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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Understanding Probability in Non-Repetitive Events
In the context of events that cannot be repeated, such as stock prices, how is probability defined? Specifically, if I develop a statistical model to predict stock prices and claim it is 90% accurate, ...
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Rating update algorithm for doubles pickleball
I'm looking to track the rating of each player in a pickleball league using a spreadsheet. A concept like ELO seems like the right approach, but I'd like to track both a mean (rating) and sigma (...
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What is the Gold Standard for Evaluating the Posterior of a Bayesian Regression Model?
Let me explain my meaning & the context:
I mean evaluating the correctness of the posterior (e.g. for approximate Bayesian inference methods).
I care mostly about Bayesian deep learning, I'd like ...
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I need help understanding importance of Bayesian Linear Regression
So, I am trying Bayesian Linear Regression. Being new to this I have tried the following things:
Equation for generated data:
$$Y = \phi(X) \cdot W + \epsilon $$ where $$W \sim \mathcal{N} ( \...
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Metropolis-Hastings Proposal Distribution multiplied by scalar. How does this affect the Hastings Ratio?
So, if I propose $\theta^*$ from $\text{Dirichlet}(\theta_i \times \tau)$, where $\tau$ is some scalar, does the Metropolis-Hastings atio need to account for that scalar i.e.
$$\text{Dirichlet}(\...
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How can I obtain the full posterior and the full conditionals from a joint normal and inverse gamma likelihood and prior?
As a complete beginner in the world of Bayesian statistics, I unfortunately have no idea of how to start this problem.
I am given that our data is distributed in: $x_n|(\mu,\sigma^2)$ ~ $N(\mu,\sigma^...
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Inference in Dirichlet process mixtures via collapsed Gibbs sampling
I need to cluster some data $\{x1, \ldots, x_n\}$ through a Dirichlet process mixture model.
Consider the following Dirichlet process mixture model, in which the base measure is a $NIW(\mu_0, \...
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Flattening a likelihood
Background
Let $y_1,y_2,\dots,y_K$ be a sequence of measurements.
I've derived a likelihood $\mathcal{L}(y|i)$ to solve a classification problem via the Bayesian classifier
\begin{equation}
p_k(i)=\...
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Is distributions defining necessary for a DAG to be a causal bayesian network?
First, let's define the following abbreviations: Directed Acyclic Graph (DAG), Bayesian Network (BN), Causal Bayesian Network (CBN), Conditional Probability Table (CPT), Conditional Probability ...
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pymc3: Updating the standard error prior
I am estimating a Bayesian multiple regression using continuous data on both the dependent variable and the regressors. My goal is to iteratively estimate the coefficient distributions as more data ...
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Elementary statistics for jurors
I have been summoned for jury duty. I am conscious of the relevance of statistics to some jury trials. For example, the concept of "base rate" and its application to probability calculations is ...
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How to solve this question about the beta distribution in a Bayesian analysis? [closed]
This question appeared in Prof. Babak Shahbaba's book (Biostatistics With R: An Introduction to Statistics Through Biological Data) in the questions of its chapter 13.
Q4. Suppose that we are ...
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A question on Bayesian credible interval vs frequentist confidence interval
The difference of Bayesian credible interval (BCI) and the frequentist confidence interval (FCI) is well explained with a nice example in this answer. Here is my own summary of the situation in the ...
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MCMC direct comparison of difference of two parameters
Say I have run a Hierarchical Bayesian model in STAN (or JAGS or BUGS) and I have the posterior samples of two slope parameters that I want to compare: $\beta_1$ and $\beta_2$. The model appears to ...
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Objective Bayesianism: Jeffreys priors vs reference priors vs principle of transformation groups
According to this answer,
José Bernardo has produced an original theory of reference priors where he chooses the prior in order to maximise the information brought by the data by maximising the ...
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Is the decomposition of a transitive kernel well-defined? [closed]
In the paper enclosed below, the authors write:
Suppose that the transition kernel, for some function (p(x, y)), is expressed as
$$
P(x, d y)=p(x, y) d y+r(x) \delta_x(d y),
$$
where $p(x, x)=0, \...
