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Questions tagged [bayesian]

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.

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Posterior expectation of a normal distribution given a "truncated" normal sample

Consider the following problem of estimating an unknown parameter, $\theta$: Suppose that $\theta \sim N(0,\tau_\theta^{-1})$, where $\tau_\theta\ge 0$ is the posterior precision. Suppose that there ...
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How to Compute the Posterior Distribution of Covariance matrix in a Matrix Normal Model with Inverse Wishart Prior [closed]

I am working on a time series model involving Kalman filters and smoothing to estimate state variables $Y_i$. The part of model is structured as follows: $Y_1, \ldots, Y_n$ are iid. $Y_i \sim \mathcal{...
Ayden Frost's user avatar
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Bayesian updating with correlated signals

I would like to ask a broad question about Bayesian updating with correlated signals. Specifically, in my context I have an agent asking some questions. The signals are the answer. Since the question ...
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Deriving the Variance of a Posterior Conditional Probability Density Function

Deriving the variance of a conditional probability density function I am trying to derive the variance of a conditional probability density function $ p_{r_s|x,y_s} $ given by: $$ p_{r_s|x,y_s} = \...
Alireza Ghazavi's user avatar
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Deriving posterior distribution with explicit constants in closed form using Jacobian method compared to Bayes' rule

Question on Bayesian Inference Deriving posterior distribution with explicit constants in closed form using Jacobian method compared to Bayes' rule I am working on a Bayesian ...
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Solving the marginal likelihood integral and approximating closed-form solutions for that

Given the likelihood function: \[ p_{y_s|r_s,x}(y_s|r_s,x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(y_s - \alpha \frac{\rho_s}{r_s^4})^2}{2}} \] and the prior distribution for \( r_s \): \[ p_{r_s}(r_s) = \...
Alireza Ghazavi's user avatar
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Question about PRML Equation 2.19 on page 73

I am confused about the justification of equation 2.19 in Pattern Recognition and Machine Learning by Christopher M. Bishop. The equation is given below: $$p(x=1|D) = \int_0^1 p(x=1|\mu)p(\mu|D)d\mu = ...
gstudent's user avatar
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Stable Kalman Filter estimator with given covariance matrices

I asked this question a while back. Essentially considering the follow basic Kalman Filter, following the Wikipedia convention. \begin{equation} \begin{split} x_k &= F_kx_{k-1} + B_k u_k +w_k\\ ...
Matt Frank's user avatar
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Bernstein–von Mises approximation of posterior density, Fisher Information

The Bernstein–von Mises approximation of posterior density theorem, states that certain under conditions, $$ d_{VT} \Big( \Pi(, |X), N \big(\hat{\theta}_n(X), \frac{I(\theta_0)^{-1}}{n} \big) \Big) \...
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Derivation of Inequality 3 from Inequality 4 Using Bayes' Theorem in "Scheduling Multithreaded Computations by Work Stealing"

In the paper "Scheduling Multithreaded Computations by Work Stealing" under the section "Atomic accesses and the recycling game", it is mentioned that inequality 4: $$ \Pr \left\{ ...
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Bayesian vs frequentist difference beyond interpretation from the algebraic or categorical standpoint

Context: I'm not a statistician at all, and I was just involved in a debate between physicists on fundamental differences between the bayesian and frequentist approaches. Someone was arguing that it's ...
Vincent's user avatar
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Intuitive explanation of "Information Filter" formation of Kalman filter

Can someone intuitively explain this "Information Filter" formation quoted from wikipedia ? In particular I struggle to understand why $\mathbf{I}_k = \mathbf{H}_k^\textsf{T} \mathbf{R}_k^{-...
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Conditional probability density function of the parameter $\lambda$, given a data set.

Consider the one-sided conditional exponential distribution: \begin{equation} f_X(x|\lambda)=\frac{\lambda}{Z(\lambda)}\exp(-\lambda x), 1\leq x\leq 20, \end{equation} where $\lambda>0$ and $Z(\...
SecretKeeper's user avatar
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Apply Bayes' Rule to a joint, prior distribution?

The problem is how to update a joint prior distribution consisting of two independent variables with observed evidence? For the prior distribution, data exists for average event frequency (i.e., ...
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Measuring Robustness in Variational Bayesian Inference and Nonlinear Filtering

I am interested in how to properly pose/measure robustness, in a qualitative or potentially quantitative manner, when inferring a probability density function (pdf) either by Bayes' rule or a ...
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How to understand likelihood function bayesian

$\mathcal{N}(W^T \cdot X, \beta^{-1})$ This is the likelihood distribution for Bayesian linear regression, right? So, the thing is, if I'm doing batch mode Bayesian regression, then: Weights (W): Size:...
Need_MathHelp's user avatar
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How to derive likelihood function

