Dear Ahmad,

  The answer contains almost nothing to do with x, H, s, n. Most people stick with what they are comfortable with, so statistical preferences often follow from exposure to mentors and colleagues who have their own history and comfort with various approaches.

   In my case, being almost equally uncomfortable either way, I would choose the technique based on what sort of answer I wish to have. Bayesian methods are about more than just priors; differences in evaluation can lead to different conclusions depending on the relationship between H and s. The way you specify and evaluate a model controls what you can reasonably conclude about the result. This is true for all methods... A frequentist approach has its own pitfalls.

   Depending on what you believe about H, the differences in method may be almost irrelevant (like if H approaches I, or 0). The differences may be large; they may even be irrelevant (like if H has underlying chaotic structure and both methods make different errors). No magic solutions on either side, and sometimes neither approach is the correct one.

Just my opinion (actual Bayesian practitioners will tend to disagree).

Dear Ahmad,

  The answer contains almost nothing to do with x, H, s, n. Most people stick with what they are comfortable with, so statistical preferences often follow from exposure to mentors and colleagues who have their own history and comfort with various approaches.

   In my case, being almost equally uncomfortable either way, I would choose the technique based on what sort of answer I wish to have. Bayesian methods are about more than just priors; differences in evaluation can lead to different conclusions depending on the relationship between H and s. The way you specify and evaluate a model controls what you can reasonably conclude about the result. This is true for all methods... A frequentist approach has its own pitfalls.

   Depending on what you believe about H, the differences in method may be almost irrelevant (like if H approaches I, or 0). The differences may be large; they may even be irrelevant (like if H has underlying chaotic structure and both methods make different errors). No magic solutions on either side, and sometimes neither approach is the correct one.

Just my opinion (actual Bayesian practitioners will tend to disagree).

The information symbol s is deterministic. However, H depends on your transmission scenario. The noise is randomly added to tge signal.

Dear Ahmad

As far as I understand from your statement..

If we consider this as a communication problem then the approach is generally to estimate noise and impulse response via experimental setup through the particular medium. However, the same may be assumed statistically based on previous well established results for the same medium.

Could you please explain what you mean by Bayesian and deterministic approach ?

I think it can be approached in both, the noise involve in the model can make the stochastic approach more desirable in this case.

I am not so sure about what I answer. What I think is as follows:

Suppose the environment is stationary. That is, the channel conditions are not varying. So, the impulse response of the channel is deterministic.

But suppose the environment is non-stationary. For example, a person in a moving car uses internet or makes a mobile call. In this case the environment is continuously varying. So, channel conditions also change. The impulse response does not remain same. So, in this situation, one has to think statistically.

I request the experts to check the validity of the answer. I may be wrong.

Dear Ahmad,

Due to the noise term "n" in your linear model, the problem necessitates a stochastic (probabilistic) approach. As far as stochastic methods are concerned, there are Bayesian and nonBayesian methods. NonBayesian methods aim for reducing the variance of the estimator while keeping it unbiased which is called minimum variance unbiased (MVU) criterion. However, Bayesian methods make some prior assumtions on the data, and given that prior information, they aim for minimizing a cost function (such as Bayes risk). Both classes of methods have advantages and disadvantages (which require a long discussion).

On the other hand, because of the noise term, it is not advisable to solve that problem in a deterministic framework.

Best wishes,

Bayesian

I agree what Bruce have said above. Your exposure to advisers and others, affect your choice.  In addition the type of problems and scenarios you are involved in can also determine which framework to adhere too. 

For me it was simple. I had an adviser who did excellent work with the Bayesian framework, since then I became advocate of the field :)

Actually our H was a random matrix usually and that is why also  we stick to Bayesian.

God luck!

The Bayesian approach is general and allows for easy inclusion of  "non Gaussian" noise (e.g. for the estimation of the errors on the results) and for the case of non stationary H. It allows for a robust solution: suppose H is a regression to be applied to the result of a medical research: rarely you can suppose your noise is Gaussian and the system stationary. The Bayesian approach is mandatory. It also allows for regularization of the problem in case of H is not full rank.  So, in a particularly simple cases, you can use the deterministic approach while in more complicated case the Bayesian is very general and, let me add the remark, pretty intuitive. In general, to minimize the investment of study, it is better to go to the Bayesian to have a more powerful tool.