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Discriminative vs generative & Bayes risk
In the context of binary classification with features denoted $X$ and labels $y$, one could specify the joint distribution of $(X, y)$ in two equivalent but different ways :
The generative way ...
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Does independence imply d-separation?
In the context of Bayesian networks if two random variables (i.e. nodes) are d-separated, they are independent. However, is there any example of random variables being independent but not d-separated?
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How to interpret model fit in posterior predictive checks between two models that both capture the observations in its 1sigma?
I have two models aimed at explaining a single observed measurement $x_{obs}$:
Simple Model with 26 parameters $f_1(\theta)$.
Complex Model with 31 parameters $f_2(\theta)$.
Both models are assumed ...
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PyMC3 implementation of Bayesian MMM: poor posterior inference
Google released a whitepaper on Media Mix Modelling (MMM) in 2017; vanilla MMM (established in the 1960s) uses multivariate regression. It's a decent mechanism to understand which of your marketing ...
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Introduction to measure theory
I'm interested in learning more about nonparametric Bayesian (and related) techniques. My background is in computer science and though I have never taken a course on measure theory or probability ...
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Generative vs discriminative models (in Bayesian context)
What are the differences between generative and discriminative (discriminant) models (in the context of Bayesian learning and inference)?
and what it is concerned with prediction, decision theory or ...
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Is forcing correlation between parameter values allowed in bayesian regression
I am using bayesian nonlinear regression in brms to fit some model parameters, and I'm running into some issues getting MCMC chains to converge. I know this is ...
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How do programs like BUGS/JAGS automatically determine conditional distributions for Gibbs sampling?
Seems like full conditionals are often quite difficult to derive, yet programs like JAGS and BUGS derive them automatically. Can someone explain how they algorithmically generate full conditionals for ...
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When (if ever) is a frequentist approach substantively better than a Bayesian?
Background: I do not have an formal training in Bayesian statistics (though I am very interested in learning more), but I know enough--I think--to get the gist of why many feel as though they are ...
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Can you use the beta-binomial distribution instead of MCMC?
So, I have a project to test the hypothesis that a marketing campaign with a new art generates more purchases than the old one, I have 2 samples of data, one using the standard ad and one using the ...
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Jeffreys' test for a single event
Assume $x\sim B(p,n)$ and using Jeffreys' prior the posterior distribution of $p$ for $x$ events and $n$ observations is $$\text{Beta}(x+\frac{1}{2},n-x+\frac{1}{2})$$ I am reading about a Jeffreys' ...
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Bayes theorem and the inverse probabilities
If I have the following data:
p(being in a fatal car accident) = 0.02%
p(driving while high) = 4.7%
p(driving while high | being in a fatal car accident) = 31.8%
I ...
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How does one usually compute the gradient and the Hessian of a proposal in a MCMC algorithm?
In some proposals of a MCMC, the mean/location vector and the covariance/scale matrix are functions of the gradient/jacobian and hessian of the log-likelihood.
I'm wondering how does one usually find ...
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Is there a way to convert a bayesian credible interval to standard deviation?
I am conducting a meta-analysis and one of the included studies only provides as a measure of dispersion the 95% credible intervals, I would like to know if there is a way to convert it to standard ...
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Monte Carlo integration methods utilizing a set of representative points given by a black box
Consider the task of integrating a function with respect to a multimodal distribution. Suppose I am given, by a black box, a set of independent but identically-distributed points in the distribution ...
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Transforming variables within MCMC to get the prior distribution to match proposal
I'm doing Bayesian MCMC where I am proposing some weights, say, a_1:a_5 from a Dirichlet distribution to ensure summation to 1. However, the prior (Beta) ...
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Spatial effects competing for variance with error term?
I am simulating a dataset from a spatial model. Each data point is distributed as follows:
$$
y_i \sim \mathcal{N}(\mu + \phi_i, \sigma^2)
$$
Here $\mu$ is a fixed value, estimated as an intercept. ...
