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Maximum Likelihood For the Normal Distribution, step-by-step!!! — Transcript · YTtoTranscript

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Maximum Likelihood For the Normal Distribution, step-by-step!!!

StatQuest with Josh Starmer · 2,484 words · 12 min read · EN

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Below is the complete, readable transcript of Maximum Likelihood For the Normal Distribution, step-by-step!!! by StatQuest with Josh Starmer on YouTube. Read the full text, copy any part you need, or generate a transcript for any video with our free tool.

We're gonna do a lot of math step by step by step by step by step StatQuest!!! Hello, I'm Josh Starmer and welcome to StatQuest Today we're going to talk about maximum likelihood for the normal distribution and it's gonna be clearly explained Not: This StatQuest follows up on the StatQuest "Maximum Likelihood Clearly Explained"

as well as the StatQuest "Probability Versus Likelihood" Lastly this StatQuest assumes you are already familiar with the normal distribution If not, check out the StatQuest "The Normal Distribution Clearly Explained" Let's start with this nasty-looking equation It's the equation for the normal distribution or normal curve It has two parameters the first parameter, the Greek character μ (mu)

determines the location of the normal distribution's mean a smaller value for μ moves the mean of the distribution to the left and a larger value for μ moves the mean of the distribution to the right the second parameter, the Greek character σ (sigma), is the standard deviation and determines the normal distribution's width

a larger value for σ makes the normal curve shorter and wider and a smaller value for σ makes the normal curve taller and narrower In this StatQuest we're going to use the likelihood of the normal distribution to find the optimal parameters for μ the mean and σ, the standard deviation, given some data x

Let's start with the simplest data set of all: a single measurement Here we've measured a single Mouse and it weighs 32 grams. Now just to see what happens We can overlay a normal distribution with μ equals 28 and σ equals 2 onto the data To determine the likelihood of the data given this curve we can plug the numbers into the likelihood function

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and then plug the numbers into this equation and here's the equation with μ equals 28 σ equals 2 and x equals 32 plugged in Now just do the math plug and chug plug and chug and the likelihood of the curve with μ equals 30 and σ equals 2, given the data, is 0.03

thus the y-axis value here is 0.03 Now we can shift the distribution a little bit to the right by setting μ equals 30 and then calculate the likelihood Again we just plug the numbers into the likelihood function and now we just plug and chug and the likelihood of the new curve with μ equals 30 and σ equals 2, given the data, is 0.12

Thus the y-axis value here is 0.12 If we decide to fix σ equals 2 so that it is a given just like the data Then we can plug in a whole bunch of values for μ and see which one gives the maximum likelihood For example, if we start with the mean of the distribution over here on the left at 20 grams

then we plug μ equals 20 into the equation and we get a crazy small likelihood and Then we can plot the likelihood on a graph The y-axis is the likelihood value and the x-axis is for the different values that we plug in for μ Each time we change μ we calculate the likelihood and plot it

We can identify the peak in the likelihood graph by determining where the slope of the curve equals zero in this case the slope equals zero when μ equals 32 Now we can fix μ equals 32 and treat it like a given just like the data And we can plug in different values for σ to find the one that gets the maximum likelihood

Note: You actually need more than one measurement to find the optimal value for σ I'll leave the reason why as an exercise for the viewer That said, if we had more data Then we could plot the likelihoods for different values of σ And the maximum likelihood estimate for σ would be at the peak, where a slope of the curve equals zero

The goal of this super simple example is to convey the basic concepts of how to find the maximum likelihood estimates for μ and σ To solve for the maximum likelihood estimate for μ We treat σ like it's a constant and then find where the slope of its likelihood function is 0 And to solve for the maximum likelihood estimate for σ

we treat μ like it's a constant and then find where the slope of its likelihood function is 0 The example with one measurement kept the math simple, but now I think we're ready to dive in a little deeper So let's use a two sample data set to calculate the likelihood of a normal distribution

Crazy Times! To keep track of things, let's call the mouse that weighs 32 grams X sub 1 And the mouse that weighs 34 grams X sub 2 And again, just to see what happens Let's overlay a normal distribution with μ equals 28 and σ equals 2 onto the data We've already seen how to calculate the likelihood for this curve given X sub 1,

the mouse that weighs 32 grams And we can calculate the likelihood for the curve given X sub 2 by plugging in 34 into this likelihood function but what's the likelihood of this normal curve given both X sub 1 and X sub 2 In other words, what's this? Because these measurements are independent (i.e. weighing X sub 1 did not have an effect on weighing X sub 2)

The likelihood of a normal distribution with μ equals 28 and σ equals 2 given the data X sub 1 equals 32 and X sub 2 equals 34 is just the likelihood of the distribution given X sub 1 times the likelihood of the distribution given X sub 2 So we just plug in the numbers and do the math

And that gives us a really small number If we had a third data point then we just add it to the given side of the overall likelihood and Add another individual likelihood term to our multiplication With n data points We add all n data points to the given side of the overall likelihood function

Then multiply together all n individual likelihood functions BAM!!! Now that we know how to calculate the likelihood of a normal distribution when we have more than one measurement (we just multiply together the individual likelihoods). Let's solve for the maximum likelihood estimates for μ and σ BRACE YOURSELF FOR A LOT OF MATH!!! But don't freak out,

we'll go through it one step at a time Here's the likelihood function without any value specified for μ and σ it equals the product of the likelihood functions for the N individual measurements and here's what the equation looks like What we need to do is take two different derivatives of this equation One derivative will be with respect to μ. When we treat σ like it's a constant

and we can find the maximum likelihood estimate for μ by finding where this derivative equals zero the other derivative will be with respect to σ when we treat μ like it's a constant and we can find the maximum likelihood estimate for σ by finding where this derivative equals zero But before we try to take any derivatives, let's take the log of the likelihood function

We do this because it makes taking the derivative way way easier In the likelihood function and the log of the likelihood function both peak at the same values for μ and σ. I know it's hard to tell that the log likelihood function for σ peeks here, but just take my word for it. I double-checked!

