Get ONMINE

Stay Ahead, Stay ONMINE

Mastering the Poisson Distribution: Intuition and Foundations

You’ve probably used the normal distribution one or two times too many. We all have — It’s a true workhorse. But sometimes, we run into problems. For instance, when predicting or forecasting values, simulating data given a particular data-generating process, or when we try to visualise model output and explain them intuitively to non-technical stakeholders. Suddenly, things don’t make much sense: can a user really have made -8 clicks on the banner? Or even 4.3 clicks? Both are examples of how count data doesn’t behave. I’ve found that better encapsulating the data generating process into my modelling has been key to having sensible model output. Using the Poisson distribution when it was appropriate has not only helped me convey more meaningful insights to stakeholders, but it has also enabled me to produce more accurate error estimates, better Inference, and sound decision-making. In this post, my aim is to help you get a deep intuitive feel for the Poisson distribution by walking through example applications, and taking a dive into the foundations — the maths. I hope you learn not just how it works, but also why it works, and when to apply the distribution. If you know of a resource that has helped you grasp the concepts in this blog particularly well, you’re invited to share it in the comments! Outline Examples and use cases: Let’s walk through some use cases and sharpen the intuition I just mentioned. Along the way, the relevance of the Poisson Distribution will become clear. The foundations: Next, let’s break down the equation into its individual components. By studying each part, we’ll uncover why the distribution works the way it does. The assumptions: Equipped with some formality, it will be easier to understand the assumptions that power the distribution, and at the same time set the boundaries for when it works, and when not. When real life deviates from the model: Finally, let’s explore the special links that the Poisson distribution has with the Negative Binomial distribution. Understanding these relationships can deepen our understanding, and provide alternatives when the Poisson distribution is not suited for the job. Example in an online marketplace I chose to deep dive into the Poisson distribution because it frequently appears in my day-to-day work. Online marketplaces rely on binary user choices from two sides: a seller deciding to list an item and a buyer deciding to make a purchase. These micro-behaviours drive supply and demand, both in the short and long term. A marketplace is born. Binary choices aggregate into counts — the sum of many such decisions as they occur. Attach a timeframe to this counting process, and you’ll start seeing Poisson distributions everywhere. Let’s explore a concrete example next. Consider a seller on a platform. In a given month, the seller may or may not list an item for sale (a binary choice). We would only know if she did because then we’d have a measurable count of the event. Nothing stops her from listing another item in the same month. If she does, we count those events. The total could be zero for an inactive seller or, say, 120 for a highly engaged seller. Over several months, we would observe a varying number of listed items by this seller — sometimes fewer, sometimes more — hovering around an average monthly listing rate. That is essentially a Poisson process. When we get to the assumptions section, you’ll see what we had to assume away to make this example work. Other examples Other phenomena that can be modelled with a Poisson distribution include: Sports analytics: The number of goals scored in a match between two teams. Queuing: Customers arriving at a help desk or customer support calls. Insurance: The number of claims made within a given period. Each of these examples warrants further inspection, but for the remainder of this post, we’ll use the marketplace example to illustrate the inner workings of the distribution. The mathy bit … or foundations. I find opening up the probability mass function (PMF) of distributions helpful to understanding why things work as they do. The PMF of the Poisson distribution goes like: Where λ is the rate parameter, and 𝑘 is the manifested count of the random variable (𝑘 = 0, 1, 2, 3, … events). Very neat and compact. The probability mass function of the Poisson distribution, for a few different lambdas. Contextualising λ and k: the marketplace example In the context of our earlier example — a seller listing items on our platform — λ represents the seller’s average monthly listings. As the expected monthly value for this seller, λ orchestrates the number of items she would list in a month. Note that λ is a Greek letter, so read: λ is a parameter that we can estimate from data. On the other hand, 𝑘 does not hold any information about the seller’s idiosyncratic behaviour. It’s the target value we set for the number of events that may happen to learn about its probability. The dual role of λ as the mean and variance When I said that λ orchestrates the number of monthly listings for the seller, I meant it quite literally. Namely, λ is both the expected value and variance of the distribution, indifferently, for all values of λ. This means that the mean-to-variance ratio (index of dispersion) is always 1. To put this into perspective, the normal distribution requires two parameters — 𝜇 and 𝜎², the average and variance respectively — to fully describe it. The Poisson distribution achieves the same with just one. Having to estimate only one parameter can be beneficial for parametric inference. Specifically, by reducing the variance of the model and increasing the statistical power. On the other hand, it can be too limiting of an assumption. Alternatives like the Negative Binomial distribution can alleviate this limitation. We’ll explore that later. Breaking down the probability mass function Now that we know the smallest building blocks, let’s zoom out one step: what is λᵏ, 𝑒^⁻λ, and 𝑘!, and more importantly, what is each of these components’ function in the whole? λᵏ is a weight that expresses how likely it is for 𝑘 events to happen, given that the expectation is λ. Note that “likely” here does not mean a probability, yet. It’s merely a signal strength. 