Questions tagged [map-estimation]
Estimation by maximizing the posterior density function
66
questions
3
votes
1
answer
219
views
Basic question about deriving MAP estimator
Say we have a random process $X(t, u)$ parametrized by $t$ and $u$ that generates data $x$. We also have a prior on $u$, $p(u)$.
Am I correct in stating that the expression to find the maximum a ...
- 43
0
votes
0
answers
15
views
How are the MLE/MAP distinction and the generative/discriminative distinction related?
What is the relationship between Maximum Likelihood Estimation versus Maximum A Posteriori Estimation and generative modeling versus discriminative modeling? Is MLE an example of a generative model ...
- 123
3
votes
0
answers
283
views
Minimum Description Length, Normalized Maximum Likelihood, and Maximum A Posteriori Estimation
TL;DR: I believe MDL using NML is a special case of the joint MAP of model and parameters, and need to verify this and find sources that have acknowledges this.
This is how I understand Minimum ...
- 197
1
vote
0
answers
21
views
MAP when the prior of the parameter is defined piecewise
As defined on Wikipedia, $$\hat{\theta}_{MAP} = \underset{\theta}{\mathrm{argmax}} f(x | \theta) g(\theta)$$
Then, to actually obtain theta-hat-MAP, we could set the derivative of the above (or their ...
- 11
3
votes
0
answers
42
views
Does the mode of MCMC samples equal the MAP of the posterior?
If I had millions of MCMC samples from a posterior, should the most frequent value among those samples (i.e., the peak of a histogram of those samples) at least in principle always equal the maximum-a-...
- 1,352
1
vote
0
answers
39
views
Laplace approximation, MAP vs MLE and wiki's notations
I was trying to understand Laplace approximation in statistics and so I was going through the wikipedia article. I don't know much about statistics and I am already getting a bit confused by the ...
- 111
2
votes
1
answer
95
views
For discrete $X_n$ if $X_n \stackrel{d}{\to} U$, does $P_e \to 1$ where $P_e$ is optimal Bayes error and $U$ is uniform
Consider the following setting. Let $ \{X_n\}_{n=1}^\infty \subseteq [-1,1] $ be a discrete random variables that converges to a continuous uniform random variable in distribution.
Let $Y_n = X_n +...
- 195
2
votes
2
answers
223
views
MAP estimation for a Gaussian mixture using EM. Concerns with the covariance update formula
I am implementing the EM algorithm for a Gaussian mixture model with prior; that is, I am using the EM algorithm to find the MAP estimate, rather than the ML estimate. As briefly discussed in section ...
- 145
0
votes
0
answers
90
views
Estimating the parameter of a Bernoulli distribution using probabilistic modeling and the MAP estimation
Suppose you tossed a coin multiple times. Sometimes you got heads and other times you got tails. You recorded your experiment in a dataset $ X$. Now you want to estimate the parameter θ (which ...
- 272
2
votes
1
answer
176
views
Why the prophet time series model uses MAP and not MLE?
I'm using prophet model for one of my time series analysis. I learnt that it optimizes the parameters by MAP approach. The fundamental idea of when to use MAP vs MLE is that when we have a strong ...
2
votes
1
answer
64
views
How can I pool Bayesian parameter estimates after multiple imputation?
After multiple imputation (imputed dataset = 20), I would like to conduct Bayesian Model Estimation with Adaptive Metropolis Hastings Sampling (amh) -- using the MCMC method.
How can I pool the ...
- 21
7
votes
2
answers
617
views
Why does the MAP differ from the MLE for the uniform prior in Laplace's Rule?
Laplace's Rule of Succession produces an estimate for the probability $p$ of a Bernoulli distribution. It starts with a $Beta(1,1)$ prior (equivalent to a uniform distribution prior on $(0,1)$), and ...
- 217
1
vote
1
answer
34
views
Repeated Bayesian inference to track a time-varying parameter online
I have trouble finding the name of the problem (and algorithms to solve it) where one needs to repeatedly estimate the value of a continuous, time-varying parameter online based on incoming ...
- 31
0
votes
1
answer
253
views
Is the posterior maximum always the same as the marginal's?
When I see plots of the conditional probabilities, marginals and joint distributions together they are mostly plot using Gaussians. It is not clear to me if this applies to every other distribution. ...
- 61
3
votes
0
answers
110
views
For log-concave densities, are joint and marginal modes consistent?
Suppose I have a probability density function $\pi(x_1, \ldots, x_n)$, which is the density of a vector-valued random variable $X$ in $\mathbb{R}^n$. Assume that $\pi$ is strongly log-concave, i.e., ...
