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Questions tagged [probability-theory]

For questions related to probability theory in the context of artificial intelligence.

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18 votes
2 answers
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Are softmax outputs of classifiers true probabilities?

BACKGROUND: The softmax function is the most common choice for an activation function for the last dense layer of a multiclass neural network classifier. The outputs of the softmax function have ...
Snehal Patel's user avatar
8 votes
1 answer
603 views

What are the main benefits of using Bayesian networks?

I have some trouble understanding the benefits of Bayesian networks. Am I correct that the key benefit of the network is that one does not need to use the chain rule of probability in order to ...
Sebastian Dine's user avatar
6 votes
5 answers
1k views

How do compute the table for $p(s',r|s,a)$ (exercise 3.5 in Sutton & Barto's book)?

I am trying to study the book Reinforcement Learning: An Introduction (Sutton & Barto, 2018). In chapter 3.1 the authors state the following exercise Exercise 3.5 Give a table analogous to that ...
MrYouMath's user avatar
  • 255
4 votes
1 answer
563 views

How is the formula for the Bayes error rate with an integral derived?

My questions concern a particular formulation of the Bayes error rate from Wikipedia, summarized below. For a multiclass classifier, the Bayes error rate may be calculated as follows: $$p = 1 - \sum_{...
EntangledLoops's user avatar
4 votes
1 answer
48 views

What do we mean by "orderly opinions" in this sentence in the context of Bayes theorem?

In this page, it's written (emphasis mine) If probabilities are thought to describe orderly opinions, Bayes theorem describes how the opinions should be updated in the light of new information What ...
MC5321's user avatar
  • 41
2 votes
1 answer
192 views

Why is the equation $r(s', a, s') =\sum_{r \in \mathcal{R}} r \frac{p\left(s^{\prime}, r \mid s, a\right)}{p\left(s^{\prime} \mid s, a\right)}$true?

I am referring to eq. 3.6 (page 49) based on Sutton's online book and can be found in an image below. I could not make sense of the final derivation of the equation $r(s, a, s')$. My question is ...
alfa_80's user avatar
  • 123
2 votes
1 answer
485 views

Why don't we also need to approximate $p(x \mid z)$ in the VAE?

In the VAE, we approximate the probability distribution $p(z \mid x)$, where $z$ is the latent vector and $x$ is our data. The reason is that $p(z \mid x)$ becomes impossible to calculate for ...
Nervous Hero's user avatar
2 votes
2 answers
443 views

Why is the equation $\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{Var} (\epsilon)$ true?

In the book An Introduction to Statistical Learning, the authors claim (equation 2.3, p. 19, chapter 2) $$\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{...
nbro's user avatar
  • 40.9k
2 votes
1 answer
158 views

How would the probability of a document $P(d)$ be computed in the Naive Bayes classifier?

In naive Bayes classification, we estimate the class of a document as follows $$\hat{c} = \arg \max_{c \in C} P(c \mid d) = \arg \max_{c \in C} \dfrac{ P(d \mid c)P(c) }{P(d)} $$ It has been said in ...
hanugm's user avatar
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1 vote
1 answer
123 views

Which formula of p(x, y) to use?

The probability distribution $p(x, y)$ can be calculated in two ways : $p(x, y) = p(y \mid x) p(x)$ $p(x, y) = p(x \mid y) p(y)$ But according to the book Deep Generative Modeling (page number 3 ...
Nervous Hero's user avatar
1 vote
1 answer
334 views

Why are today's neural networks not modeled with probability theory?

In the paper The Perceptron: A probabilistic model for information storage and organization in the brain, Rosenblatt used the probability theory to model his perceptron. My professor told me that ...
Collo's user avatar
  • 11
1 vote
1 answer
102 views

Understanding how to calculate $P(x|c_k)$ for the Bernoulli naïve Bayes classifier

I'm looking at the Bernoulli naïve Bayes classifier on Wikipedia and I understand Bayes theorem along with Gaussian naïve Bayes. However, when looking at how $P(x|c_k)$ is calculated, I don't ...
Aguy's user avatar
  • 65
1 vote
3 answers
102 views

Is it possible to compute $P( F \mid S )$ given $P(F \mid S,A)$ and $P(F \mid S, \lnot A)$ in Bayesian network?

I have a bayesian network, which has the following data: $P(S) = 0.07$ $P(A) = 0.01$ $P(F \mid S,A) = 1.0$ $P(F \mid S, \lnot A) = 0.7$ $P(F \mid \lnot S, A) = 0.9$ $P(F \mid \lnot S, \lnot A) =...
Diego C. 's user avatar
1 vote
1 answer
133 views

What is x, y, p(x), p(y) in generative model domain?

Background Generative modeling Generative modeling aims to model the probability of observing an observation x. $$ p(x) = \frac{p(y\cap x)}{p(y|x)} $$ ...
Prakhar's user avatar
  • 11
0 votes
1 answer
38 views

Book(s) recomendation for Probability Theory Foundations

I am in the process of learning ML and AI. I've been taking some courses, and I now understand the foundations of the big picture of Machine Learning. I've been using Pattern Recognition and Machine ...
John Pinto's user avatar
0 votes
1 answer
29 views

Independence of random variable in Gaussian Process context

From Bishop's Pattern Recognition and Machine Learning: $t_n = y_n + \epsilon_n$, where $\epsilon_n$ is a random noise variable whose value is chosen independently for each observation $n$. Consider ...
piero's user avatar
  • 123
0 votes
1 answer
214 views

Bayesian Perceptron: Why to marginalize over neuron's output instead of it's weights?

I found a very interesting paper on the internet that tries to apply Bayesian inference with a gradient-free online-learning approach: Bayesian Perceptron: Towards fully Bayesian Neural Networks. I ...
f_3464gh's user avatar
0 votes
0 answers
10 views

Chernoff bound for sampling flipped bits and arriving at true average

The Basic Setting Let there be an $n$-bit string generated by $n$ successive trials of random variable $X$, name the string $x_1,..,x_n$, with $x_i \in \{0,1\}$. Define sample mean as $\bar{X} = \...
xiaolingxiao's user avatar
0 votes
0 answers
72 views

How to guess a card in the game of Cambio with limited information?

I need help with a probability problem in the card game of Cambio. In this game, two players are dealt four cards each from a deck of 52 cards . At the start of the game, the bottom cards of each ...
Ali Qaqan's user avatar
0 votes
1 answer
72 views

Why isn't the evidence $p(x) = 1$ if it's an observed variable?

Every explanation of variational inference starts with the same basic premise: given an observed variable $x$, and a latent variable $z$, $$ p(z|x)=\frac{p(x,z)}{p(x)} $$ and then proceeds to expand $...
Abrrval's user avatar
0 votes
0 answers
68 views

How can the V and Q functions take the expectation over a sum where the number of summands is random?

Assume the existence of a Markov Decision Process consisting of: State space $S$ Action space $A$ Transition model $T: S \times A \times S \to [0,1]$ Reward function $R: S \times A \times S \to \...
Leroy Od's user avatar
  • 475