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 probability, and evaluates the evidence in favour of a hypothesis by combining the prior with the likelihood function of the observed data.
Questions tagged [bayesian]
2030 questions
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Posterior of normal distribution with $\mu=1$ and prior = 1
Given data, $p(y_1,...,y_n|\theta)$ normally distributed with mean, $\mu=1$ and unknown variance, $\sigma^2$ and the prior$$\theta \sim Beta \ (1,1) \implies p(\theta)=1$$ Find the posterior distribution, $p(\theta|y_1,...,y_n)$.
I tried and got…
sucksatmath
- 877
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Why can prior be swamped? (given a data set, any prior will go to the same posterior)
Is there any mathematical proof for that?
LSZ
- 179
2
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1 answer
What does Bayesian Method estimate?
I am not quite sure what Bayesian method want to infer from the data? Frequentists assume the parameters generated the data are fixed constants so they can actually estimate those constants.
Bayesians assume the parameters are random, with their…
bankrip
- 566
1
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multiplication of 2 PDFs
If I multiply the two PDFs, does the variance of the result PDF becomes narrower than the two PDFs always? In other words, if I multiply likelihood and prior to get the posterior, is the variance of the posterior narrower than that of the likelihood…
moon
- 7
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bayesian gauss prior prove
given zero mean Gaussian prior as
$\beta~ \sim(0,\Sigma p)$
inference is given by
$\log p(\beta \mid y, X)=\log p(y \mid X, \beta)+\log p(\beta)-\log (y\mid X)$
I can't understand how to get the following lines? can anyone prove it?
$-\log p(\beta…
Grace Cheng
- 31
1
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1 answer
Replace likelihood of multiple gaussian observations with likelihood of mean?
Suppose I have N independent observations from a Gaussian data generation process with know variance $\sigma$. I would assume that the Bayesian likelihood function can be written as:
$$
p(y | \mu, \sigma) = \prod_{i = 1}^N p(y_i| \mu, \sigma) =…
Willem
- 133
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Clarification on Wikipedia's entry on Variational Bayesian methods
Edit: Should I post this on Cross Validated instead?
On the Wikipedia page for Variational Bayesian methods, it is stated that
In variational inference, the posterior distribution over a set of
unobserved variables $Z = \{Z_{1}\dots Z_{n}\}$ given…
econ86
- 297
1
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1 answer
Calculate talent in Bayesian Resume Rating
Bayesian Resume Rating for sports math is explained in PDF https://www.jellyjuke.com/uploads/5/8/0/2/58022979/mathematical_explanation_of_the_bayesian_resume_rating_10-23-18.pdf
The formula used to calculate talent is:
However I am confused with…
NeDark
- 117
1
vote
1 answer
Help me to understand the part of the derivation of posterior distribution hyperparameters
I've looked on how to find hyper parameters of posterior distribution for normal distribution likelihood with unknown mean and precision.
Here is a derivation described https://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf
Im trying to understand how…
1
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1 answer
Queries on the Bayesian method
Currently I am working on a bayesian model framework and have questions related to the philosophy of using such techniques of modeling.
1.
How do I know that the prior which I have captured from the experts is valid. There are parameters of the…
Bik
- 39
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Bayes Theorem Calculating the Sensitivity of a disease test
A classic example often used for explaining Bayes theorem is through the use of the disease sensitivity example.
p(A = 1) = Have disease = 0.01
p(A = 0) = Not have disease = .99
p(T = 1 | A = 1) = Have disease given test positive = .95
p(T = 0 | A =…
Hojo.Timberwolf
- 111
1
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Difficulties in implementation of Bayesian information criterion (BIC) model selection criterion
I was reading some papers related to Bayesian model selection.
The Bayesian information criterion (BIC) model selection criterion is given by
$$\text{BIC}=-2\log f({\bf y}|\hat{\theta})+p\log n$$
where $p$ is the dimension of the model parameter…
Matata
- 2,088
1
vote
1 answer
Bayesian normal posterior in linear regression
I'm trying to show the following:
Assume that $Y_i \sim \mathcal{N}(\beta x_i, 1)$, $i=1, \dots, n$ are independent random variables.
Concretely, we consider a regression model $Y_i = \beta x_i + \varepsilon_i$,
$\varepsilon_i \sim \mathcal{N}(0,…
Lundborg
- 1,646
1
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0 answers
Question about parameter posterior in bayesian linear regression
At page 4 of cs229 section note (http://cs229.stanford.edu/section/cs229-gaussian_processes.pdf), I don't understand eq(2)
In this equation,
how do the $\theta$ is differentiated ? I think $\theta$ is just like that
$
\theta =
…
Trainer99
- 43
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1 answer
Find an expression for the posterior distribution of the change-point for this model
The simple change-point problem can be described as follows. Here it is assumed that both $p_1(y)$ and $p_2(y)$ are known completely.
$y_1,...,y_\tau|\tau $ are iid with distribution $p_1(y)$ and $y_{\tau + 1},...,y_n|\tau$ are iid as $p_2(y)$ and…
Rubicon
- 616