➕Probabilistic-Artificial-Intelligence

We have a prior $p(\text{model})$, we have a posterior $p(\text{model} \mid \text{data})$, a likelihood $p(\text{data} \mid \text{model})$ and $p(\text{data})$ is called the evidence. Classical Linear regression $$ y = w^{T}x + \varepsilon $$ Where $\varepsilon \sim \mathcal{N}(0, \sigma_{n}^{2}I)$ and it’s the irreducible noise, an error that cannot be eliminated by any model in the model class, this is also called aleatoric uncertainty. One could write this as follows: $y \sim \mathcal{N}(w^{T}x, \sigma^{2}_{n}I)$ and it’s the exact same thing as the previous, so if we look for the MLE estimate now we get ...

$$ p(\theta \mid x_{1:n}, y_{1:n}) = \frac{1}{z} p(y_{1:n} \mid \theta, x_{1:n}) p(\theta \mid x_{1:n}) \approx q(\theta \mid \lambda) $$For Bayesian Linear Regression we had high dimensional Gaussians which made the inference closed form, in general this is not true, so we need some kinds of approximation. Laplace approximation Introduction to the Idea $$ \psi(\theta) \approx \hat{\psi}(\theta) = \psi(\hat{\theta}) + (\theta-\hat{\theta} ) ^{T} \nabla \psi(\hat{\theta}) + \frac{1}{2} (\theta-\hat{\theta} ) ^{T} H_{\psi}(\hat{\theta})(\theta-\hat{\theta} ) = \psi(\hat{\theta}) + \frac{1}{2} (\theta-\hat{\theta} ) ^{T} H_{\psi}(\hat{\theta})(\theta-\hat{\theta} ) $$ We simplified the term on the first order because we are considering the mode, so the gradient should be zero for the stationary point. ...

These algorithms are good for scaling state spaces, but not actions spaces. The Gradient Idea Recall Temporal difference learning and Q-Learning, two model free policy evaluation techniques explored in Tabular Reinforcement Learning. A simple parametrization The idea here is to parametrize the value estimation function so that similar inputs gets similar values akin to Parametric Modeling estimation we have done in the other courses. In this manner, we don’t need to explicitly explore every single state in the state space. ...

While Active Learning looks for the most informative points to recover a true underlying function, Bayesian Optimization is just interested to find the maximum of that function. In Bayesian Optimization, we ask for the best way to find sequentially a set of points $x_{1}, \dots, x_{n}$ to find $\max_{x \in \mathcal{X}} f(x)$ for a certain unknown function $f$. This is what the whole thing is about. Definitions First we will introduce some useful definitions in this context. These were also somewhat introduced in N-Bandit Problem, which is one of the classical optimization problems we can find in the literature. ...

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. ...

This note extends the content Markov Processes in this specific context. One nice expansion, which treats the field a little bit more from the behavioural sciences perspectiv eis Intrinsic Motivation and Playfulness. Standard notions Explore-exploit dilemma We have seen something similar also in Active Learning when we tried to model if we wanted to look elsewhere or go for the maximum value we have found. The dilemma under analysis is the explore-exploit dilemma: whether if we should just go for the best solution we have found at the moment, or look for a better one. This also has implications in many other fields, also in normal human life there are a lot of balances in these terms. ...

Introduzione alle catene di Markov La proprietà di Markov Una sequenza di variabili aleatorie $X_{1}, X_{2}, X_{3}, \dots$ gode della proprietà di Markov se vale: $$ P(X_{n}| X_{n - 1}, X_{n - 2}, \dots, X_{1}) = P(X_{n}|X_{n-1}) $$ Ossia posso scordarmi tutta la storia precedente, mi interessa solamente lo stato precedente per sapere la probabilità attuale. Da un punto di vista filosofico/fisico, ha senso perché mi sta dicendo che posso predire lo stato successivo se ho una conoscenza (completa, (lo dico io completo, originariamente non esiste)) del presente. ...

Robbins-Moro Algorithm The Algorithm $$ w_{n+1} = w_{n} - \alpha_{n} \Delta w_{n} $$For example with $\alpha_{0} > \alpha_{1} > \dots > \alpha_{n} \dots$, and $\alpha_{t} = \frac{1}{t}$ they satisfy the condition (in practice we use a constant $\alpha$, but we lose the convergence guarantee by Robbins Moro). More generally, the Robbins-Moro conditions re: $\sum_{n} \alpha_{n} = \infty$ $\sum_{n} \alpha_{n}^{2} < \infty$ Then the algorithm is guaranteed to converge to the best answer. One nice thing about this, is that we don’t need gradients. But often we use gradient versions (stochastic gradient descent and similar), using auto-grad, see Backpropagation. But learning with gradients brings some drawbacks: ...

Gaussians are one of the most important family of probability distributions. They arise naturally in the law of large numbers and have some nice properties that we will briefly present and prove here in this note. They are also quite common for Gaussian Processes and the Clustering algorithm. They have also something to say about Maximum Entropy Principle. The best thing if you want to learn this part actually well is section 2.3 of (Bishop 2006), so go there my friend :) ...

Active Learning concerns methods to decide how to sample the most useful information in a specific domain; how can you select the best sample for an unknown model? Gathering data is very costly, we would like to create some principled manner to choose the best data point to humanly label in order to have the best model. In this setting, we are interested in the concept of usefulness of information. One of our main goals is to reduce uncertainty, thus, Entropy-based (mutual information) methods are often used. For example, we can use active learning to choose what samples needs to be labelled in order to have highest accuracy on the trained model, when labelling is costly. ...