Here's the log of the likelihood Now we're going to go, step by step, through all of the transformations that the log has on this function Just a heads up - the log is going to do a lot of transformations to this function, so hang in there!!! First the log transforms the multiplication into addition

Now we log transform the individual likelihood functions Let's focus on this one first Step 1: Move the log of the first likelihood function to the top left-hand side of the screen for reference Step 2: Convert the multiplication into addition Step 3: Convert 1 over the square root into the exponent -1/2 and convert the exponent

into multiplication Step 4: Convert the -1/2 exponent into multiplication and the ln(e)=1 So we can remove it Step 5: The log can convert the multiplication of 2πσ² into addition of two times π and σ squared and there's nothing more to do to this term so it doesn't change Step 6: Convert the exponent in the log of σ squared into 2 times the log of σ

Step 7: Lastly, the 2/2 term cancels out BAM!!! We started with this and after the log transformation ended up with this!!! I know all of this log transformation stuff looks bonkers the first time you see it Just remember, it will make it easier to take the derivative. So now we can go back to the original likelihood and turn this

into this And by following the same steps, we can transform the remaining parts of the sum into this Now let's simplify this equation Just to be clear about how we simplify, keep in mind that since we have n data points that means we have a term for the first data point, X sub 1

and that this represents the terms for the remaining n minus 1 data points. So we can combine this from the first term with the remaining n minus 1 occurrences. 1 plus n minus 1 equals n Then all n of the negative log of σ's can be combined and the last parts of each term stay the same.

This is the log of the likelihood function after simplification, and it is what we will take the derivative of. So, let's move it to the top for reference We'll start by taking the derivative with respect to μ This derivative is the slope function for the log of the likelihood curve and we'll use it to find the peak

a.k.a where the slope equals zero The first term doesn't contain μ, so it's derivative is 0 The second term doesn't contain μ either, so it's derivative is also 0 The third term contains μ, so now we have to work Specifically, the numerator contains μ and we have to apply The Chain Rule So we start with the outer derivative of negative 32 minus μ squared

and then we multiply that by the derivative of 32 minus μ and then we simplify Because the derivative is with respect to μ, σ is a constant and, thus, the denominator doesn't change. So we can combine the two pieces and the 2s cancel out and that means the derivative of this is this

Likewise, we apply The Change Rule To the remaining terms and get these! BAM!!! Now we can simplify the equation These zeros go away and we can pull the σ squared out and add the numerators together. Then we can combine the measurements and , lastly, we can combine the μs Thus, this is the derivative of the log-likelihood function with respect to μ

DOUBLE BAM!!! Now, let's take the derivative of the log-likelihood function with respect to σ. This derivative is the slope function for the log of the likelihood curve and we'll use it to find the peak, which is hard to see in this graph! So, from here on out, because they peak at the same spot

I'll show you the likelihood functions instead of the log-likelihood functions The first term doesn't contain σ, so it's derivative is zero. The derivative of the second term is just n over σ. The derivative of this term isn't tricky but it's easier to figure out when we rewrite 1 over σ squared as σ to the negative 2

Since we're taking the derivative with respect to σ, this part can be treated like a big constant. and the derivative of this is this. The two negatives cancel out and then the 2s cancel out and lastly we put σ back into the denominator by changing the sign of the exponent So the derivative of this

is this! So that's what we put into the equation. Likewise, the derivatives of the remaining terms simplify to these. BAM!!! This is the derivative of the log-likelihood function with respect to σ. We can simplify the equation by omitting the zero leaving the second term the same and we can pull the σ cubed out

and add up the remaining terms. DOUBLE BAM!!! At long last here are the two derivatives!!! To find the maximum likelihood estimate for μ we need to solve for where the derivative with respect to μ equals zero because the slope is zero at the peak of the curve Likewise to find the maximum likelihood estimate for σ

we need to solve for where the derivative with respect to σ equals zero because the slope is zero at the peak of the curve We'll start by setting the derivative with respect to μ to 0 and solve for μ We start by multiplying both sides by σ squared, that makes the σ squared go away

then we add n times μ to both sides divide both sides by n and solve. Thus, the maximum likelihood estimate for μ is the mean of the measurements. So that is where the center of our normal curve will go. DON'T GET TOO EXCITED. WE'RE NOT DONE YET!!! DANG!!! Now we need to set the derivative with respect to σ to 0

Now multiply both sides by σ add n to both sides multiply both sides by σ squared divide both sides by n and take the square root of both sides and at long last we see that the maximum likelihood estimate for σ is the standard deviation of the measurements Thus, we use the formula for the standard deviation to determine the width of the normal curve that

given the data, has the maximum likelihood In summary The mean of the data is the maximum likelihood estimate for where the center of the normal distribution should go and the standard deviation of the data is the maximum likelihood estimate of how wide the normal curve should be Note: These solutions may be obvious, but now we have the math that proves that our intuition is correct.

wait for it TRIPLE BAM! Hooray! We've made it to the end of another exciting StatQuest. If you like this StatQuest and want to see more of them please subscribe, and if you want to support StatQuest, well, please consider buying one or two of my original songs Alright, until next time QUEST ON!!!