traditional parties of the broad political centre lost ground.

Now it isn't difficult to understand why Die Linke did so well. Berlin has a genuine and measurable housing crisis. The city's own estimate says another 211,000 homes will be needed between 2026 and 2040. Die Linke made housing the centre of its campaign and was clearly seen as the strongest party on the issue, particularly among younger voters. Its proposed solutions include much more public housing, stronger rent controls and the socialisation of large private housing companies, meaning compulsory transfer into public ownership with compensation. Building new affordable public housing actually increases supply, but purchasing existing apartments would cost many billions in compensation and raises an obvious opportunity-cost question: how many new apartments, schools or transport projects could be built with the same money? Interestingly, although 57.6% backed socialisation in the 2021 referendum, only 37% supported it in the latest polling and 51% opposed it. So the election result shouldn't be read as blanket approval for every part of Die Linke's programme.

Both Die Linke and the SPD support stronger tenant protection, more social housing, progressive taxation and a strong welfare state, but their underlying philosophies differ. The SPD remains fundamentally social democratic: regulate capitalism and use a mixed economy to serve social goals. Die Linke comes from a democratic-socialist tradition and is much more willing to take essential goods and services out of private markets altogether. Their differences become larger still at federal level, particularly on defence, Ukraine and immigration: The SPD supports higher defence spending and continued military assistance to Ukraine, whereas Die Linke opposes further military build-up and weapons deliveries to war zones; on immigration, Die Linke favours much broader residence rights and opposes deportations.

Berlin is young, urban and renter-heavy. The election results there follow a broader pattern we've seen elsewhere. Increasing numbers of voters are abandoning mainstream parties they associate with failing to solve problems they experience directly, and turning instead to parties promising much more substantial change. And it isn't as though Berliners haven't tried the established parties first. Over the past 15 years they have repeatedly elected governments led by the SPD and, more recently, the CDU, while the housing crisis has continued to get worse. (To be fair, Die Linke itself was also in government from 2016 to 2023, so it cannot entirely claim outsider status either.) There's no surprise here really. This is what happens when obvious problems remain unresolved through successive governments and are just left to fester. Voters will start looking elsewhere for solutions. Whether the alternatives they now choose can actually deliver the changes they expect is a different kettle of fish.

(Sources: official Berlin election results; Berlin Senate housing planning; Infratest dimap/ARD, September 2026.)

The dragon in my garage (peer pressure edition)

Suppose I tell you there is a dragon living in my garage.

Naturally, you ask to see it.

Fair enough.

I open the garage door.

Inside are some cardboard boxes, a bicycle I have been meaning to repair since approximately the late Pleistocene, a lawnmower, several objects whose original purpose has been lost to history, and absolutely no visible dragon.

“Ah,” I say. “Of course you can’t see it. The dragon is invisible. Should have mentioned that.”

Right.

You think about this for a moment and suggest scattering flour over the floor. If the dragon is there, surely it will leave footprints.

“No. It floats above the ground.”

Very well.

You suggest bringing in an infrared camera. Dragons are creatures traditionally associated with fire, so it ought to give off some heat.

“This one doesn’t.”

“Well,” you say, “perhaps we could spray a fine mist of paint into the garage and reveal its outline?”

“Nooo. The paint passes straight through it.”

Every test you propose is followed by a new property of the dragon explaining why that particular test cannot possibly work.

“I see that you are still skeptical,” I say, disappointed.

The next day I bring twenty people to the garage. Some of them are your friends and family.

“Look,” I tell you. “All of these people believe the dragon is real. Do you believe me now?”

One of them looks genuinely puzzled by your hesitation.

“Millions of people believe in dragons like this,” she says. “Do you seriously think all those people could be wrong?”

A respected older member of the community steps forward.

“My belief in the dragon has given my life meaning,” they tell you sincerely.

Another person explains that the dragon helped her through the most difficult period of her life.

Someone else tells you that his grandmother believed in the dragon, and she was the kindest person he ever knew.

A fourth informs you that some extremely clever philosophers have also believed in dragons.

“Could we perhaps devise some kind of test to make sure...?” you say.

There is a small silence.

“You still don’t believe us?” someone asks.

Then a member of your own family, obviously embarrassed by you and hurt by your stubbornness, asks:

“Why are you the only one who cannot accept what everyone else believes is there?”

You mumble that you do not doubt anyone’s sincerity, but twenty people repeating a claim do not necessarily constitute twenty independent pieces of evidence.

Twenty people independently seeing a dragon would certainly be interesting. But if the twenty people learned about the dragon from the same tradition, heard the same stories, rely on one another’s testimony, and none of them can actually detect the dragon either, then their agreement is not twenty independent observations. It may simply be the same claim echoed twenty times over.

You say that their belief in the dragon may comfort them, inspire them, make them kinder, give structure to their lives, connect them to their parents and grandparents, or provide an answer to the rather unsettling business of finding oneself briefly alive in an enormous universe without an instruction manual.

All of that is interesting, and some of it is certainly beautiful.

But sincerity, comfort, tradition, intelligence, and a room full of people agreeing with one another are not enough to establish that the dragon is actually there.

So you return, somewhat tediously, to the original question:

“What observation could distinguish between these two possibilities?

1) There is an invisible, floating, heatless, intangible dragon in the garage.
or
2) There is actually no dragon in the garage.”

You continue, unabated:

“Because if no conceivable observation could distinguish one possibility from the other, and every failed attempt to detect the dragon can simply be accommodated by inventing another property of the dragon, then we have reached a peculiar situation. The claim has been constructed so that nothing we observe can count against it. There is no longer any obvious way to test whether it is true.”

By now nobody is particularly happy with you.

You have, after all, been invited to contemplate the remarkable properties of a dragon, and somehow you have managed to turn a beautiful evening into a boring seminar on epistemology. Boo to you.

You notice that everyone is now rather uncomfortable with your presence.

You think that, in the end, perhaps you have not learned very much about dragons, but you have certainly learned quite a lot about human beings.

You have dinner alone that evening. The others may have escaped your incessant questioning, but the questions, annoyingly, have followed you home.

* No dragons were harmed while writing this piece.

* Carl Sagan's "The Dragon in My Garage" is a famous thought experiment from his 1995 book The Demon-Haunted World that illustrates the burden of proof and why untestable claims are meaningless.