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Save acceptor conditional

so in terms of parameter simulation will consider two cases of random parameters and non-random um but of course unknown because if it is known then what is this so here it is unknown so that Lewis for example affair so he submarine underwater right so the first job is to detect then it is estimation so whatever let's say one unknown is it's a speed or if you are trying to let's say that is propeller going on here my propeller motion and it's a fee you can say because of the the the parameters from here could be random or in communities in Arnolds in processing means the communication signal a binary communication but the the sequence is so random right at small so anyway there are two normal and that means it says that there is an unknown it says nothing bad about it simply or no that's all right so the constant is something good to parameter estimation later we can help you form estimation which will go towards the end of this course so instead of a parameter we may have a waveform itself is in both the cases so unknowns could be a time dependent function or no time dependent but both could be random or so this will be a stochastic process or it could be just a random variable here and here it is unknown here it is an unknown waveform so somebody is transmitting whoa there's nothing random about it we just don't know what the waveform is so it's coming with signal so estimating it so this is a classic problem right so there is noise then an unknown signal but in the case of separate the signal that is coming from the submarine could be is usually random because it's due to that a proper love noise generation this isn't that way whatever tiny noise so that waveform itself we said it's not an intelligent signal that will correspond to here but this will come and we need to do a little bit of stochastic processes so let me here let me deal with this problem today also again either way you have you collect data of course the design of experiment is such that that you collect relevant data so how so that part is done so you collect the data so the estimator is going to be a function of the data right this is the estimator so estimator will be always a function of data and because the data is in hardly Bostic astok whether it is here or here the estimator is going to be a random variable and in this case and either way what we want is this estimator which means you wanted to be close to the actual value so the difference you can take it as error or you can again bring in a cost I hope this is easy to understand this is the penalty or cost in estimating a as a hat it's estimated as a hat so there's a cost which will depend on it'll be cost is a function of two variables there are no and your estimator now if you so one common cost is usually you take the error so one example is where you take the error I'll come to that in a second and then you minimize the average cost by chosing so the average cost is going to be expected value of C so that is C a multiplied by the density function of a and E but hey that is a function of all so this integral is on all of the white or Indian variables on off and then a a is our random variable so this is a enfold integral and we can also write this as so I can use the conditional density function and write this as in this fashion a hat is a function of R so I don't really need all here this is this is because they joined insignificance a moment this is a function of all so instead of writing like this is a joint density function of a and of this is a function of all right so the only randomness series from all the randomness is here from a so that's the joint density function and which I can also write it like this and if you want you can write this as so you can see the action is all here so if you if you want you get the what about minimization need to be done it need to be done at this stage because it's on this cost all right so we can concentrate on this function if you want to do the minimization on how to select the a hat so the whole idea is minimize this conditional variance about this conditional variance will depend on the cost function so traditionally so if you define error to be a - a hat one common cost is the mean squared error right so we'll use this is one cost function so this is known as this approach is known as and then we have to minimize it so this is which stands for minimum mean square error cry error [Music] many ways the mean square error and that this another cost function is just the absolute value of the other which will be a minus a so we'll see what that leads to and the third one is and all a cost function like this and so cost is in terms of the error but three different cost functions so I hope you can see this what it says here if the error is small there is no cost and if the error is beyond a delta then the penalty is fine so this is my cure all-or-nothing cost so let's quickly see what they what the estimators turn out to be in these three cases so we can deal with down here but that's the whole point so if I do do the Sigma is code which is a function of all so let's see what the MMSE estimator is so Sigma is 1 so I'm simply going to call it Sigma squared outright is going to be integral I am going to do this one so this is a quadratic error ei minus a hat it x if a given or B but look here R is given so this is no longer a random squared right this is distance Justin or not because the MOR is given at this point [Music] equal to zero so when you remember this is sort of it this is a constant because R is given so this leads to a head multiplied by a given all be equal to integral of a of a so what is the area this is the area under a density function so this is one so we get the unknown to be simply the conditional mean of the a what are the tossing result right so you had you get a hat MMSE equal to expected value of a given so the only catch is you need to know that density function what is your question what is your condition well I'm sorry you can see the action is actually the Hat is only coming here right this is not going to do anything to you and so it's just a deal we have too many too many unknowns here yeah is unknown obvious so we are degree of difficulty is broad warning that it or you only deal with the