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Write for me capstone project ucr online research methods for capstone for money plugged in movie review pan let's look at some Laplace smoothing examples our first example looks at Laplace smoothing for a single variable distribution we are given a set of samples from the single variable distribution the random variable is a which can take on three values a 1 a 2 a 3 and the sample that we've got them are over here we want to apply smoothing for K equals 1 okay what this means is that we're trying to find the probability for a equals a 1 the probability for a equals a 2 and the probability for a equals a 3 and we're going to find those probabilities we're going to estimate them from the sample data what maximum likelihood would do let's do that first here it would just take the counts so we have a 1 up here 1 2 3 times if it to appearing 1 2 3 times we have a 3 appearing 2 times so total of 8 samples and we get 3 over 8 for parallel for a 1 3 over 8 for a 2 and 2 over it for a 3 if we use a plus smoothing instead we will estimate this as we use Laplace moving instead well as means as follows we will take again the number of samples equal 1 so we have plus 1 and then in the numerator rather than 8 we'd have 8 plus and then it's the size of domain of a which has three values so with three times and K here is one we have four over 11 or ability of a being a 2 and the count is 3 we add 1 and then in the denominator we have a total of 8 samples and we add the number of different values the variable can take on which is 3 multiplied with K which is 1 giving us also for out of 11 the last one same story we have a count of 2 we add 1 because K equals 1 and denominator an 8 which is a total account plus 3 which is the size of the main of a times 1 and this gives us 3 out of 11 who unless we allow a smoothing is that it tends to draw your estimate of the probability distribution closer to the uniform distribution and the larger you pick K the closer you will be drawing it to the uniform distribution is that a good thing or bad thing that depends on the situation that you're facing but that's just what it does as a side effect or sometimes has a main purpose of drawing it closer to the uniform distribution you'll see that all your estimated probabilities will be nonzero let's look at another example here we have K equals to same random variable a with three possible values in the same set of samples so all that changes then K now equals two so the task remains the same we want to estimate our ability for a equals a1 equals a2 and a equals a3 are submitting from samples so it's going to be approximate where the estimated is by looking at the count with three times a1 and we add the pseudo counter consumer plus moving to and then we divide by all the number of examples 8 plus the size of the domain 3 times K which is 2 the next one a2 if also three appearances plus two from the K equals of to Laplace moving divided by 8 which is a total count of our number of samples plus three which is the domain size times two which is K and for the last one we have a count of two we have two then we divide by 8 which is a total number of samples plus 3 which is 2 the main size times 2 another way to look at this whole up plus smoothing business is to think of it as adding fake samples if we are doing Laplace smoothing with K equals 2 for each possible value of that variable can take on we add two samples plasma with K equals 2 is the same as saying oh I'm going to add a 1 twice I'm going to add a 2 twice and I'm going to add a 3 twice and then I'm just going to use the counts from this augmented set of samples as my way of finding the estimates of the probability entries let's now look at estimating a conditional distribution with Laplace mode here with conditional two here we have a conditional distribution for B given a and we use K equals 3 our domain for B is B 1 B 2 B 3 and our domain for a is a 1 a 2 so looking to estimate here is the initial probability of B given a 3 values for B 2 for a so we're looking as made 6 probabilities okay as I said in the previous slide Laplace smoothing is essentially adding fake samples to our set of real samples when we estimate a conditional probability distribution we're really estimating several probability distributions one probability distribution for each of the values a conditioning variable can take on so in this case the conditioning variable egg they found two values so really we're estimating two distributions one where a equals a1 1 where a equals a 2 for a equals a1 we have the following samples we have a 1 B 1 a 1 B 2 a 1 B 3 and again a 1 B 1 during the past moment with K equals 3 means that we are adding three instantiation of each value our random variable B here can take on this a is fixed we're conditioning on a so that means we are adding a 1 B 1 3 times we're adding a 1 B 2 3 times we are adding a 1 B 3 3 times now from this augmented set of samples we can estimate the laplace smooth estimate of the Missal meditative samples we can just use the council to get the laplace smooth estimates how often do we have B 1 1 2 3 4 5 times B 1 so B 1 if 5 times B 2 we have 1 2 3 4 times B 3 we have 1 2 3 4 times so we have that D initial distribution for B 1 given a equals a 1 is equal to 5 over 5 plus 4 plus 4 distribution for B the probability of B 2 given a equals a 1 is equal to 4 over 5 plus 4 plus 4 and then the probability for B 3 given a equals a 1 is equal to 4 over 5 plus 4 plus 4 now we can repeat this exercise for a 2 let's do that Sarah is this one here now let's look at the distribution for the case where we conditioned on a goes a 2 so Prioleau solution for be given a equals a 2 look at the samples for able Z 2 we have this one here and this one here just two samples so a 2 B 2 and another a 2 B 2 if we were to use maximum likelihood estimates we would get probability for B 2 given a equals a 2 under the maximum likelihood estimation gives us 2 over 2 which is 1 point 0 and the probability of any of the other and in sensations of B given that a equals a 2 would be 0 when we do Laplace moving we end up avoiding these zeros because we're adding fake samples for all possible instantiation of the random variable in this case B is the random variable a is being conditioned on so we add three samples where B equals B 1 so we get a 2 B 1 a 2 B 1 a 2 B 1 really the same thing for B equals B 2 and we do the same thing for B equals B 3 we now do our estimates if we need our estimates based on this augmented set of samples here we get the probability for B 2 given a equals a 2 is equal to B 2 appears 5 times 5 divided by the total number of samples which is 11 then for probability of B 3 given a equals a 2 we have B 3 appears three times total number of samples samples is 11 to 3 over 11 and then probability of B 1 given a equals a 2 a 1 appears three times so also 3 over 11 okay that's it for Laplace smoothing internet of things world 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