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In this lecture we are going to look at the code to demonstrate uncertainty estimation Using pi to which

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this lecture is going to walk you through a prepared code lab notebook.

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Although a very good exercise which I always recommend is once you know how this is done to try and

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recreate it yourself with as few references as possible as usual you can look at the title of the notebook

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to determine what notebook we are currently looking at.

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So let's begin by writing a function that generates data from it known synthetic distribution.

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This might seem weird but in fact this is exactly what we do when we run an experiment or do data collection.

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It's like we have some function called generate batch and that function is a black box which involves

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whatever data generating process happens to result in the data you're measuring.

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Of course the difference is that you don't ever know the true distribution of your data nor the underlying

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data generating process.

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And that's why we need machine learning.

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But for this example we do know the true distribution and the data generating process and hopefully

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that will give you some insight on what machine learning is trying to do.

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So what does this function look like.

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First we generate a bunch of random points for the input x from the uniform distribution between minus

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five and plus five.

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Next we write an expression for the standard deviation as a function of x.

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As you can see it's a linear function of x.

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So when x is small the standard deviation is small and when X is large the standard deviation is large.

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Next we get the target we assigned the target to be the cosine of X minus some linear function of x

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plus Gaussian noise center at zero with a standard deviation of SD.

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Then we return x and y

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if we plot the data you should be able to recognize each of these elements.

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First we can see the periodic component due to the cosine.

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That's why this thing appears like a wave.

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Second we can see that a trend downward.

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Thanks to the linear component.

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Finally we see hetero skid as this city.

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The noise seems to increase as X increases.

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In fact we know that the noise standard deviation increases linearly with X.

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Importantly we can see here that given the same X it's possible to get many different values for the

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target.

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Next we create our model.

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This is a pretty basic custom model which contains two simple antennas.

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The first CNN will be used to predicts the mean of the output and the second and then will be used to

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predict the log variance

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in the forward function.

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You can see how this is kind of the opposite of a recommender system.

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A recommender system takes in two inputs a user and an item and returns one output a predicted rating

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on the other hand.

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This takes in one input and produces two outputs.

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Next we instantiate the model and then we create our custom laws.

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So in this function we'll stick with the convention of calling the first argument outputs and the second

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argument targets inside the function.

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We know that outputs is a tuple with two components.

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So we take the first component and assign that to a variable called Mew.

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We take the second component exponential rate and assign that to a variable called V.

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Next we calculate the coefficient term of the Gaussian distribution.

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That's the log of the square root of two pi times V.

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Next we calculate the exponent term of the Gaussian distribution.

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That's zero point five divided by three times the square difference between the targets and Mew.

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Lastly we add the coefficient term and the exponent term together and take the mean.

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Next we have our optimizer and our training function.

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So this is nothing really new.

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All right.

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So here's our last per iteration.

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There should seem a little strange since what we're used to when we look at a loss per iteration is

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that the loss usually has a steady decrease downward.

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This on the other hand seems to be very noisy.

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So what is the cause of all this noise.

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Well remember that our data itself has noise and the loss function is the negative log likelihood.

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Remember that the loss doesn't only contain just the squared error anymore.

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So we shouldn't expect the loss to go down to zero since that would mean the variance is zero which

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we know is it true.

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Next we plot the model predictions in order to do this.

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We're going to generate a large batch of 1000 twenty four points after doing a scatter plot.

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We convert these points into torch is reshape them and pass them through our model.

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The model returns two things are y hats and the log variance.

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Since our model outputs the log of the variance we can exponential rate it to get the variance but that's

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not actually what we want.

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If you recall the variance doesn't have the same units as the original random variable so it doesn't

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make sense to plot them together.

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Rather we would like to plot the standard deviation which is the square root of the variance equivalently.

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We can just divide by 2 before taking the exponent since the square root is the same thing as a power

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1 1/2.

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Next we plot our prediction along with the corresponding uncertainty on top of our scatter plot of the

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data in order for a plot to look like it should we have the sort the data points first.

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This is because map plot lib joins the points you pass in by a line in whatever order you pass them

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in we can get the sword index by using the ARG sword function.

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Next we plot X versus Y.

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This should follow the center of the data points for each X.

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Next we use the fill between function to draw a transparent band with width equal to the standard deviation

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so that's why we have y hat minus SD and then y hat plus SD.

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So here's the plot and as you can see the band increases as we go from left to right and our model accurately

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predicts that the variance should increase as X increases.