I have been struggling a lot with the concept of likelihood and I'd really appreciate it if someone could verify if my understanding is correct and give input. If I understand this correcly, we pick ...
Need_MathHelp's user avatar
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Sample mean of Bernoulli trials is admissible under squared loss

Let $X_1,\ldots,X_n$ be i.i.d. Bernoulli trials with probability $\theta\in(0,1)$, and let $L:(0,1)\times[0,1]\to\mathbb{R}$ be the squared loss function, i.e. $L(\theta,a)=(\theta-a)^2$. I am trying ...
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Conjugate priors and Bayesian updates

In a paper by Cyert and Degroot (1974) (Rational Expectations and Bayesian Analysis, in Journal of Political Economy), authors use Bayesian update for an uncertain parameter. They have a model for ...
optimal control's user avatar
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Bayes classifiers with cost of misclassification

A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$: $$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...
BiasedBayes's user avatar
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Linear Regression with gaussian mixture prior

In linear regression, we assume that the output variable is Normally distributed, i.e., $p(y) = N (y | \mathbf{w}^T\mathbf{x}, \sigma^2_y)$. I want to assign a mixture of Gaussian prior to each ...
maktukmak's user avatar
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How to use argmax in Bayesian posterior estimate?

I have some yield function $f_\beta(x)$. I want to find the value $x$ that maximizes the yield. However, my function is parameterized by a parameter $\beta$. For simplicity let's assume there is only ...
Willem's user avatar
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Characterize the mixed strategy Bayesian Nash equilibria for this game. [closed]

A two-player game where Player 1 can choose either U or D and Player 2 can choose either L or R. Player 1 is either cooperative with probability $P$ or uncooperative with probability $1−P$. Player 1 ...
Toshani Singh's user avatar
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Bayes' formula for distriibutions

In my statistics course, we have discussed Bayes' theorem for distributions: $$ p(\theta | y)=\frac{p(y|\theta)p(\theta)}{\int p(y|\theta)p(\theta) \mathrm{d}\theta} $$ Where $p(\theta)$ is the prior ...
sodium-hydroxide's user avatar
1 vote
1 answer
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Calculate the posterior distribution

How can I solve the letter (a)? Discrete sample spaces: suppose there are N cable cars in San Francisco, numbered sequentially from $1$ to $N$. You see a cable car at random; it is numbered $203$. You ...
Siqueira's user avatar
2 votes
1 answer
39 views

Bayesian Inference Intractability

When looking at Bayesian posteriors $$ p(z \mid x) = \frac{p(x \mid z)p(z)}{\int p(x \mid z')p(z')dz'} $$ The denominator commonly intractable. I understand this is due to the possibility of high ...
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A conditional probability problem about a fiber cabel and bits, Bayes' theorem

I've been racking my brain about this problem for way longer than I should: "Through a fiber cabel, information is sent in the form of bits, that can take the values 0 or 1. Sometimes, there is ...
Edward Chen's user avatar
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Bayesian updating when event depends on two uncertain things

There are exponential distributions, $f(x_f)$ and $g(x_g)$, and each distribution either has a scale parameter $\lambda_0$ or $\lambda_1$, where the prior probability $f(x_f)$ has scale parameter $\...
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Bayesian network: how to speed up inference(s)?

I'm experimenting with open-source python libraries that can handle Bayesian networks easily. However the inference is slower compared to the commercial solution (SMILE). One slower inference would ...
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Derivation of Evidence lower bound and KL diveregence in VAE. I cannot understand the independence condition.

In the article "An Introduction to Variational Autoencoders" written by Diederik P. Kingma, there are this equation in page 18. \begin{align} \log p_{\theta}(x) &= \mathbb{E}_{q_{\phi}(z|...
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Combating a specific argument against the Monty Hall problem .

So to get started , I make it clear that I do know how the Monty Hall problem works and although I had my good share of problems understanding it in the past , I did manage to come to terms with it on ...
Mike Billings's user avatar
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Question on literature for contraction rates

I read in some lecture notes the following definition of contraction rate: Definition (Posterior rate of contraction) The posterior distribution $\Pi_n\left(\cdot \mid X^{(n)}\right)$ is said to ...
Grandes Jorasses's user avatar
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Defining Regular Conditional Distributions under Bayesian and Frequentist Approaches

I am trying to set up a probabilistic framework that can be easily switched between a frequentist and a Bayesian approach. Assume the measurable spaces $(\mathcal{X}, \mathfrak{X}), (\mathcal{T}, \...
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Improper distribution for spatial statistical analysis

This a question from a statistical spatial analysis. Let $\Theta = (\beta, \sigma^2, \tau^2, \phi)$. Assume that $Y | \Theta \sim \mathcal{N}(X\beta, \sigma^2R(\phi) + \tau^2I).$ Suppose that $\phi$ ...
Cristóbal Miño Morales's user avatar
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Proof of intractability of computing the prior distribution in bayesian inference

In this paper: https://www.tandfonline.com/doi/epdf/10.1080/01621459.2017.1285773?needAccess=true I do not understand why there are K^n terms in the integral in Equation (9) instead of n^K. If we have ...
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Shouldn't the product of importance sampling weight and likelihood equal 1 in PSIS?

