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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98 views

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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470 views

Trying to understand the basics of bayesian inference

This paper gives a somewhat gentle introduction to Bayesian inference: http://www.miketipping.com/papers/met-mlbayes.pdf I got to section 2.3 without much problems but got stuck in understanding that ...
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214 views

Is there an introduction to probability and statistics that balances frequentist and bayesian views?

Perhaps, roughly, I might be described as advanced undergraduate regarding mathematics. However, I have not learned statistics and have only learned elementary probability. Does there exist a book or ...
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125 views

Is there any research field dedicated to estimating a “game” itself in game theory?

Game theory stuffs usually provide how a "game" works and then tries to figure out solutions - but I am wondering if there is any research field dedicated to estimating the full rules of a game. So ...
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54 views

What is an approach for optimizing the values of a matrix?

My apologies if I get some terminology wrong, I don't have a formal math background; half my problem is articulating what I'm trying to do and identifying the domain of math that deals with this kind ...
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1answer
290 views

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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1answer
122 views

Does Bayesian probability have a different interpretation of a random variable?

Bayesian probability interprets the meaning of the probability of a random variable as some degree of belief. But does this result in any difference in the interpretation of a random variable itself? ...
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79 views

Bayesian inference

I'm a bit confused with arranging the Bayes equation to update probability. Say, I have the following data: $P(\text{blue birds in the whole study area}) = 0.16$; $P(\text{all except blue colored ...
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2answers
454 views

Application of Bayes Theorem

I am reading Nate Silver's book "The Signal and the Noise" and have a question about Bayes Theorem. I've created my own example and am trying to wrap my mind around the conclusion. Let's say, before ...
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4answers
59 views

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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2answers
80 views

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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2answers
506 views

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 ...
3
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2answers
186 views

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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1answer
483 views

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 ...
3
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1answer
489 views

Bayesian posterior with truncated normal prior

Suppose we observe one draw from the random variable $X$, which is distributed with normal distribution $\mathcal{N}(\mu,\sigma^2)$. The variance $\sigma^2$ is known, $\mu$ isn't. We want to estimate ...
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1answer
116 views

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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1answer
115 views

What is the meaning of “mean-field”?

In lots of Bayesian papers, people use variational approximation. In lots of them they call it "mean-field variational approximation". Does anyone know what is the meaning of mean-field in this ...
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2answers
77 views

Conjugate priors make calculations easier but at what cost to the model?

As I understand, when we have a parametric pdf and need to estimate the parameter based on some observed fact, we tend to choose a conjugate prior of the pdf for the parameter. Because conjugate prior ...
3
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1answer
157 views

Maximum Entropy Distribution When Mean and Variance are Not Fixed with Positive Support

I know when the mean and variance of $\ln x$ are both fixed, then the maximum entropy probability distribution is lognormal. When the mean of a random variable is fixed the MEPD is the exponential ...
3
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1answer
58 views

Implied prior with relationship $y=\text{arccot}(x)$

I'm trying to solve an exercise, which I think I have almost managed to solve but not quite. Any help would be appreciated! So, what we have is a vector which we obtain by norming the vector ...
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0answers
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If $P(B\text{ }|\text{ }A)=1-\epsilon$ and $P(C\text{ }|\text{ }B)=1$ then $P(C\text{ }|\text{ }A)\geq 1-\epsilon$ [duplicate]

If $$1=P(C\text{ }|\text{ }B)=\frac{P(C\cap B)}{P(B)}$$ then we know that $P(C\cap B)=P(B)$. If $P(B\text{ }|\text{ }A)=1-\epsilon$ for $\epsilon\geq 0$ then $$P(A)=\frac{P(B\cap ...
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1answer
92 views

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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1answer
86 views

Bayesian formula for weather exercise

If it is nice weather on one day, the probability that it is going to be nice again the next is $13/15$. If it is raining on one day, the prob. that it is going to be raining again the next day is ...
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108 views

Questions on Bayesian analysis of an opinion poll (an example in a book)

I'm sorry in advance for rather long questions. This is an example in "Bayesian logical data analysis for physical sciences" by P. C. Gregory and I have some questions about the example. In a poll ...
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228 views

What is the probability of the box?

Your box of cereal may be a contest winner! It's rattling, which 100% of winning boxes do. Of course 1% of all boxes rattle and only one box in a million is a winner. What is the probability that your ...
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2answers
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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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2answers
129 views

Is politically incorrect conclusion more likely to be true by Bayesian Logic? [closed]

We got many beliefs. Some are hidden and some are repeated. False beliefs are repeated more because people like it. True beliefs are hidden if people do not like it. So for the same amount of ...
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1answer
78 views

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 ...
2
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1answer
71 views

What is the probability of two or more from n events occuring?

