➕Probabilistic-Artificial-Intelligence

The maximum entropy principle is one of the most important guiding motives in artificial artificial intelligence. Its roots emerge from a long tradition of probabilistic inference that goes back to Laplace and Occam’s Razor, i.e. the principle of parsimony. Let’s start with a simple example taken from Andreas Kraus’s Lecture notes in the ETH course of Probabilistic Artificial Intelligence: Consider a criminal trial with three suspects, A, B, and C. The collected evidence shows that suspect C can not have committed the crime, however it does not yield any information about sus- pects A and B. Clearly, any distribution respecting the data must assign zero probability of having committed the crime to suspect C. However, any distribution interpolating between (1, 0, 0) and (0, 1, 0) respects the data. The principle of indifference suggests that the desired distribution is $(\frac{1}{2}, \frac{1}{2}, 0)$, and indeed, any alterna- tive distribution seems unreasonable. ...

DI Law of Large Numbers e Central limit theorem ne parliamo in Central Limit Theorem and Law of Large Numbers. Usually these methods are useful when you need to calculate following something similar to Bayes rule, but don’t know how to calculate the denominator, often infeasible integral. We estimate this value without explicitly calculating that. Interested in $\mathbb{P}(x) = \frac{1}{z} \mathbb{P}^{*}(x) = \frac{1}{Z} e^{-E(x)}$ Can evaluate E(x) at any x. Problem 1 Make samples x(r) ~ 2 P Problem 2 Estimate expectations $\Phi = \sum_{x}\phi(x)\mathbb{P}(x)$) What we’re not trying to do: We’re not trying to find the most probable state. We’re not trying to visit all typical states. Law of large numbers $$ S_{n} = \sum^n_{i=1} x_{i} ,:, \bar{x}_{n} = \frac{S_{n}}{n} $$$$ \bar{x}_{n} \to \mu $$ Ossia il limite converge sul valore atteso di tutte le variabili aleatorie. ...

This note will mainly attempt to summarize the introduction of some intuitive notions of probability used in common sense human reasoning. Most of what is said here is available here (Jaynes 2003). Three intuitive notions of probability Jaynes presents some forms of inference that are not possible in classical first order or propositional logic, yet they are frequent in human common sense reasoning. Let’s present some rules and some examples along them: ...

There is huge literature on planning. We will attack this problem from the view of probabilistic artificial intelligence. In this case we focus on continuous, fully observed with non-linear transitions, an environment often used for robotics. It’s called Model Predictive Control (MPC). \[...\] Moreover, modeling uncertainty in our model of the environment can be extremely useful in deciding where to explore. Learning a model can therefore help to dramatically reduce the sample complexity over model-free techniques. ...

The main difference between reinforcement learning and other machine learning, pattern inference methods is that reinforcement learning takes the concept of actions into its core: models developed in this field can be actively developed to have an effect in its environment, while other methods are mainly used to summarize interesting data or generating sort of reports. Reinforcement learning (RL) is an interdisciplinary area of machine learning and optimal control concerned with how an intelligent agent ought to take actions in a dynamic environment in order to maximize the cumulative reward. ~Wikipedia page. ...