I'm reading the 'Overthinking: Pareto-smoothed cross-validation' in Chapter 7.4 of Richard McElreath's textbook Statistical Rethinking 2nd edition. The author said: Cross-validation estimates the ...
EndlessHomework's user avatar
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1 answer
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(Bayesian probability) Show that $P(H|E) = \frac{h (c +a \overline c)}{hc+a\overline c}$

The question, briefly How does the calculation $P(H|E) = \frac{h (c +a \overline c)}{hc+a\overline c}$ work? Some background I'm trying to work through the proof of the following theorem in ...
snofelet's user avatar
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29 views

Bayes Method How to draw coefficients summing to 1 and each of them follow exponential distribution

There is a question that how to draw the estimators (ratio estimator) if we know the prior for the numerator or the ratio estimator. Assume that we have three coefficients: $a_1,a_2$ and $a_3$ . And ...
Mike's user avatar
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2 votes
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Metropolis-Hastings Algorithm. How do we know that it converges to target distribution?

In the Metropolis-Hastings algorithm, we choose to accept our sample with probability: $$\rho =min\left\{1, \frac{p(x')g(x|x')}{p(x)g(x'|x)}\right\} $$ Where $x$ refers to the current state of the ...
Matheo Xenakis's user avatar
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Bayesian updating on expectation with many candidates

I have an individual selected for a job. He was competing against 10 candidates. All individuals are independently drawn from uniform distribution U(0,1). Him being chosen means he is better than the ...
Elina Gilbert's user avatar
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Maximizing a function with binary indicators

I am an econ undergrad trying to understand how to maximize this payoff function, which includes binary components. I want to solve this equation using backwards induction, so I want to maximize the ...
trynacode's user avatar
1 vote
1 answer
32 views

Probability Density Function of $Y \vert X$ when both $X$ and $Y$ are conditional on $\theta$

In the Bayesian setting, suppose that we know the PDF of $X$ given $\theta$ is $p_X(x \vert \theta)$ and the PDF of $Y$ given $\theta$ is $p_Y(y \vert \theta)$. In the standard fashion, we may assume ...
YessuhYessuhYessuh's user avatar
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Optimal Calibration for a Test Set

I want to calibrate the parameters $\theta$ of a known forward model $y=f(\theta, x)$, i.e., I want to identify offsets of around +- 10% from a nominal model $\theta_0$ I can not measure $y$ directly, ...
scleronomic's user avatar
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Multivariate Gaussian distribution derived from random effects

There's this proposition I encountered in a paper on Bayesian data analysis. I am trying to use it to ease some computations for a mixed effects model. It goes as follows: $\textbf{Proposition 1.}$ ...
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Factoring integral term on joint expectation

I am reading a paper titled Transformer can do Bayesian inference. I am lost in proof of insight 1, in which they derive (equation 3): $$-\int_{D,x,y}p(x,y,D)\log q_{\theta}(y|x,D) = -\int_{D,x}p(x,D)\...
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Interpreting $P(\alpha|\text{data})\propto P(\text {data} | \alpha)\cdot P(\alpha)$ [closed]

In the context of posterior and prior probabilities, one has $P(\alpha|\text{data})\propto P(\text {data} | \alpha)\cdot P(\alpha)$. What confuses me here is that probability is defined for events, ...
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1 vote
2 answers
113 views

What Is the Probability The Second Kid Is a Boy? [duplicate]

Okay, so I was asked this question in an interview on a machine learning expert position. To be honest, the question itself (and the hint by the interviewer) seemed quite ill-phrased, which probably ...
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Is the optimal strategy to find the special coin the greedy strategy

Problem. There's $n$ coins, $n-1$ are fair and one always shows heads, but all coins look the same. You are allowed to flip $m$ coins, where $m < n$ and the choice of the $i$th flip can depend on ...
jojo's user avatar
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1 answer
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Question on determining the posterior pdf

Can someone tell me how the pdf of noise (w) is equivalent to the conditional pdf of observations (x) given A, assuming noise is independent of A for the equation x[n]=A+w[n] where A is the mean (and ...
Math_noob's user avatar
3 votes
1 answer
64 views

Is this reasoning about Bayesian Inference on the Lewis Carroll's pillow problem correct?

The Lewis Carroll's Pillow Problem's solution was previously addressed here and made me wonder if it makes any sense to reason that in this context the Bayesian framework allows us to estimate "...
Luciano Dourado's user avatar

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