All Questions
17
questions
12
votes
1
answer
742
views
Steps to figure out a posterior distribution when it might be simple enough to have an analytic form?
This was also asked at Computational Science.
I am trying to compute a Bayesian estimate of some coefficients for an autoregression, with 11 data samples: $$ Y_{i} = \mu + \alpha\cdot{}Y_{i-1} + \...
- 2,352
8
votes
0
answers
302
views
Time evolution of a Bayesian posterior
I have a question regarding the time evolution of a quantity related to a Bayesian posterior.
Suppose we have binary parameter space $\{ s_1, s_2 \}$ with prior $(p, 1-p)$,
The data generating ...
- 3,348
6
votes
2
answers
175
views
Posteriors and Sample Sizes
Suppose we have a two dimensional parameter $\theta=(\mu,\sigma^2)$, and a prior distribution $p(\theta)$. Let our sample come from a normal distribution with mean $\mu$ and variance $\sigma^2$. The ...
- 207
3
votes
1
answer
1k
views
What is this "trick" for finding posterior distributions?
I am new to Bayesian statistics. I am trying to understand a certain passage in my course notes.
Excerpt:
Discussion:
I don't understand much about the above excerpt. I think that the goal here is to ...
- 581
3
votes
1
answer
5k
views
Beta-Binomial conjugate proof
Can someone explain this proof to me? I get stuck on the transition from the third line to the last line. Namely:
Is the integral being evaluated or not?
How does the entire expression reduce to a ...
- 3,382
3
votes
1
answer
203
views
How can I plug in the value of parameter found by Maximum a Posterior?
Suppose I have 1 heads and 4 tails from 5 coin tosses. To find out the probability of 1 heads and 4 tails in my coin toss experiments, I decided to use Binomial Probability Mass Function for the ...
- 3,525
2
votes
2
answers
72
views
How can I get the probability of a predicted outcome conditional on the posterior in bayesian regression?
Assume that I am running the following regression:
$\hat{y_t} = \beta_0 + \beta_1 \cdot x_t$
where as $\hat{y_t}$ is a continuous variable.
Lets assume a gaussian likelihood and nonconjugate priors ...
2
votes
1
answer
102
views
Help with the prior distribution
The question is as follows:
Consider an SDOF mass-spring system. The value of the mass is known and is equal to 1 kg.
The value of the spring stiffness is unknow and based on the experience and ...
- 225
2
votes
1
answer
88
views
Regarding the bayes rule derivation of posterior distribution, $p(\omega|x,y),$ for a given dataset $D$ over $\omega.$
So I was going through this paper and under Uncertainty modeling it says
So I tried deriving it on my own and I got
$p(\omega | X, Y) = \frac{p(Y | X, \omega) \cdot p(X,\omega)}{p(Y | X) \cdot P(X)}$
...
- 138
2
votes
0
answers
186
views
When does this prior dominate likelihood?
This is a simple Bayesian inference problem, where we are trying to infer some weight parameter $w$. Our posterior distribution is
$$ P\propto \exp\left(-\frac{1}{\sigma^2} w^Tw\right) \exp\left(-f(w)\...
- 281
2
votes
0
answers
161
views
How does Bayes' rule on two exponentials suggest a sigmoid?
In Platt's 1999 paper on turning support vector machine output into a probabilistic score, he says
Bayes rule on two exponentials suggests using a parametric form of a sigmoid
where he cites this ...
- 141
2
votes
1
answer
83
views
In deriving the parameter of a posterior, is it necessary to use the likelihood over $n$ samples?
In a test I had to derive the posterior of the multinomial distribution with the conjugate Dirichlet prior. I used common relation $$p(\mu|X;\alpha) \propto P(X|\mu) P(\mu|\alpha).$$ I did, however, ...
- 6,624
1
vote
1
answer
54
views
Range of integration for joint and conditional densities
Did I mess up the range of integration in my solution to the following problem ?
Consider an experiment for which, conditioned on $\theta,$ the density of $X$ is
\begin{align*}
f_{\theta}(x) = \...
- 35
1
vote
0
answers
394
views
Why logit transformed binomial proportion is approximately normal?
What is the argument to prove asymptotic normality of logit transformed of binomial proportion which follows Beta distribution?
$\theta$ has beta prior and model $y|\theta$ follows $Binom(\theta,n)$. ...
- 1,445
0
votes
1
answer
359
views
Posterior probability - change in Beta hyperparameters
Can you explain, how does $\text{B}(\alpha, \beta)$ transfrom to $\text{B}(s+\alpha, f+\beta)$ in the following equation?
$$
\begin{align*}
p(\left. q=x \right| s,f)
&= {{{s+f \choose s} x^{s+\...
