# Questions tagged [map-estimation]

Estimation by maximizing the posterior density function

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### 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 ...
17 views

### Is it ok to just use Add One Smoothing for any sufficient statistics on Counts/Probabilities?

I know that Add-one smoothing is due to $$\theta_{\text{MAP}}=\arg\max_\theta \log(P(\theta|D))$$ when the posterior is a Binomial/Bernoulli with a $\text{Beta}(2,2)$ prior. Now, I am implementing ...
1 vote
30 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 ...
57 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. ...
79 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., ...
90 views

### Does the beta negative binomial (BNB) distribution have a conjugate prior?

BNB distribution is constructed using negative binomial and beta distributions, which are both exponential family, so my guess would be yes, there shoudl exist a conjugate prior in theory. But what is ...
28 views

### Estimating parameters for a set of related random variables

Suppose I have some random variables $$X_i \sim Dist(\theta_i)$$ for $i = 1, ..., n$ where $Dist$ is some known probability distribution family and $\theta_i$ are some parameters which may vary ...
1 vote
189 views

### MLE ≠ MAP under Gaussian Prior?

I saw a post on why MLE and MAP yield the same result when under uniform prior. But, I was wondering about the case when they are under Gaussian Prior. I suppose they are different in this case but I ...
1 vote
69 views

### Is MAP a point estimator? Why?

In Bishop's Pattern Recognition and Machine Learning book, page 30, section 1.2.6, it says that the MAP estimator for the curve fitting model derived in section 1.2.5 is a point estimator of the model ...
55 views

### Hessian of log marginal likelihood is rank deficient

I am implementing the Bayesian MAP estimation algorithm presented in [1, Sec. 4.4]. More specifically, I am trying to estimate the hyper-parameters of the main MAP estimator by optimizing the Bayes ...
1 vote
26 views

### Is there an analog of Lehman-Scheffe theorem for Bayes/MAP/biased estimators?

In statistics, the Lehmann-Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. ${ }^{}$ The theorem states ... 41 views

### Bayesian statistics

Assuming I have that $Y_i\mid \mu$ is an iid ~ $N(\mu,\sigma^2)$, for $i \in (1,\dotsc,n)$ with $\sigma_i$ known and improper prior $\pi(\mu)=1$ for all $\mu$. i. How can I derive a formula for the ...
34 views

### MAP estimation with hierarchical prior

I have a hierarchical prior with the probabilistic model as $$p(Y,H,Z|G)=p(Y|H,Z)p(H|Z,G)p(Z|G)$$ and I was wondering if MAP estimation for H and Z together is possible here by maximizing the log ...
1 vote
224 views

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### Deriving the posterior distribution over the model parameters: are the model parameters and data independent?

We are told (in Section 9.2.3, Deisenroth et al.: Mathematics for Machine Learning) that we can compute the posterior over a model's parameters $\boldsymbol\theta$ (here in the context of linear ...
289 views

### The prior in MAP and Bayesian inference

We can use a Normal distribution as a prior when handling a Normal distribution as likelihood in Bayesian inference However if we want to do MAP given a Bernoulli as likelihood can we use Normal ...
70 views

### Does it make sense to condition on fixed values of some parameters before doing MCMC on the other parameters?

I have a Bayesian model with a large number of parameters (around 50), and as usual my goal is to infer the posterior distribution for the parameters, with MCMC. However, I am only interested in the ...
904 views

### Shouldn't log likelihood always be normalized by data size in bayesian estimation?

This is very interesting problem. I wonder if the whole bayesian statistics is neglecting it or if I am super confused. I will illustrate it on a bayesian Maximum a Posteriori (MAP) estimation and ...
1 vote
23 views

### Finding the MAP for a function whose conditioning depends on an exponential integral

Let $X$ be such that $X \sim exp( \lambda = 1)$ and let $Y$ be such that $Y \sim U[0,x]$, where $x$ is the realization of $X$. Given that information I know that: $f_{X}(x) = e^{-x}$ for $x \geq 0$...
1 vote
26 views

### MAP, MLE and parametrised data

It is often said that maximum likelihood is used to obtain estimates of distrubtion's parameters. However, what is unclear is whether it will produce consistent estimate parameters other than those of ...
201 views

1 vote
87 views

### Assign an error to the parameters of MAP estimate

Through a MCMC Gibbs sampler I obtain $M$ chains of the parameters vector $\mathbf{\theta}$, meaning that each component of $\mathbf{\theta}$ is the value of one parameter at a given iteration. ...
1 vote
222 views

### Discriminative Models with Class Priors

In discriminative models, we model $p(Y|X)$ directly while in generative models we model $p(X|Y)p(Y)$ where $X$ is the input and $Y$ is the output variable. I am confused when the parameters and ...