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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Combining Dirichlet and Gamma-Normal distributions

I have a model that describes 2 dimensional data where each data points is define as d = [category, x]. The category dimension can take 3 different values with respective probability $p_1$, $p_2$ and $...
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How to write down a 3-level hierarchical model with a level-1 equation (reduced form)?

Suppose longitudinal data on $y_{rct}$: the average temperature of region $r$ in country $c$ at time $t$. You $y_{rct}$ fit a 3-level hierarchical linear model with country-year and country deviations ...
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Hierarchical Bayesian modeling with count data (PYMC): how to specify this model?

I'm completely new to Bayesian statistics and tried to get a grasp of the fundamentals for a specific case I'm working on. However, I feel like I've led myself down a blind alley and I'm still ...
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What parameters can be fixed in a hierarchical model when the outcome is standardized?

Suppose I have unstandardized data and estimate a 2-level Bayesian hierarchical model with varying intercepts and no covariates. The grand mean $\alpha_i$ and within-$i$ variance of the outcome $\...
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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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Why do Bayesians not model the input as a random variable in discrminative models?

From this question: Example of a Discriminative Bayesian Model There is this table comparing Frequentist/Bayesians and Discriminative/Generative models. $$\begin{array}{l|c|c} & \text{Frequentist}...
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The gradient vector in Hamiltonian Monte Carlo (leapfrog method)

Let $x_{t}, \omega_{t} \in \mathbb{R^{d}}$ The Hamiltonian Monte Carlo says this: Deterministic: it relies on the Hamiltonian dynamics so given an initial state, at any time $t$, specified by the ...
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How does maximizing ELBO in Bayesian neural networks give us the correct posterior predictive distribution?

In Bayesian/variational neural networks one often uses the Evidence Lower BOund (ELBO) as the objective function to optimize with respect to the model parameters. That is if $D=\{y_i,x_i\}_{1\dots n}$ ...
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Conditional Probability expression

I am trying to understand the next expression about a conditional probability: $P(D|T)=P(V|T)P(D|V)+(1-P(V|T))P(D|\neg V)$ It seems a variation of the Bayes theorem, but I can't see how this ...
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Incorporate partial information about Y into predictions

I have a linear regression model predicting exports of toys from the United States on an annual basis. This initial model is based on a few factors: toy companies' demand projections, toy production ...
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RetroBayesianism or? I have access post-hoc to "surprised\shocked" assessments only. What math to use? [closed]

This is a basic alternative"fringe" statistics question as to who has studied this under what name(s), or what you recommend as a personal\undocumented workaround or adaptation\combination ...
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Clarification on the connection between deep ensembles and bayesian neural networks

I'm not sure if I understand the relationship between deep ensembles and Bayesian interpretation well. can you tell me if i am right or wrong? Suppose I have an ensemble of L neural networks trained ...
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Interpreting posterior with Half-Normal shape

I am building a Marketing Mix Model in PyMC and am not sure how to interpret the posteriors, especially those with half-normal priors (sigma=1). I’ve chosen this prior because media could not have a ...
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Why is bambi returning multiple posterior mean predictions per (chain,draw)? [closed]

Let's say I want to calculate posterior for conversion rate per group. y denotes successes, n trials. ...
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Is it practical to derive the prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"?

Is it practical to derive the optimal prior distribution by dividing the posterior by the likelihood and multiplying by the "evidence"? Suppose you assume a probability distribution. You ...
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Reasoning about modelling uncertainty w.r.t input

I am trying to build up my reasoning about uncertainty modelling and ways of modelling it. What I am trying to essentially get at is how changes in input variables results in different posteriors(...
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Finding the scale of a $t$ distribution given information about CDFs

I'm using this 2017/18 exam paper to prepare for a course in Bayesian Statistics. It has a question, Assume $\{x_1,\dots,x_n\}$ is a random sample from a Gaussian distribution with mean $\mu$ and ...
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How do I multiply two discrete distributions together?

As part of my PhD research, I am simulating a system that comprises multiple sensors, and I want to perform sensor fusion. My sensors give me sets of measurements (values between 0 and 1 mm) that I ...
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Using the bootstrap estimates to compute the acceleration parameter for BCa bootstrap CI

I have noticed different software use different method of calculating the acceleration parameter, $a$, for computing the bias-corrected and accelerated (BCa) bootstrap confidence interval (CI). In ...
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How to interpret BayesFactor::contingencyTableBF BF results in R?

Say we build the following table: ...
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Using Bayes' Formula, how to obtain class prior and other components from the data

I am trying to solve the following problem using Bayes' formula 'Given visitors have visited a page, what is the probability they will convert?' I have a website with which people interact. It has 50 ...
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Bayesian Model selection vs Model comparison

Is anyone aware of any articles or book chapters about the distinction between model selection and model comparison in bayesian multilevel modeling? I am fitting several competing growth models using ...
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What is the distinction between models with multivariate priors and (multiplicative) interaction terms?

In some models, groups of parameters are assigned multivariate priors with a covariance matrix where (at least some) elements are estimated. The motivation is that the parameters depend on each other. ...
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When to specify multivariate versus univariate priors on parameters?

Suppose a linear regression model: $$y \sim Normal(\beta X, \sigma)$$ For our purposes, assume $y$ is a univariate outcome and $X$ is a design matrix containing an intercept and one additional ...
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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 ...
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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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Can I calculate precision from prevalence and misclassification rate?

If I know $P(D+)=P(disease+)=0.001$, a test is accurate (meaning the test agrees with the truth) 99% of the times, could I calculate the precision $P(D+|T+)=P(disease+|test+)$? $P(D+|T+)=\frac{P(T+|D+)...
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Finding likelihood given uniform data

I'm using this past paper to pre-study for a module in Bayesian Statistics. It has a question, A precision weighing device yields unbiased measurements within half a gramme, which can be modelled as $...
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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))?
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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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Why not use the same distribution for the prior in Bayesian statistics?

I am wondering why introductory books on statistics use a conjugate distribution family for the prior instead of using the same pdf of the one we are trying to infer the parameters? For example, the ...
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