# Questions tagged [bayesian]

Bayesian inference is a method of statistical inference that relies on treating the model parameters as random variables and applying Bayes' theorem to deduce subjective probability statements about the parameters or hypotheses, conditional on the observed dataset.

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### Calculate probability of alarm and the posterior probability of this alarm being false over different frequencies of output

I have the following information for an automatic detection system that output a warning when a signal is detected: Specificity: .99 (i.e. a false positive rate $FP = .01$) Sensitivity: .9 (i.e. a ...
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### updating dependent parameters using Bayesian method

I want to update multiple parameters using Bayesian with informative prior. I assume my parameters are independent-p(θ_1,θ_2)=p(θ_1)p(θ_2). After observing i.i.d observations, the posterior~ prior x ...
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### Ask for rationale of finding the corresponding prior from regularizer by taking exponential of negative regularizer

In equation (5.112) of textbook "Pattern Recognition and Machine Learning" by Christopher M. Bishop, the simple regularizer takes the form $\frac{\lambda}{2}{\bf w}^T{\bf w}$. The author ...
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### Estimating moments of censored data with multiple bounds

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations. Each time I draw a random sample, my data is censored, meaning ...
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### How does blocked Gibbs Sampling change the interpretation of the generative author-topic (LDA) model

The author topic model is a version of a Latent Dirichlet Allocation model which looks to estimate a set of author to topic, and topic to word distributions to model how authors combine to produce ...
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### How to check if variation of intercepts and slopes between random effect is significant (linear mixed model)?

I have conducted a linear mixed effect regression for the day of green-up in the Arctic. Regions are random effects, weather variables are kept as fixed effects. My data, after scaling to center, is ...
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### Bayesian Posterior distribution for binomial distribution with uniform prior

Suppose we have two independent binomial distribution given p, i.e. $X_1|p \sim Bin(n_1, p)$, $X_2|p \sim Bin(n_2, p)$. We also know the prior distribution for p is $p \sim U(0,1)$. Now I would like ...
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### How to maximize the ELBO in coordinate ascent variational inference

In the lecture by D.Blei: https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/variational-inference-i.pdf Variational inference is explained and he shows how to derive the optimal ...
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### Family of distributions dominated iff posterior dominated by $\sigma$ finite measure

I'm not sure how to prove the following: Let $S$ be a sample space, and $\Theta$ the space of parameters. Show that if $(S,\mathcal{S})$ and $(\Theta,\mathcal{B})$ are standard Borel spaces, then the ...
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### Does the following "approach" for integrating over the data space makes sense?

Suppose that we have the following posterior and prior distributions $p(\mu|x,m_{1}(x),s_{1}(x)) = Normal(\mu;m_{1}(x),s_{1}(x))$ and $p(\mu|m_{2},s_{2})$ The $m_{1}(x),s_{1}(x)$ indicate that the ...
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### Marginal Likelihood Computation for Bayesian Linear Model

Given a simple Bayesian linear model with $N$ observations $y = X\beta + \varepsilon \quad \quad \varepsilon \sim \mathcal{N}(0, \Sigma)$ with known error variance-covariance matrix $\Sigma$ and ...
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### Finding posterior with Ga prior, Exp likelihood

I'm using this 2017-2018 exam paper to pre-study for a module in Bayesian Statistics. It has a question, Assume that the waiting time, $t$, of a client in a bank can be modelled with an exponential ...
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### In what sense is "Bayesian cross validation" Bayesian?

In cross validation, we repeat training the model with resampled training data and measure average of the errors from the different resampled training samples. So cross validation is basically a ...
1 vote
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### Connection Between Bayesian Prior and Variable Selection in Lasso [duplicate]

I am interested in learning more about the Bayesian interpretation of the Lasso model. The Lasso model assumes a Laplace distribution of coefficients and the optimal coefficients maximize the ...
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### Applying Bayes Theorem to combine probability mass functions of time series changes

I apologize. I'm not well trained in formal statistics, so feel free to gently correct my terminology and methods. For a univariate continuous real-valued time series X, I've calculated probability ...
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### Is it possible to use a scaled version of a beta distribution to represent lifetime?

I'm a beginner in statistics, so I'm not sure if this has been asked before. I've looked, but I couldn't find an answer. So I'm trying to use Bayes' theorem to build a probability distribution ...
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### Frequentist method for random samples from unknown urn

Say you have two urns with a large number of red and blue marbles each and you know the proportion of red and blue marbles in each urn. Now we choose one urn at random (but don't know which) and ...
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### How to determine interventional distributions from observational data?

How do we compute/query interventional distributions from observational data (i.e. without knowledge of the causal graph such as a Structural Causal Model (SCM))?
1 vote
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### Difference between sequential and one-batch Bayesian update

I learnt that sequential Bayesian update and one batch (all at once)update will give the same result if the observations are i.i.d. I tried to test this theory using my model which contains 4 ...
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### Computing Joint Distribution from Marginal Distributions, and Vice Versa

I'm trying to learn bayesian networks by doing probability computations by hand. Given the probability distribution P(D),P(I),P(G|I,D), I want to compute ...
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### Updating prediction by combining confidence values of incoming events

I want to predict a binary variable $y$. I assume a prior probability of $p(y=1) = 0.5$. Now there is evidence from incoming independent “events” that should increase or decrease my overall prediction ...
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### Modelling probabilities of a sum of binomials with different probabilities and trials

I have the following example data, where each row is an independent observation: A B C Y 10 22 6 2 4 60 2 0 12 8 10 3 ... $A$, $B$, $C$ and $Y$ are all positive integers. The variables $A$, $B$ ...
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### How do I include 0-rated items while ranking items with variable number of ratings?

I have a list of items and ratings from 0 to 10, with decimal ratings so that possible ratings are 0, 0.5, 1, 1.5 ... 9.5, 10. I am using https://www.evanmiller.org/ranking-items-with-star-ratings....
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### Bayes rule with two X variables

I'm trying to work out the proper updating of two (independent) random variables ($\pi_a$, $\pi_b$), given an observation $y$. Each $\pi$ is Bernoulli, corresponding to a hidden state. ($\pi_a$, \$\...
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