What is the Cramer Rao Lower Bound?

C.R. Rao passed away earlier this week, and this is a name that is familiar to anybody who has completed a course in statistics at the Masters level. Well, ought to be familiar, at any rate. And if you ask me, even undergraduate students ought to be familiar with both the name, and the many achievements of India’s best statistician.

Yes, that is a tall claim, but even folks who might disagree with me will admit that C.R. Rao makes for a very worthy contender. (And if you do disagree, please do tell me who your candidate would be.)

Here’s a lovely write-up about C.R. Rao from earlier on this year:

Professor C R Rao, the ageless maestro in mathematical statistics, was recently in the news because he received the 2023 International Prize in Statistics. The news made headlines because the Prize was called the ‘Nobel Prize in Statistics’.
There’s no Nobel Prize in Statistics, but if there had indeed been one, Calyampudi Radhakrishna Rao would have got it long ago. Perhaps in 1960.
C R Rao entered the field of statistics when it wasn’t sufficiently ‘mathematical’. Classical statistics, at that point, was about gathering data, obtaining counts and averages, and estimating data variability and associations.
Think, for example, of the mammoth census exercise. Rao was among the first to bring serious mathematics into the mix. He asked questions like: When does a ‘best’ statistical estimate exist? If it does, how can we mathematically manoeuvre to make a given estimate the best? In situations where no best estimate exists, which among the candidate estimates is the most desirable?


By the way, the very last anecdote in this post was my favorite, so please make sure you read the whole thing.

You could argue for days about which of his many contributions were the most important, but there will be very little disagreement about which is the most famous: the Cramer-Rao Lower Bound.

In estimation theory and statistics, the Cramér–Rao bound (CRB) relates to estimation of a deterministic (fixed, though unknown) parameter. The result is named in honor of Harald Cramér and C. R. Rao.


Plow through the article if you like, but I would advise you against it if you are unfamiliar with statistical theory, and/or mathematical notation. You could try this on for size instead:

What is the Cramer-Rao Lower Bound?
The Cramer-Rao Lower Bound (CRLB) gives a lower estimate for the variance of an unbiased estimator. Estimators that are close to the CLRB are more unbiased (i.e. more preferable to use) than estimators further away. The Cramer-Rao Lower bound is theoretical; Sometimes a perfectly unbiased estimator (i.e. one that meets the CRLB) doesn’t exist. Additionally, the CRLB is difficult to calculate unless you have a very simple scenario. Easier, general, alternatives for finding the best estimator do exist. You may want to consider running a more practical alternative for point estimation, like the Method of Moments.


But even this, I would say, isn’t a great way to make oneself familiar with the subject. Try your luck with YouTube, maybe? Lots of very good videos in there, but none that I would call simple enough.

Which, of course, brings us to ChatGPT.

Here’s what I started with:

“Give me the simplest possible explanation of what the cramer-rao lower bound is, why it is important, and one example of a simple application. Thank you!”

Your mileage may will vary, because you might end up using ChatGPT 3.5. Even if you were to use ChatGPT4, there’s no guarantee that the answer will be the same. It will, in fact, almost certainly be different. Bu that’s fine! Copy this prompt and run it.

This was my follow-up question:

“Lovely, this was helpful. Can you walk me through its theoretical derivation? Keep the explanation as simple as possible, and note that I am not very familiar with matrix algebra.”

I wanted as simple an explanation as possible from ChatGPT, which is why I added the bit about not being familiar with matrix algebra. You could say you’re not at all familiar with it. You could say you’re not familiar with statistics, or mathematical notations, or any other qualifiers you think are necessary. Don’t be shy about telling ChatGPT what you don’t know – this is important because it will tailor its answers accordingly.

This is what I went with next:

“Now imagine I have to explain this to, say, a very motivated 12 year old. This student is very willing to learn, but is unfamiliar with statistical concepts. She has, in fact, had no classes in statistics at all, but has learnt the basic concepts in statistics, and wants to run a real life experiment. In my conversations with her, the CRLB came up, and she wanted to know more about it. How would you explain the derivation of the CRLB to her?”

