Questions tagged [probability-distribution]

For questions related to AI theory that relies on the knowledge of a distribution of probabilities across one or more dimensions affecting probability. Such a distribution may be in discrete buckets, such as quartile, octile, or percentile conventions or continuous functions based on some closed form (algebraic formula). Distributions of probability are key in planning, natural language handling, and other AI objectives.

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11 views

Transforming neural network target values before training

Consider the scenario in which I am measuring certain $f(a,x)$, which i want to be the target value for some related input $g(a,x)$. In other words, I am trying to map $$g(a,x)\Rightarrow f(a,x)$$ I ...
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34 views

When should one prefer using Total Variational Divergence over KL divergence in RL

In RL, both the KL divergence (DKL) and Total variational divergence (DTV) are used to measure the distance between two policies. I'm most familiar with using DKL as an early stopping metric during ...
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38 views

Estimating $\sigma_i$ according to maximum likelihood method

Let be a Bayesian multivariate normal distribution classifier with distinct covariance matrices for each class and isotropic, i.e. with equal values over the entire diagonal and zero otherwise, $\...
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27 views

How to calculate v min and v max for C51 DQN

Background: In C51 DQNs you must specify a v-min/max to be used during training. The way this is generally done is you take the max score possible for the game and set that to v-max, then v-min is ...
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32 views

Why is the mean used to compute the expectation in the GAN loss?

From Goodfellow et al. (2014), we have the adversarial loss: $$ \min_G \, \max_D V (D, G) = \mathbb{E}_{x∼p_{data}(x)} \, [\log \, D(x)] \\ \quad\quad\quad\quad\quad\quad\quad + \, \mathbb{E}_{z∼p_z(...
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Are there any deep learning tools so that can be used to estimate long-tail distributions?

Are there any deep learning tools so that can be used to estimate long-tail distributions? update: my context is there are is there is a long-tail distribution (probability distribution) in my data of ...
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53 views

In continuous action spaces, how is the standard deviation, associated with Gaussian distribution from which actions are sampled, represented?

I have a question about implementing policy gradient methods for problems with continuous action spaces. Assume that actions are sampled from a diagonal Gaussian distribution with mean vector $\mu$ ...
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1answer
43 views

Intuitively, why can the training of a neural network be formulated as a probability estimation problem?

Neural network training problems are oftentimes formulated as probability estimation problems (such as autoregressive models). How does one intuitively understand this idea?
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23 views

Is the generator distribution in GAN's continuous or discrete?

I have some trouble with the probability densities described in the original paper. My question is based on Goodfellow's paper and tutorial, respectively: Generative Adversarial Networks and NIPS ...
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81 views

How does $\mathbb{E}$ suddenly change to $\mathbb{E}_{\pi'}$ in this equation?

In Sutton-Barto's book on page 63 (81 of the pdf): $$\mathbb{E}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_t=s,A_t=\pi'(s)] = \mathbb{E}_{\pi'}[R_{t+1} + \gamma v_\pi(S_{t+1}) \mid S_{t} = s]$$ How does $...
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What is the most efficient data type to store probabilities?

In ML we often have to store a huge amount of values ranging from 0 to 1, mostly being probabilities. The most common data structure to do so seems to be a floating point? Indeed, the range of ...
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61 views

Can a variational auto-encoder learn images composed of random noise at each pixel (each drawn from the same distribution)?

Can a variational auto-encoder (VAE) learn images whose pixels have been generated from a Gaussian distribution (e.g. $N(0, 1)$), i.e. each pixel is a sample from $N(0, 1)$? My gut feeling says no, ...
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38 views

What is the difference between model and data distributions?

Is there any difference between the model distribution and data distribution, or are they the same?
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54 views

What does a joint probability density function have to do with Stochastic Optimal Control and Reinforcement Learning?

I stumbled upon a job offer from a company that was looking for someone who was good with Reinforcement Learning (applied to finance) and something in their offer caught my eye. It goes something like ...
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70 views

How can I use the success and failure data to estimate parameters of a Dirichlet distribution?

I have used Beta function to estimate the performance of the agent. I have failure and success data of the task that runs on the agent. The parameter $\alpha$ is a number of successful tasks, while $\...
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1answer
103 views

Why do we regularize the variational autoencoder with a normal distribution?

When we define the loss function of a variational autoencoder (VAE), we add the Kullback-Leibler divergence between the sample taken according to a normal distribution of parameters: $$ N(\mu,\sigma)...
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89 views

Generalization error: Inputs drawn from distributions

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.2 Capacity, Overfitting and Underfitting, the authors say the following: Typically, when training a machine ...
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1answer
43 views

Solving the supervised learning problem of learning $p(y \vert \mathbf{x})$ by using traditional unsupervised technologies to learn $p(\mathbf{x}, y)$

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.1.2 The Performance Measure, $P$, the authors say the following: Unsupervised learning and supervised learning ...
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106 views

Many of the best probabilistic models represent probability distributions only implicitly

I am currently studying Deep Learning by Goodfellow, Bengio, and Courville. In chapter 5.1.2 The Performance Measure, P, the authors say the following: The choice of performance measure may seem ...
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1answer
51 views

Why does the machine learning algorithm need to learn a set of functions in the case of missing data?