assumptions the assumption is a right here so this is the classic result innovation of the mean squared error leads to leads to the conditional mean of the unknown given the data so I'm going to just write the result here and then you want to be arrested so he had MMSE is a I mean conditional mean of a given or is the whole data and if you want to find out the cost associated with that so you take this estimator and put it into the previous expression so if you recall the cost turned out to be expected value of the cost of cost was was too awesome we are dealing with quadratic so this is double integral a minus expected value of a given or this is our best estimator squared multiplied by the conditional density function of a given R so this is the conditional variance of a or x f4 so the cost will be if you want an expression so that just the expected so that would be the minimum mean squared error in the signal suitcase oh but this is the look at here this is the conditional mean is the best estimator so you plug it in back then this is the main - variance density function that they expect that's the variance but condition now multiplied by the density function that will be expected where this right so let us look at the second one maybe let me see how far he can do here so here the cost will be since we're dealing with the three problems this is the first one so remember what this is this is the minimization of the mean squared leads to this as the best estimator it's a classic result only problem here is you just need to know the density function so sometimes this density function is not known and if this is not known you can't do it so here of course it is integral of a - a hat a multiplied by a so a and remember I'm just taking this inner integral here so in so the quadratic I am going to put this error function look here this is absolute value of B here are these Aurora's are not - in the estimator so this now I can write this as minus infinity to a head or a - room body variable is a so a minus a this need to be positive so in this region this is positive so I don't need the positive sign if a given all plus a hat or to plus infinity this is absolute value so this will be a hat - a F now we want to minimize this so minimize session means this Sigma squared you take the derivative of this quantity with respect to the unknown they are notice again a hat so we can use the same principle via date so remember here the unknown recently offers a level of the integral so this is the derivative of this which is 1 then you substitute the derivative in do into memory a is appearing okay yeah all right so that's why I am showing you a rather than heat here it's just this one what does it mean they whatever is this error daughter is this quantity should be positive right so one way to realize this is nobody a is the variable a hat is what it is yes so you can here we'll go from minus infinity to plus infinity so initially initially so think of it this way air moves this way a hat is somewhere here so first first let me integrate this VJ minus infinity to so we have a had in this region right so I would like so maybe this is what it should be the other way this is a head - a I don't know whether that's what you would let me finish it by your argument this should be positive right whatever is this quantity so there's only two cases yes if a succeeding a head is if this quantity is here - a head if it's not a head there should be a head - name so in this region each one is lottery a hat is larger because there is somewhere here right so I'm going to write it as a hat - a in this region which is a sequin is here is longer than a head so I'll write it as a minus a and so this expression is exactly the same as this except this is different this I can differentiate rather than jump in here with the sine function and all that stuff right so the derivative of this is now the derivative this is generative the top limit that's 1 then you substitute into this wherever the a is appearing so that will be 0 here yes the review of the bottom limit 0 so there is no contribution term is leave the limits as they are take the derivative of this with respect to your variable a hat so what's the derivative of this with respect to your head what yeah this is what so just f of a given B F now we go to this and do exactly the same thing derivative of the top limit 0 minus derivative of the bottom limit which is 1 but then when you substitute the bottom limit into this that's 0 so the third term is integral of integral stays the same the derivative of this quantity with respect to a head what do you get - F of a given hour I put the - outside this should be equal to zero so we get the condition we get the condition that you have minus infinity to a hat f of a given R b/a should be equal to a head to infinity F of a given all so if you have the density function you can see if the density function is like this f of a given R what it says is that the area up to a hat if a hat is here this area this area must be equal so how much is each part 1/2 what do you call such a point anybody and by a point on the x-axis where the two areas are equal media so the simple answer is a head turns out to be the median I mean it is robust because see if the if a signal is like this you transmit one right and with the noise suppose it becomes like this etc the median so in other words some values are small some values are large and if you rearrange and look at the median it will turn out to be this one right because look at these I just eyeballed it right median is not the middle value but if you rearrange like this so BD will be this which corresponds to this right so the idea is that small noise is not going to disturb and the D values will be the that's one also if you have a perfect image that you have pepper noise you do a median filter on small blocks you can get up the tiny values because it just goes for the sum so if you bought a society if you want to see what's the it's always the median maybe not the extremely poor not extremely rich but if you arrange the income or something in their level what is so this value it will be the median of F of this density function here this turns out to be the beam right so this is the conditional mean of no more conditional mean of a given are ever thing is of data is the king or