Goodness of Maximum Likelihood Estimator for linear regression

Summary

The script delves into the concept of linear regression, discussing both the optimal solution and the probabilistic perspective. It highlights the equivalence of the maximum likelihood estimator to the solution derived from linear regression under the assumption of Gaussian noise. The focus then shifts to evaluating the estimator's performance, introducing the idea of expected deviation between the estimated and true parameters. The script concludes by emphasizing the impact of noise variance and feature relationships on the estimator's quality, inviting further exploration into improving the estimator's accuracy.

Takeaways

• 📚 The script discusses the problem of linear regression and the derivation of the optimal solution W*.
• 🧩 It explains the probabilistic viewpoint of linear regression, where y given x is a result of W transpose x plus Gaussian noise.
• 🔍 The script highlights that the W* from linear regression is the same as the maximum likelihood estimator for W, denoted as ŴML.
• 🤖 The maximum likelihood estimator ŴML is shown to be equivalent to assuming a squared error in the noise, which is a Gaussian assumption.
• 🔑 The script introduces the concept of using the estimator ŴML to understand its properties and gain insights into the problem.
• 📉 The importance of estimating the true W is emphasized, and the script discusses how to measure the 'goodness' of ŴML as an estimator.
• 📊 The script explains that ŴML is a random variable due to the randomness in y, and thus its estimation quality can vary.
• 📈 The concept of expected value is used to understand the average performance of ŴML in estimating W, considering the randomness in y.
• 📐 The script introduces the Euclidean norm squared as a measure to quantify the deviation between the estimated ŴML and the true W.
• 📘 It is mentioned that the average deviation of ŴML from the true W is proportional to the variance of the noise (sigma squared) and the trace of the inverse of the covariance matrix of the features.
• 🔄 The script suggests that while we cannot change the noise variance, we might be able to improve the estimator by considering the relationship between features.

Q & A

What is the problem of linear regression discussed in the script?

The script discusses the problem of linear regression, focusing on deriving the optimal solution W* for the model and exploring the probabilistic viewpoint of linear regression where y given x is modeled as W transpose x plus noise.

What is the relationship between W* from linear regression and ŴML from maximum likelihood estimation?

The script explains that W* derived from linear regression is exactly the same as ŴML, which is derived from the maximum likelihood estimation, given the assumption of Gaussian noise.

Why is it beneficial to view the linear regression problem from a probabilistic perspective?

Viewing linear regression from a probabilistic perspective allows for the derivation of the maximum likelihood estimator for W, which provides insights into the properties of the estimator and helps in understanding the problem itself.

What does the script suggest about the properties of the maximum likelihood estimator ŴML?

The script suggests that ŴML has nice properties that can provide insights into extending the linear regression problem, particularly in terms of how good it is as an estimator for the true W.

What is the significance of assuming Gaussian noise in the linear regression model?

Assuming Gaussian noise is equivalent to assuming that the error is squared error, which simplifies the derivation of the maximum likelihood estimator and leads to the same solution as the optimal solution from linear regression.

How is the quality of the estimator ŴML measured in the script?

The quality of ŴML is measured by looking at the expected value of the norm squared difference between ŴML and the true W, which is derived from considering the randomness in y.

What does the script imply about the randomness of y and its impact on ŴML?

The script implies that since y is a random variable, ŴML, which is derived from y, is also random. This randomness affects the quality of ŴML as an estimator for the true W.

What is the expected value of the norm squared difference between ŴML and the true W?

The expected value of the norm squared difference is given by sigma squared times the trace of the matrix (XX^T) pseudo-inverse or inverse, indicating the average deviation of ŴML from the true W.

How does the variance of the Gaussian noise (sigma squared) affect the quality of ŴML?

The variance of the Gaussian noise directly affects the quality of ŴML, as a larger variance means more noise is added to W transpose X, leading to a larger average deviation between ŴML and the true W.

What role do the features in the data play in the quality of ŴML?

The features in the data, represented by the trace of the matrix (XX^T) inverse, also affect the quality of ŴML. The relationship between the features influences the estimator's goodness.

Can the quality of ŴML be improved by manipulating the features in the data?

The script suggests that while we cannot change the variance of the noise (sigma squared), we might be able to improve the quality of ŴML by manipulating the features in the data, although it does not specify how this can be achieved.

