Skip to main content
16 votes

Understanding the Difference Between MAP Estimation and ML Estimation

Maximium A Posteriori (MAP) and Maximum Likelihood (ML) are both approaches for making decisions from some observation or evidence. MAP takes into account the prior probability of the considered ...
Mark Borgerding's user avatar
8 votes

Understanding the Difference Between MAP Estimation and ML Estimation

You have a set of message set $m_i$, $0 \le i \le N-1$. (For example, QPSK will be $N=4$). For the transmitted message $m_i$, the corresponding symbol vector is $\textbf{x}_i$, and the received symbol ...
jithin's user avatar
  • 2,263
6 votes
Accepted

cramer lower bound, MAE, and MSE

The topics you bring up are at the heart of estimation theory. I highly recommend reading Steven Kay's Fundamentals of Statistical Signal Processing: Estimation Theory for a detailed background and ...
Ash's user avatar
  • 915
5 votes
Accepted

How to remove the modulation before doing frequency offset estimation?

For lower-cardinality PSKs like the QPSK, the "classical" way is to take the signal to the $M$th power, $M$ being the number of constellation points. From the shape of an $M$-PSK modulation, ...
Marcus Müller's user avatar
4 votes

Understanding the Difference Between MAP Estimation and ML Estimation

A brief, non-mathy explanation: ML assumes that all hypothesis are equally likely. MAP does not make this assumption. MAP is the optimum criterion, but under some conditions ML is optimum too. When ...
MBaz's user avatar
  • 15.3k
3 votes

Maximum Likelihood for Colored Noise

Let's have a look on the following model: $$ y \left[ n \right] = \left( h \ast x \right) \left[ n \right] + \left( g \ast w \right) \left[ n \right] $$ Where $ x \left[ n \right] $ is the signal of ...
Royi's user avatar
  • 19.7k
3 votes
Accepted

MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)

I looked this up on Wikipedia: A complex Gaussian random variable $V = \mathfrak{Re}(V)+j\mathfrak{Im}(V)$ is said to be zero mean circularly symmetric $\mathcal{CN}(0,\Gamma)$ if the random vector $[\...
Atul Ingle's user avatar
  • 4,134
3 votes

MLE formulation -- confusion regarding the terms in the equation (Part1)

Hi: I don't have time to carefully look at everything right now and I'm unfamiliar with circular complex random variables. But, if it's similar to regular normal rv's and the two variables are ...
mark leeds's user avatar
  • 1,117
3 votes
Accepted

Why Is The Maximum Likelihood Estimation (MLE) Method Taken as the Benchmark for Comparing with Other Methods?

Coming up with a "good" estimator for a parameter of interest is not an easy task because it is important to define what good means. There are many ways of defining it, depending on your application. "...
Atul Ingle's user avatar
  • 4,134
2 votes
Accepted

Maximum likelihood estimator of active time delay and passive time delay

Very intuitively, the Generalized Cross-Correlation is a "standard" cross-correlation of the windowed signals (I'll restrict myself to window-GCC, I'm pretty certain there's others, too!). Windowing ...
Marcus Müller's user avatar
2 votes

Cramer Rao Lower Bound for Cross Correlation (Time Shift Estimation)

CRLB is also a function of signal bandwidth (effective bandwidth or second derivative of xcorr peak): $$\sigma=\dfrac{1}{2\pi F_e\sqrt{\frac{2E}{N_0}}} $$ Also note,...
Linas Svilainis's user avatar
2 votes

MLE parameter estimation -- confusion regarding some terms in the pdf of complex normal r.v (Part 2)

Hi: It's worded more clearly now in that you're estimating A and there's only one RV which makes more sense. Consider the first link I sent in the other message and go to where it says the "likelihood ...
mark leeds's user avatar
  • 1,117
2 votes

Difference between Cramer Rao bound and mean absolute error MAE

The Cramer-Rao bound is not a metric for the quality of a certain estimator, but rather can be calculated for a certain estimation problem and gives you a theoretical lower bound for the variance of ...
mateC's user avatar
  • 338
2 votes

Maximum Likelihood for Colored Noise

Royi's answer is excellent. I'd like to discuss a different way of arriving at the same answer, one that tells you how to find the matrix $G$ if you don't know it. In many applications this matrix ...
orchi_d's user avatar
  • 587
2 votes
Accepted

How can I write the likelihood of this system

Looking at the histograms you provided, it is very possibly correct that the data is exponentially distributed. However, the model $Z_m(\omega) = F(\omega, \Theta) + \eta_m(\omega)$ does not make ...
mateC's user avatar
  • 338
2 votes
Accepted

