10 votes
Accepted

What loss function to use when labels are probabilities?

Actually, the cross-entropy loss function would be appropriate here, since it measures the "distance" between a distribution $q$ and the "true" distribution $p$. You are right, ...
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6 votes

How can a probability density value be used for the likelihood calculation?

The probability density is used to 'measure how good' the parameters are because it is a natural way of quantifying if these parameters are good for the observed data. Also, as the notation often ...
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4 votes
Accepted

Are the authors of the VAE paper writing the PDFs as a function of the random variables?

When it comes to notation/terminology, often, people in machine learning are (a bit?) sloppy, which causes a lot of confusion, especially for newcomers to the field or people not very math-savvy. I ...
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  • 33k
3 votes

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

I think that the normalisation factor is assumed to be non-zero. So, in practice, I guess, you must eventually check that $P(z \mid b, a)$ is non-zero (even though, I guess, it will likely never be ...
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  • 33k
3 votes
Accepted

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

Also, in general, in the conditional expectation, which distribution do we compute the expectation with respect to? From what I have seen, in $\mathbb{E}[X|Y]$, we always calculate the expected value ...
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3 votes

What is the difference between model and data distributions?

Yes. In Machine Learning we consider that the samples in your training set are sampled from an underlying distribution called the data generating distribution. Generative models classify the samples ...
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  • 3,083
3 votes
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In deep learning, do we learn a continuous distribution based on the training dataset?

Well, there are some questions here... Does it (Deep Learning) try to learn a continuous distribution based on the training-set and its corresponding mappings, and map unseen examples from this ...
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3 votes
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What is a probability distribution in machine learning?

Random variables You do not necessarily need to understand the concept of a random variable (r.v.) to understand the concept of a probability distribution, but the concept of a random variable is ...
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  • 33k
3 votes

How can we "draw i.i.d" from any probability distribution?

As far as I know, it doesn't make sense to say that a probability distribution is i.i.d., as you're saying. The property i.i.d. is a property of a sequence of random variables. In your case, the ...
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  • 33k
2 votes

What is a probability distribution in machine learning?

A probability distribution in ML is the same as a probability distribution elsewhere. A probability distribution (or probability function, or probability mass function, or probability density ...
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2 votes
Accepted

How does maximum approximation of the posterior choose a distribution?

Introduction: MAP finds a point estimate! As opposed to your apparently current belief, in maximum a posteriori (MAP) estimation, you are looking for a point estimate (a number or vector) rather than ...
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  • 33k
2 votes
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Why is Jensen-Shannon divergence preferred over Kullback-Leibler divergence in measuring the performance of a generative network?

Lets start with question 1) how does JS-divergence handles zeros? by definition: \begin{align} D_{JS}(p||q) &= \frac{1}{2}[D_{KL}(p||\frac{p+q}{2}) + D_{KL}(q||\frac{p+q}{2})] \\ &= \frac{...
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  • 2,229
2 votes
Accepted

What does "the expectation is taken across different possible inputs, drawn from the distribution of inputs we expect the system to encounter" mean?

The language used here is confusing me, because it is discussing a "distribution", as in a "probability distribution", but then refers to inputs, which are data gathered from outside of any ...
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  • 9,316
2 votes
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Solving the supervised learning problem of learning $p(y \vert \mathbf{x})$ by using traditional unsupervised technologies to learn $p(\mathbf{x}, y)$

This is the definition of conditional probability + Total probability decomposition formula: $p(y|x) = \frac{p(y,x}{p(x)} = \frac{p(x,y)}{\sum_{y'}p(x,y')}$. The idea is to use some unsupervised ...
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  • 371
2 votes
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Many of the best probabilistic models represent probability distributions only implicitly

The probabilistic models that represent distributions implicitly are, for example, the GANs. (Goodfellow is one of the authors of the original GAN model). In the paper Variational Inference using ...
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  • 33k
2 votes

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

@The Pointer the $2^n$ came from the question: How many function do we need to have if each of the $n$ inputs can be missing? example: $f_1(\text{missing}, x_2, x_3, \dots, x_n)$ for $x_1$ missing $...
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2 votes
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What is emperical distribution in MLE?

The idea behind this kind of reasoning is that there is a "true" distribution (unknown to us, mere mortals) and that the data is generated following this distribution. But what we don't ...
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  • 278
2 votes

Why is KL divergence used so often in Machine Learning?

In ML we always deal with unknown probability distributions from which the data comes. The most common way to calculate the distance between real and model distribution is $KL$ divergence. Why ...
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2 votes

Why is KL divergence used so often in Machine Learning?

This question is very general in the sense that the reason may differ depending on the area of ML you are considering. Below are two different areas of ML where the KL-divergence is a natural ...
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  • 196
2 votes

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

To add to nbro's answer, I'd say also that much of the time the distance measure isn't simply a design decision, rather it comes up naturally from the model of the problem. For instance, minimizing ...
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  • 961
2 votes
Accepted

Binary vector expected value

Some lines above the author says By the law of large numbers we can also assume that the total input to the node has a Gaussian distribution hence we can assume $X \sim \mathcal{N}(0,1)$ with ...
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2 votes
Accepted

Is it abuse of notation to use tilde operator in this context?

The notation $p(x)$ is widely used in machine learning (e.g. here) and even statistics (e.g. here). People often use $p(x)$ to refer to a probability distribution (either pmf, pdf, or cdf) rather than ...
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2 votes

Is knowing underlying probability distribution mandatory for deciding iid property of random variables?

The point is even you know the distribution, sometimes you can't prove that the sampled data is i.i.d. or not! (more details in https://stats.stackexchange.com/q/130381/144441). Hence, without ...
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  • 1,663
2 votes

Are the authors of the VAE paper writing the PDFs as a function of the random variables?

Machine learning papers are often somewhat confused about the distinction between a distribution and its probability density. I would rewrite this The process consists of two steps: (1) a value $\...
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  • 746
1 vote
Accepted

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

VAE's try and model the distribution of your data. So it's not going to learn " images composed of random noise at each pixel" per se (though, if overfitting, it could remember them). But it would be ...
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  • 246
1 vote
Accepted

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

I did not read those two specified linked/cited papers and I am not currently familiar with the total variation distance, but I think I can answer some of your questions, given that I am reasonably ...
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  • 33k
1 vote
Accepted

How to calculate v min and v max for C51 DQN

If you're using a discount factor less than 1, you should be able to compute a maximum return (likewise, a minimum return) based on the max (min) reward you can earn at each timestep. However, this ...
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  • 961
1 vote
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Intuitively, why can the training of a neural network be formulated as a probability estimation problem?

Consider the case of binary classification, i.e. you want to classify each input $x$ into one of two classes: $y_1$ or $y_2$. For example, in the context of object classification, $y_1$ could be "...
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1 vote

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

Extracting a joint PDF just means that you create a model that models the behavior of several variables combined instead of in isolation. If these variables aren't independent and your loss ...
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