No twelve year old student asked me this question, to be clear. But this is just one of many, many ways in which you can try and get ChatGPT to simplify stuff for you. This answer is worth pasting in full:

“Alright, let’s use a more intuitive approach!

Setting the Scene: Imagine you’re trying to guess the number of candies in a jar. Each time you guess, you might be a little too high or a little too low, but on average, you’re pretty close. The CRLB is like a magic number that tells you the smallest amount you can be off by, on average, no matter how good your guessing strategy is.Step-by-Step Explanation:

  1. Making a Guess: Every time you make a guess about the number of candies, you’re using some clues or information. Maybe you’re looking at the size of the jar, the size of the candies, or how full the jar is. All these clues help you make a better guess.
  2. Score Function – The Clue Meter: Imagine you have a “clue meter” that measures how good each clue is. If the needle on the meter moves a lot when you change your guess a little, that means the clue is really helpful. If the needle barely moves, the clue isn’t very useful.
  3. Expected Value – Average Clue Power: Now, if you average the readings of the “clue meter” over many guesses, it should point to zero. This means that sometimes the clues tell you to guess higher and sometimes lower, but on average, they don’t favor any particular direction.
  4. Information Matrix – Clue Strength: The strength of your clues is like how much the needle on the “clue meter” wiggles around its average (zero). If it wiggles a lot, your clues are strong. If it barely wiggles, your clues are weak.
  5. The Magic Number – CRLB: Now, the CRLB is like a magic number that tells you the smallest amount you can be off by, on average, based on how strong your clues are. If your clues are super strong (the needle wiggles a lot), then the magic number is small, and you can guess really accurately. If your clues are weak (the needle barely wiggles), then the magic number is big, and your guesses might be off by a bit more.

Conclusion: So, when you’re trying to guess the number of candies in the jar, the CRLB is like a challenge. It’s the universe saying, “Bet you can’t guess better than this!” And if you can guess with an error smaller than that magic number, then you’re doing an amazing job! If not, maybe there’s a better way to use your clues.”

Much better. But I had questions!

“Is it possible for the guess to be lower than the CRLB? If yes, I’m confused about why it should be called the “lower bound””

Here is its answer:

“Great question! Let’s clarify this.The Cramér-Rao Lower Bound (CRLB) is a theoretical lower bound on the variance (a measure of how spread out the estimates are) of any unbiased estimator. It doesn’t directly bound the error of a single guess or measurement. Instead, it bounds the average squared error (variance) of an estimator over many, many guesses or measurements.In our candy jar example, the CRLB is used in a slightly simplified and illustrative manner. Here’s what it means:

  1. Single Guess: In any single guess, your error (the difference between your guess and the actual number of candies) can certainly be less than the CRLB. For instance, you might even guess the exact number of candies and have an error of zero!
  2. Average Error Over Many Guesses: If you were to make many guesses (or if many people were to guess), and we calculated the average of the squared differences between all those guesses and the actual number, that average would typically be above the CRLB. The CRLB tells us that we can’t consistently get an average squared error below this bound with any guessing strategy.

So, when we say “Try to get within 5 candies of the actual number” in the game, it’s a simplification for illustration. In a real-world application of the CRLB, it’s about the average performance of an estimator, not the performance on a single estimate.In essence, while individual errors can be below the CRLB, the average squared error of a good estimator shouldn’t be below the CRLB. If it is, either the estimator is biased, or there’s some other issue with the assumptions or calculations.”

Next obvious (to me, at any rate) step is to ask it to come up with a working demo:

“Great! Can you help me build a very small demo of this experiment you just described in javascript that I can load as an HTML page?”

What a time to be alive! If you are learning statistics for the very first time, and especially if you have (in your opinion) a not-so-great stats prof, you can now make the world’s best lemonade from the lemon you’ve been handed! Here is its answer:

“Of course! Let’s create a simple demo where:

  1. A random number of candies (between 1 and 100) is placed in a virtual jar.
  2. The user tries to guess the number of candies.
  3. After guessing, the user is told how close their guess was to the actual number.
  4. We’ll also display the CRLB (for simplicity, we’ll just use a fixed value as our CRLB).”

