# Tagged Questions

The approach and interpretation of probability associated with Bayes theorem; usually used as opposed to the frequentist approach. It can be seen as an extension of logic that enables reasoning with propositions whose truth or falsity is uncertain. A Bayesian probabilist starts with some prior ...

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### Bayes Theorem Example in Nate Silver's The Signal and the Noise

In his book The Signal and the Noise, Nate Silver presents this example application of Bayes's Theorem on pp. 247-248: Consider a somber example: the September 11 attacks. Most of us would have ...
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### Is Entropy = Information circular or trivial?

I have seen several "maximum entropy distributions" used in the mathematical and statistical literature, often with the justification that they are "minimally informed" beyond the assumptions and data ...
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### Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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### What is the most general formalism for machine learning?

Most of the literature I can find in the field of machine learning is extremely practical, listing many techniques you can use like neural networks, SVMs, random forests, and so on. There are lots of ...
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### Bartlett's paradox in Bayesian evidence

I've come across Bartlett's "paradox" (not to be confused with Lindley's paradox, also known as the Lindley-Bartlett paradox) in Bayesian statistics. The paradox originates from Bartlett's 1957 paper, ...
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### hint with Bayes rule problem

The pirate Captain Queequeg has a lazy crew and suspects they are planning to stage a mutiny. Captain Queequeg's solution is to have every member of the crew roll Queequeg's lucky die. If the roll is ...
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### Why is $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$?

Could anyone derive or explain why the formula $P(X,Y|Z)=P(Y|X,Z)P(X|Z)$ is true? I understand conditional probability definition, but this formula confuses me and makes my head hurt x) Here's ...
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### Bayes, two tests in a row

I came up with a standard Bayesian example as to point out my confusion. There is an epidemic. A person has a probability $\frac{1}{100}$ to have the disease. The authorities decide to test the ...
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### Why would I use Bayes' Theorem if I can directly compute the posterior probability?

I fully understand the mechanics of Bayes' Theorem. However, I am wondering when do I need to use it? If I am able to compute the posterior probability directly from measured data, why would I need to ...
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### Bayes factor and Posterior odds

Consider the following posterior odds \begin{equation*} \frac{P(H|D_1,D_2)}{P(\overline{H}|D_1,D_2)}=\frac{P(D_2|H,D_1)\times P(D_1|H)P(H)}{P(D_2|\overline{H},D_1)\times P(D_1|\overline{H})P(\overline{...
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### A house is guarded by two alarms

I am trying to wrap my head around the following problem A house is guarded by two alarms. If Alarm 1 fires, p(theft) = 80% If Alarm 2 fires, p(theft) = 70% If both alarms fire at the same time, ...
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### Differentiating the posterior distribution function

I am learning about Bayesian statistics and I'm currently doing loss functions. Let $f(\theta | \mathbf{x} )$ be a posterior pdf . Let $F(\theta | \mathbf{x} )$ be the associated distribution ...
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### Jaynes' taxicab problem

I am currently reading Jaynes' Probability Theory, The Logic of Science and am still trying to absorb everything. On page 190, he poses the following intriguing question, paraphrased here. Suppose ...
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### What is the extension of Bayesian Network into cyclic graph?

The wikipage of Bayesian Network says "Formally, Bayesian networks are directed acyclic graphs whose nodes represent random variables in the Bayesian sense" But in the model I need to build, cyclic ...
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### Explanation of Aumann's “agreeing to disagree” in modern notation

I'm attempting to understand Aumann's classic 1976 paper Agreeing to Disagree, which claims, under certain assumptions, that if two Bayesian agents share knowledge of each others' posteriors then they ...
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### How does the posterior of a dirac prior look like?

Edit for the Moderators: Should this question migrate to stats.stackexchange? I have a very basic question concerning updating from a prior to a posterior in bayesian statistics. Setting: I ...
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### Bayesian versus Classical (frequentist) Statistics

Very often in text-books the comparison of Bayesian vs. Classical Statistics are presented upfront in a very abstract way. For example, in the current book I'm studying there's the following ...
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### What am I doing wrong in calculating Fisher Information of Triangular Distribution?

I am trying to find Jeffrey's prior for the Triangular distribution which has the following probability density function: f(x\mid \theta) = \begin{cases} \dfrac{2x}{\theta} & : x \...
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### Not sure how to solve Bayesian parameter learning problem

I could use some help solving a problem about a Dirichlet prior. We have a multinomial distribution over an alphabet of 27 symbols parameterized by $\mathbf{\theta}=(\theta_1, ..., \theta_{27})$. We ...
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### How to do Bayesian updating on biased information?

You have a coin that you can flip, but you can't see. It's a weighted $3$-sided coin taken (uniformly) randomly from some small known collection of $100$ weighted coins. However, we don't know how ...
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### Bayesian Inference in Measure Theory

What's the deal. How does this work, or can you point me to some references? I tried $\mu(A|B) = \mu(A \cap B) / \mu(B)$ and got stuck on $\mu(B) = 0$. Edit: Sorry for being lazy. My background is ...
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### How do I calculate the aposteriori probability distribution for someone's answer to a poll being an approval?

Imagine I'm polling a random sample from the population and it asks them if they approve of the President or not. I also ask them some categorical demographic questions (age-bracket, race, gender, ...
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### Can every factorization be represented by a Bayesian network?

A Bayesian network is defined as a directed acyclic graph with a set of random variables as its nodes, and it satisfies two axioms, 1) Root nodes (nodes without parents) are independent. 2) Given ...
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### Calculate Conditional Probability for program that does not crash

I got these 2 questions in exam, but unfortunately i failed to solved these. 1) you want to buy a computer. The probability that you can run the probabilistic program $X$ on it is $97$% ...
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### Determine how likely it is that a set of boolean data is produced by a distribution

Suppose we have a collection of independent Boolean random variables $X_i$ and $Y_i$ (for $1 \le i \le N$), and are told $p_i = P(X_i = 1)$ for all $i$. We are now given a set of values $x_i$ that was ...
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### Difference between Frequentist and Bayesian approach to Statistics

What is the difference between the Frequentist vs. the Bayesian approach to Statistics? Would someone be so kind to come up with a simple example that shows how the approaches and possibly the ...
I'm trying to learn Bayes's formula, and am coming up with some poker problems to learn this. My problem is as following: given a $H4,H5$ ($4$ of hearts, $5$ of hearts) hand, what are the odds that I'...