Questions tagged [probability-theory]
For questions related to probability theory in the context of artificial intelligence.
11
questions
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1answer
58 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 ...
1
vote
1answer
23 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 ...
1
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1answer
46 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 ...
2
votes
2answers
152 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)
$$\mathbb{E} \left[ (Y - \hat{Y})^2 \right] = \left(f(X) - \hat{f}(X) \right)^2 + \operatorname{...
0
votes
1answer
79 views
Bayesian Perceptron: Why to marginalize over neuron's output instead of it's weights?
I found a very interesting paper on the internet that tries to apply Bayesian inference with a gradient-free online-learning approach: Bayesian Perceptron: Towards fully Bayesian Neural Networks.
I ...
4
votes
1answer
181 views
How is the formula for the Bayes error rate with an integral derived?
My questions concern a particular formulation of the Bayes error rate from Wikipedia, summarized below.
For a multiclass classifier, the Bayes error rate may be calculated as follows: $$p = 1 - \sum_{...
1
vote
1answer
68 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 ...
0
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0answers
59 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 \...
1
vote
3answers
95 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) =...
6
votes
0answers
167 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 ...
2
votes
1answer
118 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 ...