Here’s a basic HTML page with embedded JavaScript for the demo it came up with:

<!DOCTYPE html>
<html lang="en">
    <meta charset="UTF-8">
    <meta name="viewport" content="width=device-width, initial-scale=1.0">
    <title>Candy Guessing Game</title>

<h2>Candy Guessing Game</h2>
<p>Guess the number of candies in the jar (between 1 and 100):</p>

<input type="number" id="guess" min="1" max="100">
<button onclick="checkGuess()">Submit Guess</button>

<p id="feedback"></p>
<p id="crlb">Cramér-Rao Lower Bound (CRLB): 5 (This is our magic number! Try to get within 5 candies of the actual number.)</p>

    // Randomly determine the number of candies in the jar
    let candies = Math.floor(Math.random() * 100) + 1;

    function checkGuess() {
        let guess = document.getElementById("guess").value;
        let difference = Math.abs(guess - candies);
        if (difference === 0) {
            document.getElementById("feedback").innerText = "Amazing! You guessed it exactly right!";
        } else if (difference <= 5) { // Using 5 as our CRLB for simplicity
            document.getElementById("feedback").innerText = "Great job! You're within the magic number! The jar has " + candies + " candies.";
        } else {
            document.getElementById("feedback").innerText = "Keep trying! The jar has " + candies + " candies. You were off by " + difference + ".";


What do you do with this code? Here’s ChatGPT again:

“You can save the above code as an .html file and open it in a web browser. The user can input their guess, and upon submission, they’ll receive feedback on how close their guess was to the actual number of candies. The CRLB is set to 5 for simplicity, so if the user’s guess is within 5 candies of the actual number, they’re doing great!”

What’s the point of having learnt an idea if you can’t discuss it in class or with like-minded friends, batchmates and peers? And so I had one final question for the best research assistant I’ve ever had:

“Thank you! One last question. Can you please give me five good questions regarding the CRLB that I can either ask in class, or use as a starting point for discussions with my batchmates or seniors. Please optimize the questions for conceptual clarity. That is, our attempts to answer these questions should make the concept, its applicability and its importance clearer to us.”

“Certainly! Here are five thought-provoking questions regarding the Cramér-Rao Lower Bound (CRLB) that can stimulate discussion and deepen understanding:

  1. Foundational Understanding:
    • “Why is the CRLB considered a ‘lower bound’ on the variance of an unbiased estimator? What does it imply about the efficiency of other estimators?”
  2. Practical Implications:
    • “In what real-world scenarios might the CRLB be particularly crucial for ensuring the accuracy of measurements or predictions? Can you think of any industries or fields where a slight improvement in estimation accuracy could have significant consequences?”
  3. Assumptions and Limitations:
    • “The CRLB is derived under certain assumptions, such as the estimator being unbiased. How might the CRLB change if some of these assumptions are relaxed or not met? Are there situations where the CRLB might not be applicable or might be misleading?”
  4. Comparison with Other Bounds:
    • “How does the CRLB compare to other bounds or benchmarks in statistics? Are there scenarios where another bound might be more appropriate or informative than the CRLB?”
  5. Deepening Conceptual Understanding:
    • “If the CRLB provides a lower limit on the variance of an unbiased estimator, what would it mean if an estimator consistently has a variance below the CRLB? What might this indicate about the estimator or the data it’s working with?”

These questions are designed to encourage exploration of the foundational concepts behind the CRLB, its practical applications, and its relationship with other statistical tools and concepts. Discussing them should provide a more comprehensive and nuanced understanding of the CRLB and its significance in the field of statistics.”

There is always at least one student who says something along the lines of “No way can my college prof answer these, lol”, or “No way are my friends going to want to talk about this”, or “<Insert variant of choice here>”

Well, what about feeding these back to ChatGPT?

I’ve waited for years to write this sentence, and fellow students of econ will allow me to savor my moment of sweet, sweet revenge.

This is left as an exercise for the reader.

Honor the great man’s work by learning about it, and with ChatGPT around, you have no excuses left. Get to it! Oh, and by the way, the meta lesson, of course, is that you can do this with all of statistics – and other subjects besides.