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Learning operating probabilities from interval data
Suppose I have a machine. When the machine is active (operating), it runs for at least $\mu > 0$ time. I know that at some point in the time interval $[l, h]$ ($l, h \in \mathbb R_{\ge 0}, l < h$...
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Bayesian Learning: Finding the variance of noise
Suppose $x_i \sim N(10,4)$ - ie, the distribution is known.
There is a noisy signal $s_i \sim N(x_i, \sigma_e^2)$ and I want to estimate $\sigma_e$.
I see some pairs ($s_i, x_i$) but they are not '...
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How can I model this Bayesian learning process of two types of coins?
(As suggested on the comment, I slightly changed my previous question.)
I have $N$ coins and I am testing them one by one if it is fair or not. I know that, if it is unfair, the probability of head ...
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Specific step in the proof of conjugate prior for normal distribution with unknown mean and variance
I'm struggling to follow a specific step in the proof that
$$
\tau \sim \text{Gamma}(\alpha, \beta), \quad \mu | \tau \sim \mathcal{N}\left(\nu, \frac{1}{k\tau}\right)
$$
is a conjugate prior ...
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How to make Bayesian-style inference for a Poisson process?
I am working on a fleet management software recently. Normally, the arrival of merchant request is a Poisson process. That is to say, on average we have a new merchant request every 10 minutes, but ...
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Which proposal function should be used in this particular case of the Metropolis-Hastings algorithm?
As part of my research, I would like to apply the Metropolis-Hastings in order to sample from some posterior distribution. More precisely, the data comes from a multivariate normal distribution in the ...
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Fitting nonlinear Bayesian regression with a summation term in brms
I'm trying to fit parameters for a Holling type II curve for multiple prey items. This takes the form:
$$
\frac{dP_i}{dt} = \frac{a_iP_i}{1 +\sum_j{a_jh_jP_j}}
$$
where $P_i$ is density of prey ...
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Do MCMC chain lengths affect model comparison?
Say I have two Bayesian regression models fit to the the same data. For whatever reason, the second model takes longer to converge, so I run the MCMC sampling chain twice as long for the second model ...
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Multilevel model where skew of random effect depends on an independent variable
I am trying to construct a model where the skew of the distribution of a random effect changes with an independent variable. I'd eventually like to fit this using ...
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How to determine p(a) in bayes' formula with this example of symmetric conditional probabilities?
Why is P(A) in this example 1?
In class, we were shown an illustrative example of bayes that no one including the professor could understand.
P(A|B) = P(A) * P(B|A) / P(B)
Question is:
Use Bayes ...
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Bayes estimate of upper limit of uniform distribution with exponential prior [closed]
Let $X_{1}, . . . , X_{n} > 0$ be a random sample from $U(0, \theta)$. Suppose $\theta$ has the prior $\pi(θ) = e^{-\theta} ; \theta > 0$. Find the Bayes estimate of $\frac{1}{\theta}$ with ...
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Pros and Cons Using Bayesian Metrics or Time Series Approach?
I have two years' worth of sales data for a range of products, including:
Sales: Total sales revenue
Usage Time: Total usage time in hours
Reviews: Numeric ratings and textual reviews from users
I'm ...
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non-negative constraints and interactions in the ensemble model
In the context of prediction problems using regression models, suppose I have $K$ different models all trained (fitted) on the same targets (observations). These models are different - low correlation ...
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Adding a magnitude penalty to a GAM
This is a follow-up to a previous question of mine, explaining the problem in more detail in the hopes of getting more precise advice.
Consider the following structured additive regression model or ...
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Problem with bayesian implementation of a Time-lagged Linear Model in PyMC3
I am trying to build a GLM of a time-series y(t) with 2 predictor time series x1(t) and x2(t), where t is in days. But the second time-series influences y(t) with an unknown lag of l days. I was ...
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Probability of winning a game: frequentist vs Bayesian approach
Alice and Bob play the game - the rules of the game are not important, and after 8 rounds Alice has 5 points and Bob has 3 points. Every round one of 2 players gets 1 point and the winner of the game ...