the queen whatsoever ever or the intensity function is modified by the data and you look at the median turns out to be the best estimator for mean squared error and if you take the absolute value the median is the best estimate alright so let's go to the third one so they here the cost function is like this so you have error which is a minus a hat and the cost function is not descriptive but again nonlinear cost [Music] so this is where we are right from the previous expression I take this over here I take this is the cost function and if you so I take this the inner integral this is what we want to minimize over a hat so what is the best thing that's the customer safety so look a if or if if this one thing was born what would be the value of this integral anybody suppose the scene II was just a line it'll be one point so you can write this as 1 minus integral from right because this is this wall so I can write this as this minus this right so this is going to be area under the density function is 1 so it's simply yes so this cost should be minimized that means this should be maximized might so if the density function is like this so suppose this is the density function of a given R then here you have to think physically I want to maximize this quantity my what is my only option my only option is remember this is on epsilon so epsilon is a minus a hat so translating this to I remember this is a so the limits will be what epsilon is a minus a hat so epsilon can go from minus Delta 2 plus Delta so what will be relieved it's only here anybody yeah yeah so limit Sonia rather will be a had plus Delta and minus Delta so we are known as a hat nobody so your sweet career you are moving this window along a the question is where should I put the a hat so I can put a hat anywhere I like where should I put a head so that the this region the area under that this area is Maksim should I put it here or should I put it here so the answer is obvious if you are moving this a hat this a hat should be put where this apostrophe D of Peaks so this is called the maximum a posteriori estimate so this is maximized by so the correct choice for a hat will be maximum so there's so this is known as math ma P so this is map estimator so it's a simple idea so the to summarize the map estimator is one so everything is everything is based on the conditional the modified density function of the unknown parameter given the data so you may have been you have been because it's a random variability as they oppose a priori density function then you collect the data then you have of course the a posteriori density function of the same random variable but modified and data the data itself is random so you need to obviously figure out how to but this is what we are using not this and then it's it's a peak corresponds to the map estimator that's what we are so physically you can see the of course if it it could have you know it need not be unimodal can have multiple peaks so even if the density function is like this this will be anything unless these two peaks are equal then you can take them with whatever probabilities like also maximum value of f of a given all is the map estimate so but this you can write it as are you one a want to print by FA you will f off so instead of taking the maximum value of this you can take the maximum value of the logarithm so this comes out to be log of so to get a hat of map if the density function is differentiable and so on you can do this derivative of the log of F a given with respect to a which is the same as instead of doing that you can also do it relative the log of F or you need if the totem is irrelevant because there is no a here and then you equate this to 0 at a equal to a hat man and you so put this where you know the cost function that he can compute the minimum so I gave you three three different estimation procedure for for random parameters and just to do a problem so for example get this always are poison with the parameter lambda but let's say land right surface exponentially it's some parameter V so lambda is a random variable so the density function of RI equal to K given lambda is e raised to minus lambda learn back to the power K over K effectively so that's this part here I mean this is just one observation if you have multiple observations and a given lambda if they are independent then this becomes the product except alright so if you say if you go to that step so the total is going to be product of P of all right even carry but given lambda so this will be the product of e raised to minus lambda lambda I to the power K lambda to the power K I over K effective but it doesn't change in a different so you just see either one is fine so this is e raised to minus lambda so that's the first spot and then we can take its log of them so to find the map estimator I am saying I can here do this but this is usually hard so we find that look at what I'm doing the given data conditional density function of are given the parameter x so let me take this so that that part I did so as when you take the logarithm of this it's going to be the logarithm of this logarithm of this so that's going to be minus n lambda plus Sigma K I log lambda minus log K a so then then I need to add B to that I need to add this portion logarithm of the density function of lambda so lambda is given to be f lambda is given to be erased - it's a 1 over mu e raised to minus lambda mu B that's what we mean by a so when I take it logarithm log of F lambda is going to be minus log of U minus lambda over me so I am going to simply add it here I'm going to add the second term which is log of F lambda it's going to be minus log mu minus lambda over me now I need to take the derivative of this with respect to what what is the parameter here is lambda right so I need to take the derivative of this whole thing with respect to lambda and try to save some space all right so that if I do this here is going to be minus n plus Sigma ki for lambda but minus 1 over mu equal to 0 so you can solve for lambda so lambda is going to be so you have Sigma ki over lambda is in mu plus 1 over mu right so lambda hat math is okay and remember all the excise of conditionally poison so this is the summer voice oh nice again conditionally poison with parameter in lambda acceptor so this is the this is the map estimator for