Outlines

00:00

📚 Introduction to Linear Regression and Estimation

This paragraph introduces the concepts of linear regression and the derivation of the optimal solution W*. It discusses the probabilistic perspective of linear regression, where y is expressed as W transpose x plus noise. The maximum likelihood estimator for w is derived, and it is shown that this estimator is equivalent to the solution obtained from linear regression. The paragraph also touches on the assumption of Gaussian noise and its implications on the error estimation, setting the stage for further exploration of the estimator's properties.

05:02

🔍 Evaluating the Estimator's Performance

The second paragraph delves into evaluating the performance of the maximum likelihood estimator (W hat ML) for W. It discusses the concept of measuring the estimator's accuracy by examining the norm squared difference between W hat ML and the true W. The importance of considering the average performance over multiple datasets due to the randomness of y is emphasized. The paragraph introduces the idea of calculating the expected value of this norm squared difference to understand the estimator's average deviation from the true W, highlighting the need for a deeper understanding of the estimator's properties.

10:06

📉 The Impact of Noise and Data Features on Estimation

This paragraph explores how the variance of the Gaussian noise (sigma squared) and the relationship between data features affect the estimator's performance. It explains that the average deviation of W hat ML from the true W is influenced by both the noise variance and the trace of the inverse of the covariance matrix of the features. The paragraph suggests that while the noise variance is a given factor, the structure of the features in the dataset could potentially be manipulated to improve the estimator's performance, although it acknowledges that this is a complex and uncertain proposition.

Keywords

💡Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In the context of the video, it discusses deriving the optimal solution W star for linear regression, which is the set of parameters that best fits the data according to the least squares method.

💡Probabilistic Viewpoint

The probabilistic viewpoint in statistics involves considering the likelihood of observing data given a model. In the script, this concept is applied to linear regression by assuming that the dependent variable y, given the independent variables x, is normally distributed around the linear relationship defined by W transpose x, with some variance sigma squared.

💡Maximum Likelihood Estimation (MLE)

Maximum Likelihood Estimation is a method of estimating the parameters of a statistical model. The MLE of a parameter is the value that maximizes the likelihood function, which measures the probability of observing the given sample. In the video, MLE is used to derive the estimator for W, which is found to be the same as the solution from linear regression.

💡Gaussian Noise

Gaussian noise, also known as white noise, is a type of statistical noise that has a probability distribution of a normal bell curve. The script mentions Gaussian noise with a mean of 0 and some variance sigma squared, which is added to the linear model to represent the random error in the observed data.

💡Estimator

An estimator is a statistic used to infer the value of an unknown parameter based on a sample of data. In the script, W hat ML (W hat MLE) is the estimator for the true parameter W, and the discussion revolves around its properties and performance in estimating the true W.

💡Euclidean Norm

The Euclidean norm, also known as the L2 norm, is a measure of the length of a vector in Euclidean space. It is used in the script to measure the goodness of the estimator W hat ML by calculating the squared norm of the difference between W hat ML and the true W.

💡Expected Value

The expected value of a random variable is the long-term average value of repetitions of the experiment it represents. In the video, the expected value is used to understand the average performance of the estimator W hat ML over all possible datasets, given the randomness in y.

💡Trace

The trace of a square matrix is the sum of its diagonal elements. It is used in the script to describe a property of the matrix XX transpose, which is related to the features of the data and affects the average deviation of the estimator from the true parameter.

💡Pseudoinverse

The pseudoinverse of a matrix, often denoted as A^+, is a generalization of the inverse for non-square matrices. It is used in the script to describe the mathematical operation involving the matrix XX transpose in the context of deriving the estimator W hat ML.

💡Covariance Matrix

A covariance matrix is a square matrix that contains the covariance between each pair of variables in a dataset. In the script, the inverse of the covariance matrix (or a similar concept) is mentioned in the context of the trace, which is related to the features' relationships and their impact on the estimator's performance.

💡Mean Squared Error (MSE)

Mean Squared Error is a measure of the quality of an estimator; it is the average of the squares of the errors, that is, the average squared difference between the estimated values and the true value. Although not explicitly mentioned in the script, the discussion about the average deviation of W hat ML from W is related to the concept of MSE.

Highlights

Derivation of the optimal solution W* for linear regression.

Introduction of a probabilistic viewpoint for linear regression.

