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A Markov model includes the probability of transitioning to each state considering the current state. "Each state" may be just one point - whether it rained on specific day, for instance - or it might look like multiple things - like a pair of words. You've probably seen automatically generated weird text that almost makes sense, like Garkov (the output of a ...


8

Yes and no! There's no inherent reason that machine learning systems can't deal with extreme events. As a simple version, you can learn the parameters of a Weibull distribution, or another extreme value model, from data. The bigger issue is with known-unknowns vs. unknown-unknowns. If you know that rare events are possible (as with, say, earthquake ...


3

No, you can't extract any probability from a fuzzy membership grade. The uncertainty expressed by fuzzy logic is about partial truth, not about probability. $ \mu_S(x) = 0.9 $ doesn't mean that "$ x $ is tall" is true with a probability of 0.9, but that "$ x $ is tall" is 90% true (notice the difference in semantics). You have to think ...


3

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 over distribution $X$. No, for $\mathbb{E}[X|Y]$ we take expectation of $X$ with respect to the conditional distribution $X|Y$, i.e. $$\mathbb{E}[X|Y] = \...


3

This formulation/interpretation can indeed be confusing (or even misleading), as the output of a neural network is usually deterministic (i.e. given the same input $x$, the output is always the same, so there is no sampling), and there isn't really a probability distribution that models any uncertainty associated with the parameters of the network or the ...


3

The intuitive explanation is that there are many equally good "optimal" policies. This is mentioned at the end of the example problem description you posted. My gut says that the family of optimal policies would be any policy from the double/nothing family. So, for example, if you bet 25 on the first bet instead of 50, I think your overall chances of winning ...


3

The function $r(s,a,s')$ gives the expected reward in each scenario, but not the distribution of rewards that lead to values $r_{search}$ and $r_{wait}$ The text explains that reward is $+1$ for each can found, and that different distributions of numbers of cans are expected when waiting as opposed to searching. However, it does not give any description of ...


3

This equation and more information of it can be found in Expectation Maximization Wikipedia site and the explanation there was as follows (formula there in two parts): Some more explanation from same page: In statistics, an expectation–maximization (EM) algorithm is an iterative method to find maximum likelihood or maximum a posteriori (MAP) estimates of ...


2

I think Minsky deprecated the suggestion that probabilistic models could be surrogates for component models for intelligence that he suggested were grounded in principles and processes that interact (i.e. Society of Mind). But I don't believe he ever referred to probabilistic models as dead ends. All intelligence models must employ awareness of likelihoods ...


2

When considering effective approaches to AGI, one must extrapolate outwards to the types of modelling (and therefore inputs) that would be necessary to achieve any general utility. One consideration might be the fundamental "building blocks" of our physical world, and understanding the movements of these, can lead to accurate predictions of (all) occurrences....


2

(this was intended as a comment, but turned out long and longer) A couple of points to elaborate on Ben's answer: It is possible to generate different models (out of existing data!) and then look for the model that best fit new data (e.g. with knn). Example: States = {sleep, eat, walk, work} Model 1: Most probable sequence on weekdays, say: sleep → sleep ...


2

This means "Parameterized by". First, we all agree on the idea of conditional probabilities: $$P(X | Y) = P(X,Y) / P(Y)$$ That is, the probability that X happens given that we've seen Y happen, is the fraction of worlds in which Y happened that also contain X. This is uncontroversial. If you're a Bayesian, you might view parameters themselves as ...


2

At first, like Neil Slater says, I thought this could only be solved using the expected rewards instead of actual rewards, or else there wasn't enough information to solve it. But now I think there might be a way to solve this question. Here is my thinking on this problem (I would be curious for anyone's thoughts, as I am working through this book myself). ...


2

Another specific way to do this if one uses a neural network for this. Use a dropout a layer in your network and instead of scaling the activations at test time, one can sample the activations (just like in training-time) and predict multiple times for a given input, then look at distribution of your outputs. Intuitively this would add "probabilistic, ...


2

Welcome to AI.SE @vdbuss, and great first question! This point is touched on in Section 15.2.3 (page 576 in my copy), in the second paragraph, and there's a good exercise at the end of the chapter (15.4) that is designed to get you to think through exactly why these are different procedures. If you want to really absorb it, I suggest trying to work out that ...