Difference between Likelihood Estimation and CRLB Estimation for Cooperative Radar

They are not the same thing. This is a longer version of PeterK's answer. Let's start with the ML estimator, using your notation. For a fixed target $p$ and assuming radar model that is characterized ...
AlexTP's user avatar
  • 6,605
1 vote

Why this second part of the integral given by the gradient of the log-likelihood is zero

I suspect it's because $$ m(t) = A \cos(\omega_c t + \phi) $$ and $T$ is a single period of the cosine (or an integer multiple of the period). Then $$ \frac{d}{d t}\left(A \cos(\omega_c t + \phi)\...
Peter K.'s user avatar
  • 25.9k
1 vote
Accepted

Maximum Likelihood Estimation of Phase

I'm not sure that there's a closed-form solution for the ML estimate of phase, but if you recast the definition of $x[n]$ to $$x[n] = a \cos (\omega n) + b \sin (\omega n) + w[n], \; \; n= 1,...,N $$ ...
TimWescott's user avatar
  • 12.9k
1 vote
Accepted

Does Maximum Likelihood detector (QAM symbol detection) enhance BER?

If your error vector magnitude is below the decision threshold between adjacent symbols then there would be no impact on the BER. Thus there will be a point where you can continue to reduce the error ...
Dan Boschen's user avatar
  • 52.3k
1 vote
Accepted

ML estimation - solve for x

It looks okay to me. If you define empirical mean $\hat{\mu}_d = \frac 1N \sum \tilde{d}[n]$ and empirical second moment $\hat{\gamma}_d = \frac 1N \sum \tilde{d}^2[n]$ then you effectively have an ...
Florian's user avatar
  • 2,463
1 vote

Bayesian Information criterion is independent of prior - What does 'prior' mean here?

"Prior" refers to the "already known" probability (or its probability distribution function) of an event happening
Dsp guy sam's user avatar
  • 2,620
1 vote
Accepted

Equivalence of Maximum Likelihood (ML) and Discrete Fourier Transfrom (DFT) Peak Finding for Single Tone Estimation

Maximum Likelihood under the assumption of Additive White Gaussian Noise (AWGN) is always equivalent to finding the hypothesis with the minimum distance to given data. Since minimizing distance is ...
Royi's user avatar
  • 19.7k
1 vote
Accepted

Maximum Likelihood Detection of Signal Vectors in Gaussian Noise

The probability of error of the ML detector is equal to the probability that the received vector $\mathbf{y}$ is closer to $\mathbf{x}_1$ than to $\mathbf{x}_0$, which is equal to the probability that ...
Matt L.'s user avatar
  • 90.5k
1 vote
Accepted

Why Isn't the ML Estimator (MLE) in MIMO Spatial Multiplexing Obtained by the Least Squares Solution?

You must attention to the written text. It doesn't say the ML isn't optimal, what it says is that the problem isn't regular LS problem but Least squares problem with Constraints. The constraints ...
Royi's user avatar
  • 19.7k
1 vote

Maximum Likelihood for Colored Noise

First, I think that the expression $ A = h\star h\star c $ is not correct. Actually, $ A $ is a convolution matrix whose elements contains $ h\star h\star c $. The dimensions of $ A $ will depend on ...
JohnMarvin's user avatar
1 vote

Differences Using Maximum Likelihood or Maximum a Posteriori for Deconvolution / Deblur

You can think of MAP as a regularization of the ML. Just like you have regularization for Least Squares Problem (They can be built, mostly, as MAP problem). The nice thing is that, as always, the ...
Royi's user avatar
  • 19.7k
1 vote
Accepted

Notations to Use in Formulating of Maximum Likelihood Estimation

1) What notations to use for probability density function is it the one below: $\mathsf{P}_y(y_n|{\mathbf{u}_n})$ $\mathsf{P}_z(z_n|{\mathbf{u}_n})$ what goes in the subscript if I want to use $z$? ...
Atul Ingle's user avatar
  • 4,134
1 vote

Why Is The Maximum Likelihood Estimation (MLE) Method Taken as the Benchmark for Comparing with Other Methods?

There are 3 major reasons in my opinion which makes the Maximum Likelihood Estimator so popular: Intuition It is very clear what's the logic behind this method and what you maximize. It makes sense ...
Royi's user avatar
  • 19.7k

Only top scored, non community-wiki answers of a minimum length are eligible