Identification of y given x as W transpose x plus Gaussian noise.

Observation that W* from linear regression equals ŴML from maximum likelihood estimation.

Assumption of Gaussian noise leading to squared error assumption.

Discussion on the benefits of viewing ŴML as an estimator.

Understanding ŴML as an estimator for the true W in a probabilistic model.

Exploration of the properties of the estimator ŴML.

Importance of the Euclidean norm in measuring the goodness of ŴML.

Concept of expected value in assessing ŴML's performance.

The randomness of y affecting the estimator ŴML.

The impact of data variability on the estimator's quality.

The derivation of the expected value of the deviation between ŴML and the true W.

The expression of the average deviation in terms of sigma squared and the trace of XX^T.

The role of noise variance in the quality of the estimator.

The influence of feature relationships on the estimator's performance.

The potential for tweaking feature relationships to improve the estimator.

Transcripts

 Hello.  

So for, we have looked at the problem of  linear regression, and we derive the optimal  

solution W star for linear regression.  And then we said that we can look at a  

probabilistic viewpoint for linear regression. And we said that the y given x will be some W  

transpose x plus 0 mean noise, and derive the  maximum likelihood estimator for w. And we  

observed that the w star that comes from linear  regression is exactly same as w hat ML, which  

comes from the maximum likelihood estimation.  So, this comes from maximum likelihood.  

So, which gave us the idea that, if we assume that  the noise is Gaussian, then that is equivalent to  

assuming that the error is squared error. So, then  we ask the question, what else is the benefit of  

viewing this as an estimator? That is W hat ML is  an estimator for W and what other properties can  

we look at of this estimator that will help us  gain some insights about this problem itself.  

And now, what we are going to look at is one  nice property of this estimator which will  

give us some insights into extending the linear  regression problem in a particular direction.  

For that, let us try to understand  how good as an estimator W hat ML is.  

Remember, there is some w such that y  given x is W transpose X plus some epsilon,  

where epsilon we said is, is a Gaussian noise  with mean 0 and some variance sigma squared.  

This is what our assumption was. So, which means that for every x in our data,  

the y was generated using some w transpose x,  some unknown but fixed w, which we do not know,  

and then adding a 0 mean Gaussian noise, which  means that the y given x itself can be thought of  

as a Gaussian with mean w transpose x and sigma  squared. This is what we have seen so far.  

Now, our W hat ML is trying to estimate what  is this W? It is a guess for this W. So,  

now we can ask the question, how good is W hat  ML as a guess for the true w. Now, how can we  

measure this goodness, remember, W is some vector  which is unknown, but is in some d dimension. So,  

where d is the number of features. So, it is W hat ML, W hat ML is also  

in d dimension, because it is an  estimator for w. However, W hat ML  

is derived out of an optimization problem. That  is we maximize the likelihood and get W hat ML as  

the solution of an optimization problem. And this  optimization problem involves data, which involves  

x and y. And we are assuming y is random. That is y has some random component. So,  

the y is a random variable. So, if y is  random, then the likelihood estimator,  

the best estimator for w is also going  to be random, because it depends on y.  

So, if you are trying to see which w maximizes  our chance of seeing all the y's y1, y2 tell yn  

and so the w is going to be some function of the  underlying y, and y because y is random. So, it is  

W. So, now what could happen is that in if we  try to understand W hat ML for some realization  

of our data points, let us say you have some  data x1 to xn, x1, y1 let us say till xn, yn.  

Now of course, y is random x, we not  assuming anything about x, y is random.  

Now, it might be the case  that the y's that we have got  

are not in representative of the underlying W. In  other words, we might have had a very extremely  

unlucky day that these noise values when you are  computing y's, so y1 is W transpose x1 plus some  

noise epsilon 1, maybe this noise is too high. So, in our data that we get, and so this y's are  

not really giving us enough information about W  which means that if y’s are very noisy, then the w  

that we are going to calculate is also going to be  kind of bad. So, we cannot decide how good is our  

W hat ML with respect to just a single data set.  Instead, we can ask, well, because y’s are random  

so is W hat ML, we can ask on an average. How does  our random W hat ML, perform in estimating W.  

So, one way to capture this is by looking at  the following quantity, we can we want a way  

to understand the goodness of W hat ML,  how good W hat ML is in estimating W.  