2

Parzen was a statistician, who worked in spectral analysis and stochastic processes. I don't know if he invented them, but those windows and probability density esimation methods are named after him. See also his Wikipedia entry.


2

Although this question is slightly primarily opinion-based and too broad (and I will probably close it as such) and a good answer will necessarily depend on your background, I will list some of the main theoretical prerequisites that everyone should ideally be familiar with before diving into TensorFlow Probability (TFP). I am familiar with TFP, given that ...


1

I don't see where it's implied that G is a probability distribution. G is a function, whose output conditioned on one variable has a probability distribution, but it isn't one. z is random noise which is G's source of randomness. y is something that isn't random. We call G over and over with the same y and different random z's and look at the distribution of ...


1

Let $A$ and $B$ be two events. In general, the probability that either $A$ or $B$ occurs is defined as $$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$ If $A$ and $B$ are disjoint, i.e. they cannot happen at the same time, then $P(A \text{ and } B) = 0$, so the above formula becomes $$ P(A \text{ or } B) = P(A) + P(B) $$ If the probability of ...


1

I don't know if $N(X)$ has a name or has any applicability in AI, but I can comment on how this function varies as the $H(X)$ based on your equation $$N(X) = \dfrac{1}{2^{H(X)}}$$ which looks correct to me (just apply the $\log_2$ to both sides). In the case of a Bernoulli random variable (which is a categorical r.v. that can take 2 values, $0$ or $1$, which ...


1

Your call to model.predict() is returning the logits for softmax. This is useful for training purposes. To get probabilties, you need to apply softmax on the logits. import torch.nn.functional as F logits = model.predict() probabilities = F.softmax(logits, dim=-1) Now you can apply your threshold same as for the Keras model.


1

Figure 3 in the original WGAN paper is actually quite helpful to understand the difference between the score in WGAN and the probability in GAN (see screenshot below). The blue distribution are real samples, and the green one are fake samples. The Vanilla GAN trained in this example identifies the real samples as '100% real' (red curve) and the fake samples ...


1

You could maybe do something like this, it's a bit hackish \begin{equation} y = C_1\cdot 1 + C_2 \cdot 0.5 + C_3 \cdot 0 \end{equation} $y$ represents the output and its bounded $\in [0, 1]$. $C_i$ is probability for class $i$. This way when $C_1 \approx 1, C_2 \approx 0, C_3 \approx 0$ you have \begin{equation} y \approx 1\cdot 1 + 0.5 \cdot 0 + 0 \cdot 0 \...


1

You make a valid point, vanilla neural networks cannot give you more than a point estimate of class confidence. If one wanted to actually gain an idea of variance, you need a framework that allows such a mechanism. A popular methodology to this is Bayesian modeling. In other words given some data, $\Omega$, you want to create some form of descriminative ...


1

Predicting with confidence: the best machine learning idea you never heard of by Scott Locklin might provide you an idea. The name of this basket of ideas is “conformal prediction.”


1

Model input: 1 mean scaled input for each emitter 1 distance value for each distance Multiple input You mentioned there is noise. If the noise is constant, ie you test it in place A and the values returned are always the same, then it means training in different places. If you place it in a place and the first reading is different from the second reading....


1

In the announced problem, most of the transitions aren't possible, so most the terms of equations (3.3) and (3.4) from the book will end up being 0. In my understanding, $$ \begin{align} p(s'= high | s = high, a = search) &= \sum_{r \in \{0, -3, r_{search}, r_{wait}\}} p(s'=high, r | s = high, a = search) \\ &= p(s'=high, r =0 | s = high, a = ...


1

You are right, that pseudocode is not correct. In particular, the definition of $H_t^i$ in line $11$ should be changed; all the way on the right-hand side, it should have $3N - 3j$ columns of $0$s, rather than $3j$ columns of $0$s. With that change, every matrix $H_t^i$ will have the same number of columns: $$6 + 3j - 3 + 3N - 3j = 3 + 3N,$$ which ...


1

Seems like recurrent neural networks (RNN) should work for your use case. An excellent introduction is available at: The Unreasonable Effectiveness of Recurrent Neural Networks


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