I agree with Olcay Akay. It means you have a linear model with some random elements and you need some statistical approach to estimate parameters of the model, and it can be Bayes estimators, or Minimum Variance Estimators, or Robust Estimators, and so on. For sure, I cannot be deterministic.

THIS  IS USUALLY THE  (MAINLY  INFORMATICS)  ENGINEER DILEMMA WHEN HE HAS TO TAKE INTO ACCOUNT AN "UNTAMED" NOISE. INEVITABLY THE PROBLEM  BECOMES  BAYESIAN. IF ONE CAN TAME THE NOISE (IN THE HARDWARE SENSE) THEN IT BECOMES DETERMINISTIC.

AFTER ALL WE ALL NOW KNOW THAN NOT EVEN RANDOMNESS IS PURELY RANDOM BUT  ALSO THERE IS NOTHING 100% DETERMINISTICIN OUR REALITY!

since n is random

Ok , this is a cliche !

Noise sources are many (in transmission lines) and accountability for the sources is a necessity. Source noises , line and detector  hence deterministic ...

Receiver designs - say BER depends on probability (SNR /  Modulation techniques ( ON-OFF keying say) ) then may be Bayesian to estimate as close as possible to the actual values so that bandwidth is better utilized, Q-factors ...

Hi,

Bayesian/ statistical models will work for noise added cases. The noise is random in nature. For random signals, how deterministic one will give good results...

Thanks & Regards,

Sumaiya.

The deterministic model usually provides a simpler solution for detection, estimation problem such as the GLRT detector and Cramer-Rao Bound. However, the bayesian may not.

The Bayesian framework may provide a better ability of parameter identification because it assumes the signal is a random, and thus only provide a range of the parameters, not a  point.  This property means that for Bayesian model, we just need fewer samples to estimate unknown parameters than the deterministic one, in order to make all the parameters identifiable. An good example of this is that the Fisher matrix for deterministic unknown parameter is not invertible, while for bayesian model it can. This means the Bayesian model is better for parameter identification.

Bayesian model take fewer samples to estimate the required parameters

Dear Mr. Bruce, 

I got your point, and I like it.

Thank you for your help.

Dear Mr. Mohamed El-Tolba,

You say "The symbol s is deterministic,".. Ok, this is when you take a deterministic approach. But you allow it to be random when you choose to be a Bayesian.

Thanks for your help.

Dear Mr. Imran,

Bayesian is when you choose H and s to be random and you try to estimate some of their statistical characteristics. 

Deterministic is when you say "Oh wait, H and s are not random, but they are unknowns*.

Thanks for your help.

Dear Mr. Elisha,

Thanks for your help.

Hi Mr. Rajeesh,

I see your point there.

Thanks for your help.

Hi Mr. Olcay,

What you say is very clear. I like it. 

But could you emphasise on why is it not advisable to use deterministic when the noise term is present.

Thanks a lot for the help.

Hi Mr. Mario,

Could you elaborate ?

Hi Mr. Solomon,

I like what you just said :-) 

Indeed, it makes sense to choose a Bayesian framework if H is a random matrix.

Thanks for your help.

Hi Mr. Gaetano,

Thank you for your help.

Hi Mr. Ramon,

Thank you for your helpful comment.

Hi Mr. Dimitri,

Thanks for your help.

Hi Mr. Thorston,

Thanks for your help.

Hi Ms. Aparna,

Thanks for your help.

Hi Sumaiya,

Thanks for your help.

Hi Mr. Yikai,

Thank you for your helpful comment.

Hi Mr. Amr,

Thank you for your helpful comment.

Dear Ahmad,

I will try my best to answer your question "why is it not advisable to use deterministic when the noise term is present?".

Noise is, by its very nature, unpredictable. For example, samples of white noise are totally uncorrelated with each other. Hence, observing past noise samples is of no help for predicting its future samples. 

On the other hand, by definition, behavior of deterministic signals, processes, algorithms, etc, is entirely determined by their initial states and inputs. Hence, they cannot be considered as random or stochastic. A deterministic algorithm, for example, if given the same input information, will always produce the same output information.

We can surmise from the above two paragraphs that the terms "noise" and "deterministic" can possibly be considered as oxymorons. Noise is not deterministic and anytyhing that is deterministic cannot contain noise. Therefore, noise-corrupted signals should be dealt with stochastic-based techniques such as Bayesian methods and deterministic methods should solely be reserved for deterministic processes. 

Best wishes,