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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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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4k views

Coin toss - probability of a tail known that one is heads

A friend of mine tossed a fair coin twice. Suppose I ask him whether he got a head in the two tosses, and he says yes. What is the probability that one toss is tail? Now suppose instead that I happen ...
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241 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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Bayes rule with multiple conditions

I am wondering how I would apply Bayes rule to expand an expression with multiple variables on either side of the conditioning bar. In another forum post, for example, I read that you could expand ...
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Is my interpretation of Bayesian probability and inference correct?

I have the following interpretation of the Bayesian probability and inference (without referring to Measure Theory, I am still at the very beginning of learning it): Let's say we have five random ...
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34 views

Finding the marginal posterior distribution of future prediction, $y_{n+1}$

Assume the following bivariate regression model: $y_i = \beta x_i + u_i$ where $u_i$ is i.i.d $N(0, \sigma^2 = 9)$ for $i = 1, 2, ..., n$. Assume a noninformative prior of the form: $p(\beta) ...
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70 views

Kolmogorov's paper defining Bayesian sufficiency

I'm looking for a translation to either English, French or German of Kolmogorov's Russian paper Kolmogorov, A. (1942). Sur l’estimation statistique des paramètres de la loi de Gauss. Bull. Acad. Sci. ...
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question related to Bayes' rule and Bays' risk.

Let $X_1, X_2, X_3, \ldots, X_n$ be a random sample for $N(e,1)$. Let the prior p.d.f. of $e$ be $N(0,\sigma^2)$ under the square error loss function $L(e,d)={(d-e)}^2$. Find the Bayes' decision rule ...
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Separate expression $c(x + y)^2 e^{yz}$

Considering the following expression, with $x, y, z, c \in \mathbb{R}$, is such expression separable into $f(x, y, z) = g(x, z)h(y, z)$? $c(x + y)^2 e^{yz}$ If not, why isn't it separable? Note: ...
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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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Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
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Finding Probability of P(S|W) at Bayesian Network of Rain Problem

I am studying Bayesian Networks. Given that variables: $W$: Wet grass $R$: Rain $S$: Sprinkler I know the probabilities of: $P(C)$ $P(S | C)$ $P(S | !C)$ $P(R | C)$ $P(R | !C)$ $P(W | R,S)$ $P(W | ...
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1answer
173 views

A confusing excersice about Bayes' rule

The following is from a textbook one bayesian stats. that I can't understand some deduction. It is relevant about multiple parameters to be estimated. The jth observation in the ith group is denoted ...
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53 views

Monty Hall Problem Solve Using Detailed Algebra

I have been searching the monty hall problem for two days now and I generally understand it but I am having a very hard time solving the monty hall problem using Bayes's theory. I do not know what ...
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1answer
68 views

combining conditional probabilities

I've come across a physics paper in which pdf $$ p(a|b) $$ is desired, but only $$ p(a|c)\\ p(c|b) $$ are known. It is claimed that $$ p(a|b)=\int p(a|c)p(c|b) dc. $$ Is this correct wlog? I can't ...
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93 views

Two definitions of Bayes Sufficiency

"Bayes Sufficiency" is defined in two ways. Are they equivalent? Setting A statistical experiment $S$ is a triplet $\left(\left(\Theta,\mathcal{F}\right),\left(\Omega,\mathcal{A}\right),P\right)$, ...
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29 views

Coin tossing - Two tosses, one is a head, probability other is a tail? [duplicate]

A friend of mine tossed a fair coin twice. Suppose instead that I happen to see the result of one of his tosses, and it is a head. What is the probability that the other toss is tail?
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107 views

Does it become more likely that ZFC is consistent, the more time we explore it without finding a contradiction?

Intuitively, the more time we spend exploring ZFC without finding a contradiction, the higher the (subjective) probability that ZFC is consistent. Is this intuition sound? If not, why not?
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How to prove if P(A|B)>P(A) then P(B|A)>P(B) [closed]

How to prove that If P(A|B)>P(A) then P(B|A)>P(B)