A number of independent events, say $A$, $B$, $\ldots\,$, $E$, can happen with associated probabilities $P(A)$, $P(B)$, $\ldots$ For each event that happens I have to pay £10. The likelihood I have ...
2
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1answer
246 views

I am confused about Bayes' rule in MCMC

Bayes' rule appears to bevery simple at first sight, but when studied deeply I find it is difficult and confusing, especially in MCMC applications when multiple parameters need to be estimated. For ...
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3answers
346 views

13 DOF Kalman filter

I'm trying to develop a system with the following characteristics: Inputs: 3-axis accelerometer [3 DOF] 3-axis gyroscope [3 DOF] GPS with three parameters (lat, lon, altitude) [3 DOF] Barometric ...
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3answers
102 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
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2answers
89 views

Beta distribution quesions

Just a simple beta distribution question just to be sure that I understand it. Say we do experiments, and we expect a proportion $\theta$ of people having a specific property (which means $\theta ...
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2answers
113 views

Comparing uniform priors

The background of the problem is this: Assume that we have a parameter vector $\Theta$ which satisfies $\Theta^\prime\Theta=1$. If we let this vector have the uniform prior, the density of the prior ...
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1answer
47 views

Bayesian Problem… I think

Let X be the number of coin tosses until heads is obtained. Without knowing that the coin is fair, I assume that the probability of heads is uniformly distributed. How would I find the distribution ...
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1answer
221 views

Bayes Estimator

Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf: $f_{X}(x|\alpha,\beta) = ...
2
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1answer
63 views

On Finding Means of Distributions

If I have a distribution which depends only on one variable, I usually find the mean by: (Continuous Case) $$\mu=\displaystyle \int xf_X(x).dx$$ What happens in the following cases: Conditional ...
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1answer
90 views

Conditional Probability Question

There are three events, $A$, $B$ and $D$. I know that $P(D)=0.2$, $P(A)=0.34$ and $P(B)=0.43$. I have calculated that $P(D\mid A)=0.5294$ and $P(D\mid B)=0.44186$. Now I need to calculate $P(D\mid ...
2
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1answer
52 views

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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2answers
68 views

Coin toss with unknown probability – Bayesian interpretation

I have observed a coin being tossed $n$ times. I do not know whether the coin is fair or not, but in every single toss I observed, the coin came up heads. What should my belief about $p$ (the ...
2
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1answer
111 views

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 ...
2
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2answers
153 views

Posterior Distribution of a Prior Variable

Let $X_{1},\dots,X_{n}$ be a random sample from an exponential distribution with density $f(x;\theta)=\theta e^{-\theta x}$, $x>0$ (having mean $1/\theta$). Assume a prior density for $\theta$ ...
2
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1answer
28 views

Determine which parameter has correlation with result and which is not

sorry for probably silly question, it's the first time when I need to do such work. I have large data set with regarding clicks on some element on web page. It contains some characteristics of such ...
2
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1answer
56 views

What's the posterior for mutivariate lognormal with covar known?

I know the univariate case but not the multivariate case. Suppose we have a multivariate lognormal dist: $$ \boldsymbol{X} \sim \text{lognormal }(\boldsymbol{\mu}, \boldsymbol{\Sigma}) $$ where ...
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1answer
19 views

Bayesian statistics, bivariate prior distribution

I've got a simple question buy I'm not sure how to solve it. It's a bit long. Suppose you've got $n$ iid random variables $X_1$, $\dots$, $X_n$ from the normal distribution with unknown mean $M$ and ...
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2answers
49 views

Bayesian learning for input “If A, then B.”

Can anyone point me to literature on Bayesian learning when the new information has the form “If A, then B”? I’m familiar with the rule that after one learns X, posterior probability P(Y) equals prior ...
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1answer
61 views

Certainty that one has found all of the socks in a pile

Suppose that I have a pile of $n$ socks, and, of these, $2k$ are "mine." Each of the socks that is mine has a mate (so that there are $k$ pairs of my socks) I know $n$, but not $k$. Assume that all ...
2
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1answer
310 views

Sleeping Beauty paradox - fair prior

I've been reading the Sleeping Beauty problem wiki. The contradicting answers, to me, appear to stem from frequentist and Bayesian interpretations: The "thirder" solution refers to a limit in an ...
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1answer
143 views

What's the difference between Maximum a posteriori and Bayes' rule?

What's the difference between Maximum a posteriori and Bayes' rule? They look similar, except that I do understand Bayes' rule and I don't understand MAP. The people I asked - who work in math and ...