While you might expect AI to be the epitome of cold, logical reasoning, researchers now suggest that they might be even more illogical than humans. Researchers from University College London put seven of the top AIs through a series of classic tests designed to test human reasoning. Even the best-performing AIs were found to be irrational and prone to simple mistakes, with most models getting the answer wrong more than half the time. However, the researchers also found that these models weren't irrational in same way as a human while some even refused to answer logic questions on'ethical grounds'. Olivia Macmillan-Scott, a PhD student at UCL and lead author on the paper, says: 'Based on the results of our study and other research on Large Language Models, it's safe to say that these models do not'think' like humans yet.'

Do large language models (LLMs) display rational reasoning? LLMs have been shown to contain human biases due to the data they have been trained on; whether this is reflected in rational reasoning remains less clear. In this paper, we answer this question by evaluating seven language models using tasks from the cognitive psychology literature. We find that, like humans, LLMs display irrationality in these tasks. However, the way this irrationality is displayed does not reflect that shown by humans. When incorrect answers are given by LLMs to these tasks, they are often incorrect in ways that differ from human-like biases. On top of this, the LLMs reveal an additional layer of irrationality in the significant inconsistency of the responses. Aside from the experimental results, this paper seeks to make a methodological contribution by showing how we can assess and compare different capabilities of these types of models, in this case with respect to rational reasoning.

Let's get the basic definition out of the way: Markov Chain Monte Carlo (MCMC) methods let us compute samples from a distribution even though we can't compute it. Let's back up and talk about Monte Carlo Sampling. What are Monte Carlo methods? "Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results." And you are tasked with determining the area enclosed by this shape.

In order to reason about probabilistic knowledge, we must reason about time and actions as well. When we say, for example, that "the probability of'heads' after a coin toss is 50% and that of'tails' is 50%", we implicitly assume that there is an action (in this example, tossing a coin) which can bring the world to different states in the next moment in time. The uncertainty lies in the state transition: the world may end up in a state where the coin shows heads or in a state where it shows tails. Despite the evident dependence of our informal notion of probability on the notions of action and time, the formal mathematical languages that we use to talk about probabilities rarely support mentioning action and time explicitly. Kolmogorov's probability theory, for example, merely defines probability as the measure function in a measure space with total measure 1 [9]. The task of modeling time-dependent actions and their possible outcomes in terms of events in a probabilistic space remains informal. While this informality is not problematic in the simplest situations (e.g. when we are interested in the possible outcomes of a single action, or when multiple actions are independent of each other), slightly more complex situations may already lead to confusion and difficulty. A famous example is the Monty Hall problem [10]. Another inconvenience of dealing with probabilities just in terms of a measure space is that its set-theoretic language (where events are represented as subsets of the sample space) is rather limited.

The Monty Hall problem was first featured on the classic game show "Let's make a Deal". In the final segment of the show, contestants were presented with a choice of three different doors. Behind two of the doors would be a goat, and behind the third would be an extravagant prize such as a car. The contestant begins the game by picking one door. The host, Monty Hall, would then open one of the remaining doors.

The Monty Hall problem was first featured on the classic game show "Let's make a Deal". In the final segment of the show, contestants were presented with a choice of three different doors. Behind two of the doors would be a goat, and behind the third would be an extravagant prize such as a car. The contestant begins the game by picking one door. The host, Monty Hall, would then open one of the remaining doors.

How about trying to find any use of the famous Monty Hall problem in a stock index context? First of all, some of you may be confused because neither "Monty Hall problem" nor "Let's make a deal" are familiar to you so I will refresh you what these names are concerned to. Monty Hall was a TV presenter for "Let's make a deal", a famous American show in the sixties. Suppose you're on this game show and you're given the choice of three doors: behind one door there is a prize; behind the others, there is nothing. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which results to be empty.

The Monty Hall problem is the TV game scenario where you, the contestant, are presented with three doors, with a car hidden behind one and goats hidden behind the other two. After you select a door, the host (Monty Hall) opens a second door to reveal a goat. You are then invited to stay with your original choice of door, or to switch to the remaining unopened door, and claim whatever you find behind it. Assuming your objective is to win the car, is your best strategy to stay or switch, or does it not matter? Jason Rosenhouse has provided the definitive analysis of this game, along with several intriguing variations, and discusses some of its psychological and philosophical implications. This extended review examines several themes from the book in some detail from a Bayesian perspective, and points out one apparently inadvertent error.