I am currently studying the textbook Deep Learning by Goodfellow, Bengio, and Courville. Chapter 5.1 Learning Algorithms says the following: Classification with missing inputs: Classification ...
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59 views

Formulation of a Markov Decision Process Problem

Given a list of $N$ questions. If question $i$ is answered correctly (given probability $p_i$), we receive reward $R_i$; if not the quiz terminates. Find the optimal order of questions to maximize ...
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20 views

Why we multiply probabilities with support to obtain Q-values in Distributional C51 algorithm?

In 'Deep Reinforcement Learning Hands-On' book and chapter about Distributional C51 algorithm I'm reading, that to obtain Q-values from the distribution I need to calculate the weighted sum of the ...
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43 views

Why am I getting the logarithm of the probability bigger than zero when using Neural Spline Flows?

I am using a normalizing flow (Neural Spline Flows) to approximate a probability. After some training, the average loss is around 0.5 (so the logarithm of the probability = -0.5). However, when I am ...
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23 views

Deciding std. deviation for policy network output?

When I try to fit a Normal Distribution to the output of a policy network, for a continuous action space problem, what should be its standard deviation? mean for the distribution will directly be the ...
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350 views

What is probability distribution in machine learning?

If we were learning or working in machine learning field then we frequently come across this term probability distribution. I know what probability, conditional probability and probability ...
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1answer
65 views

How does maximum approximation of the posterior choose a distribution?

I was learning about the maximum a posteriori probability (MAP) estimation for machine learning and I found a nice short video that essentially explained it as finding a distribution and tweaking the ...
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115 views

Why is Jensen-Shannon divergence preferred over Kullback-Leibler divergence in measuring the performance of a generative network?

I have read articles on how Jensen-Shannon divergence is preferred over Kullback-Leibler in measuring how good a distribution mapping is learned in a generative network because of the fact that JS-...
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1answer
71 views

In deep learning, do we learn a continuous distribution based on the training dataset?

At least at some level, maybe not end-to-end always, but deep learning always learns a function, essentially a mapping from a domain to a range. The domain and range, at least in most cases, would be ...
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18 views

Probabilistic classification - normalize results

I have a probabilistic classifier that produces a distribution over my 3 classes - C1, C2, C3. I want to compare some new points I'm classifying to each other, to see which one is the best fit for a ...
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71 views

Why is the entire area of a join probability distribution considered when it comes to calculating misclassification?

In the image given below, I do not understand a few things 1) Why is an entire area colored to signify misclassification? For the given decision boundary, only the points between $x_0$ and the ...
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1answer
119 views

What to do when PDFs are not Gaussian/Normal in Naive Bayes Classifier

While analyzing the data for a given problem set, I came across a few distributions which are not Gaussian in nature. They are not even uniform or Gamma distributions(so that I can write a function, ...
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1answer
41 views

Why is the expectation calculated over finite number of points drawn from a probability distribution?

This is from the book Pattern Recognition by Bishop. Why is expectation here a simple average? Why is $f(x)$ not being multiplied by $p(x)$?
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43 views

Which loss functions for transforming a density function to another density function?

I am looking at a problem which can be distilled as follows: I have a phenomenon which can be modeled as a probability density function which is "messy" in that it sums to unity over its support but ...
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1answer
71 views

How are the parameters of the Bernoulli distribution learned?

In the paper Deconstructing Lottery Tickets: Zeros, Signs, and the Supermask, they learn a mask for the network by setting up the mask parameters as $M_i = Bern(\sigma(v_i))$. Where $M$ is the ...
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65 views

Unit integral condition on the output layer

I want to train a neural network on some input data from a probability distribution (say a Gaussian). The loss function would normally be $-\sum\log(f(x_i))$, where the sum is over the whole data (or ...
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1answer
191 views

Why can we approximate the joint probability distribution using the output vector of an LSTM?

In the paper, Contextual String Embeddings for Sequence Labeling, the authors state that \begin{equation} P(x_{0:T}) = \prod_{t=0}^T P(x_t|x_{0:t-1}) \end{equation} They also state that, in the LSTM ...
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1answer
81 views

Standard deviation of the total input to a neuron

Raul Rojas' Neural Networks A Systematic Introduction, section 8.2.1 calculates the standard deviation of the output of a hidden neuron. From: $$ \sigma^2 = \sum^n_{i=0}E[w_i^2]E[x_i^2] $$ When I ...
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67 views

Binary vector expected value

Raul Rojas' Neural Networks A Systematic Introduction, section 8.2.1 calculates the variance of the output of a hidden neuron. Raul Rojas says that "for binary vectors we have $E[x_i^2] = \frac{1}{3}$...
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464 views

What loss function to use when labels are probabilities?

What loss function is most appropriate when training a model with target values that are probabilities? For example, I have a 3-output model. I want to train it with a feature vector $x=[x_1, x_2, \...
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1answer
45 views

Can the normalization factor for the belief state update be zero?

In order to update the belief state in a POMDP, the following formula is used: $$b'(s')=\frac{O(a, s', z) \sum_{s\in S} b(s)T(s, a, s')}{\mathbb{P}(z \mid b, a)}$$ where $s$ is a specific state in ...
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59 views

Can we derive the distribution of a random variable based on a dependent random variable's distribution?

In the diagram below, there are three variables: X3 is a function of (depends on) X1 and X2, ...
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53 views

What kind of distributions can be used to model discrete latent variables?

If we take the vanilla variational auto-encoder (VAE), we $p(z)$ is a Gaussian distribution with zero mean and unit variance and we approximate $p(z|x) \approx q(z|x)$ to be a Gaussian distribution as ...