Two math word problems (Bayes) that are similar to me but different to the LLMs
It's easy to write a math world problem that stumps GPT-4, but I'm not surprised by that: I'm totally accustomed to writing buggy code!
When I noticed that the FDP paused their exam due to AI (see my Financial certifications: Is AI the beginning of the end?), I was curious about the nature of their multiple choice questions (MCQs). Their first published sample question is a trivial application of Bayes Theorem; it’s typical of a question any CFA/FRM/CFP/etc ought to be able to answer:
”Frank plays pool on Simonis tablecloth 75% of the time. His overall potting success rate is 80%, which increases to 90% when he plays on Simonis tablecloth. What is the probability that Frank is playing on Simonis tablecloth if he has successfully potted a ball?”
— Question #1 in FDP’s Limited Sample of Multiple-Choice Questions (April 8 - April 22, 2004)1
ChatGPT 3.5 gets it wrong, but ChatGPT 4.0 gives a perfect answer. Note the instructive code snippet (this never gets old to me, amazing tech)!
Can I write a similar Bayes question that is natural to a human reader but stumps the LLM?
I wondered how hard it would be to write a human-easy but machine-hard Bayes question. It only took one attempt. And my first try was not trying to be difficult; notice that I’ve redundantly included the parenthetical, which is a habit I’ve developed writing/answering thousands of MCQs, when I reflexively am compelled to give clear assumptions:
Jeff follows a 3:2 hybrid work schedule, with two days at home and three days in the office each week. He avoids gossip with a 90% success rate overall (i.e., 9 out of every 10 days), which reduces to 84% on office days. If Jeff gossiped today, what is the probability that he is working from home?
GPT 4.0 gives an incorrect answer and, to me, it’s an unexpected fail: it assumes P(B|A) = 10% when I made the extra effort to assume that 90% is an unconditional probability; i.e., if P(B’) = 90% then P(B) = 10%. Because perplexity allows me to select the model (including Claude 3 Opus), I tried to get the answer from perplexity—with various models—but could not even after adding clarity to the problematic unconditional probability assumption (emphasis mine):
Jeff follows a 3:2 hybrid work schedule, with two days at home and three days in the office each week. Overall, across all days, he avoids gossip with a 90% success rate (i.e., 9 out of every 10 days), which reduces to 84% on office days. If Jeff gossiped today, what is the probability that he is working from home?
What’s stunning to me is how excellent is the response except for the final, trivial arithmetic:
P(H|G) = P(G|H) * P(H) / P(G)
= 0.01 * (2/5) / 0.1
= 0.008 / 0.1
= 0.08
Until then, it’s a brilliant sequence!
Of course I asked GPT 4 to re-write my question to avoid this mistake and then it gives a correct answer. But perplexity fails, and on the same step as before!
What do I make of this?
I don’t nearly know. I’m learning LLMs from both sides: as a user (easy) and from the technical side (much harder and slower). So far, I’ve reached only a few tentative beliefs including:
Contrary to Ethan Mollick’s advice to anthropomorphize the LLM2, I think it’s important to understand that to prompt is to write a natural language code snippet. We are not talking to a person here, we’re using a new tool. Prompt engineering is a new variation on writing code. It’s way easier on the front (few bugs) but this looseness and flexibility masks the underlying challenges.
Language is not intelligence3. I’ve written/responded to enough MCQs that I’m somewhat confident my own gossip question above is sufficiently clear, especially after my instinctive insertion of the redundant, clarifying clause—“(i.e., 9 out of every 10 days)”—so it’s a fine question for a human. But that’s me imagining an intelligent (human) counterparty, my natural state. I’ve given the assumptions, the reader will understand. But there’s insufficient basis to assume that succeeds for the language model, which is clearly not intelligent. As if. As far as the language model is is concerned, I’ve written a buggy program.
Yesterday Jurgen Gravestein wrote something (based on this paper) that strikes me as important. He explains why “language is the culprit” that leads to our confusion in relation to the oxymoronic artificial intelligence (and machine learning):
You see, as people we treat language fluency as a proxy for intelligence. Simply put, we judge how smart people are based on the way they talk … we tend to overestimate the capabilities of AI systems like GPT-4, due to their linguistic fluency, and perceive them as smarter than they really are.
My Bayes example is a word problem, the sort that many students hate and for good reason: those with mathematical tendencies are likely to prefer a purely symbolic setup. So much more to explore …
I’ve just read his book Co-intelligence that includes four rules (aka, principles) for working with AI: 1. Always invite AI to the table; 2. Be the human in the loop; 3. Treat AI like a person (but tell it what kind of person it is); 4. Assume this is the worst AI you will ever use. Admittedly, he’s fully aware that anthropomorphism is a “sin … so let’s start sinning.” His practical point is to get learners started with personas.
On Fridays, I babysit a precious and precocious 19-month old who is supremely intelligent but does not yet speak any language that is widely recognizable. At best, language proficiency may signify (and correlate with) intelligence but the inverse is not necessarily true. Note: the social thumb is not my own adorable tender charge (I didn’t get his permission), it’s just the Midjourney response to a prompt (“toddler at the home office, operating a wacom tablet, and writing a phrase, but the text is intelligible”).