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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 ...


7

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

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

In the first sentence of the paper, Nilsson states that [s]everal artificial intelligence (AI) applications require the ability to reason with uncertain information. Nothing (well, almost nothing) is ever just true or false, and binary logic is not enough to model a complex world. So we need more powerful means of describing logical relationships that go ...


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

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] = \...


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 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

Welcome to AI.SE @rudreshdwivedi! This is a great question, and I hope to see many more like it. Demster-Shafer Theory and Bayesian Networks were both techniques that rose to prominence within AI in the 1970's and 1980's, as AI started to seriously grapple with uncertainty in the world, and move beyond the sterilized environments that most early systems ...


2

Position Detection In a traditional data acquisition and control scenario, with some assumptions, the relation between sensors signals $s_i$, emitters drive $\epsilon_j$, distances $x_{ij}$, and calibration factors is modelled as follows. $$ \forall \, (i, j) \text{,} \quad \frac {s_i} {v_i} = \frac {\epsilon_j} {v_j \, x_{ij}^2} $$ The assumptions include ...


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

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

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 ...


2

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 ...


2

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 function) is any function that accepts as input elements of some specific set $x \in X$, and produces as output, real-valued numbers between 0 and 1 (inclusive), such ...


2

Bayes Error Rate For the general case of K different classes, the probability of classifing x instance correctly is: \begin{equation} \label{eq1} \begin{split} P(correct) & = \sum_{i=1}^{K} p(x \in H_i, C_i) \\ & = \sum_{i=1}^{K} \int_{x \in H_i} p(x,C_i) \, dx\\ & = \sum_{i=1}^{K} \int_{x \in H_i} P(C_i|x)p(x)\,dx \\ \end{split} \end{equation} ...


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

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

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

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 strictly connected to the concept of a probability distribution (given that each random variable has an associated probability distribution), so, before proceeding,...


1

The main difference between a variational auto-encoder (VAE) and an auto-encoder is that the VAE is a generative and statistical model while an auto-encoder (AE) is just a data compressor and decompressor (it is just a function approximator), so an AE is not a statistical model. They both have an encoder and a decoder and they both convert the inputs to a ...


1

That really depends on the nature of the problem. I will assume that you asked this question on the AI stackexchange because you thought that there was a type of AI that would solve the problem. By giving a certain chance that something happens, you also imply that if the number of samples increases, the relative frequency of an outcome will converge to the ...


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

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

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

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). ...


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

Computational Learning Theory gives us an interesting framework to understand what statistical learning is doing. The gist of it is, we can model the process of statistical learning as one of formal deduction. The learning itself does not require a random element. This shouldn't be too surprising. Consider a classic decision tree learner like C4.5 or ID3: ...


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