So, one way to understand this is to look at W hat  ML minus w. And because these are d dimensional  

vectors, you can look at their norm squared. So, when I say norm, it is like the Euclidean  

norm, the distance between these two vectors.  But now, again, because we cannot just look at  

one sample, one data set and decide this,  we want to understand this on an average,  

how does this happen, which means we need to  look at the expected value of this quantity.  

What is the expected value? Well, the  expected value is the average expectation  

over the randomness in y that is, it is as  if one way to think about this is that let  

us say we create one data set where their y’s  are derived random according to W transpose  

x plus Gaussian 0 mean Gaussian noise. Now, you calculate the W hat ML for this  

data set and see how far away W hat ML is with  respect to W. Now, you do the same experiment  

many-many times and then average the distance  between W hat ML’s that you get with respect  

to the true W, so the distance squared. Now, this average will slowly converge to  

the expected value of this random variable. So,  this is one way to think about what we are trying  

to compute. So, on an average, where the average  is over all possible data sets that you can get,  

so, we are all possible randomness in y, what is  the average deviation of W hat ML with respect  

to the true W, we can try to ask this. So, this is a quantity of interest for us.  

I will not really derive this quantity. Because I  mean, it is not too hard to derive this, but let  

us not go and derive this right now. Instead, I  will tell you what this quantity is? So, actually,  

it is not too hard to derive this because once you  know what is the quantity, what is W hat ML as a  

function of x and y, which we already know is  just X transpose, X, X transpose we know that W  

hat ML is just X X transpose pseudo inverse x y. So, this is something that we have already seen.  

Now, y is the random part and you can use the  properties of y and so on to derive what this  

value is. But what I am going to do here is just  to give you the value, those who are interested  

can try and actually compute this value for  yourself. It is a bit of not too much of an,  

I mean just an algebraic derivation. So, what is this value? This value turns out to be  

an interesting quantity, and we will see why it is  interesting. I will first put down what this value  

is? This value turns out to be sigma squared  into the trace of XX transpose, pseudo inverse  

or inverse. So, both are okay. So, this is  telling us something very interesting.  

The average deviation between your estimated W hat  ML and W, the true W is a product of two terms.  

One term is sigma squared, which is the variance  of the noise that you add to W transpose X.  

Now, this variance should somehow affect our  quality of W hat ML. That is very clear. So,  

because if I add noise, if I add more noise. Which means that if I add a 0 mean noise with the  

larger variance, which means that I am confusing  my W hat, W transpose X value with more noise. And  

so my quality should go down. So, which means that  the more the variance is, the more the average  

deviation between W hat ML and true W would be,  so, the sigma squared, remember sigma squared  

is just the variance of the Gaussian noise. So,  that we add to W transpose X. So, that should  

directly affect our quality of our estimator  which is fine, which is what is happening.  

But what this interestingly tells us is  that it not only depends on sigma squared,  

it also depends on the data features themselves.  So, it depends on the features via the trace of  

the matrix X X transpose inverse which is like  the inverse of your covariance matrix. So,  

we can even for simplicity, we can just  assume that this is an inverse that is still,  

which means now it is not. So now, depending on how the  

features themselves are related to  each other will also affect our,  

Estimators Goodness. So, it is not just the  noise that y has with respect to, W transpose  

X plus whatever noise you add, of course, that is  going to affect but that is not the only thing.  

It is also how the features are related to each  other, which is an interesting thing to observe.  

Well, there is some data generating process. So, we are given the data, and we are assuming  

that is this data generating process, which  is adding this Gaussian noise to our labels  

and then giving us the y's. So, we cannot change  this data generating process, which means that we  

do not have any control over sigma squared as  such. But then the features are given to us.  

So, now the question is, we cannot really  touch sigma squared, because it is some  

noise that gets added and then it is given to  us. But we are given X and then the features  

themselves somehow affect the goodness of W  hat ML. So, can we somehow tweak this or play  

around with this to get a better estimator, which  perhaps will have lesser mean squared error?  

We do not know that is even possible. But if  at all, it is possible, it is going to be,  

you have to touch this term, because  sigma squared is anyway out of question,  

because that is a noise that is in some sense, you  have to suffer that much. Can we somehow get rid,  

I mean, not get rid of can you somehow at least  reduce this part or at least focus on this part  

is the next valid question that we can ask.  And let us see what we can do with this.