𝑘! is a combinatorial correction so that we can say that the order of the events is irrelevant. The events are interchangeable. 𝑒^⁻λ normalises the integral of the PMF function to sum up to 1. It’s called the partition function of exponential-family distributions. In more detail, λᵏ relates the observed value 𝑘 to the expected value of the random variable, λ. Intuitively, more probability mass lies around the expected value. Hence, if the observed value lies close to the expectation, the probability of occurring is larger than the probability of an observation far removed from the expectation. Before we can cross-check our intuition with the numerical behaviour of λᵏ, we need to consider what 𝑘! does. Interchangeable events Had we cared about the order of events, then each unique event could be ordered in 𝑘! ways. But because we don’t, and we deem each event interchangeable, we “divide out” 𝑘! from λᵏ to correct for the overcounting. Since λᵏ is an exponential term, the output will always be larger as 𝑘 grows, holding λ constant. That is the opposite of our intuition that there is maximum probability when λ = 𝑘, as the output is larger when 𝑘 = λ + 1. But now that we know about the interchangeable events assumption — and the overcounting issue — we know that we have to factor in 𝑘! like so: λᵏ 𝑒^⁻λ / 𝑘!, to see the behaviour we expect. Now let’s check the intuition of the relationship between λ and 𝑘 through λᵏ, corrected for 𝑘!. For the same λ, say λ = 4, we should see λᵏ 𝑒^⁻λ / 𝑘! to be smaller for values of 𝑘 that are far removed from 4, compared to values of 𝑘 that lie close to 4. Like so: inline code: 4²/2 = 8 is smaller than 4⁴/24 = 10.7. This is consistent with the intuition of a higher likelihood of 𝑘 when it’s near the expectation. The image below shows this relationship more generally, where you see that the output is larger as 𝑘 approaches λ. The probability mass function without the normalising component e^-lambda. The assumptions First, let’s get one thing off the table: the difference between a Poisson process, and the Poisson distribution. The process is a stochastic continuous-time model of points happening in given interval: 1D, a line; 2D, an area, or higher dimensions. We, data scientists, most often deal with the one-dimensional case, where the “line” is time, and the points are the events of interest — I dare to say. These are the assumptions of the Poisson process: The occurrence of one event does not affect the probability of a second event. Think of our seller going on to list another item tomorrow indifferently of having done so already today, or the one from five days ago for that matter. The point here is that there is no memory between events. The average rate at which events occur, is independent of any occurrence. In other words, no event that happened (or will happen) alters λ, which remains constant throughout the observed timeframe. In our seller example, this means that listing an item today does not increase or decrease the seller’s motivation or likelihood of listing another item tomorrow. Two events cannot occur at exactly the same instant. If we were to zoom at an infinite granular level on the timescale, no two listings could have been placed simultaneously; always sequentially. From these assumptions — no memory, constant rate, events happening alone — it follows that 1) any interval’s number of events is Poisson-distributed with parameter λₜ and 2) that disjoint intervals are independent — two key properties of a Poisson process. A Note on the distribution:The distribution simply describes probabilities for various numbers of counts in an interval. Strictly speaking, one can use the distribution pragmatically whenever the data is nonnegative, can be unbounded on the right, has mean λ, and reasonably models the data. It would be just convenient if the underlying process is a Poisson one, and actually justifies using the distribution. The marketplace example: Implications So, can we justify using the Poisson distribution for our marketplace example? Let’s open up the assumptions of a Poisson process and take the test. Constant λ Why it may fail: The seller has patterned online activity; holidays; promotions; listings are seasonal goods. Consequence: λ is not constant, leading to overdispersion (mean-to-variance ratio is larger than 1, or to temporal patterns. Independence and memorylessness Why it may fail: The propensity to list again is higher after a successful listing, or conversely, listing once depletes the stock and intervenes with the propensity of listing again. Consequence: Two events are no longer independent, as the occurrence of one informs the occurrence of the other. Simultaneous events Why it may fail: Batch-listing, a new feature, was introduced to help the sellers. Consequence: Multiple listings would come online at the same time, clumped together, and they would be counted simultaneously. Balancing rigour and pragmatism As Data Scientists on the job, we may feel trapped between rigour and pragmatism. The three steps below should give you a sound foundation to decide on which side to err, when the Poisson distribution falls short: Pinpoint your goal: is it inference, simulation or prediction, and is it about high-stakes output? List the worst thing that can happen, and the cost of it for the business. Identify the problem and solution: why does the Poisson distribution not fit, and what can you do about it? list 2-3 solutions, including changing nothing. Balance gains and costs: Will your workaround improve things, or make it worse? and at what cost: interpretability, new assumptions introduced and resources used. Does it help you in achieving your goal? That said, here are some counters I use when needed. When real life deviates from your model Everything described so far pertains to the standard, or homogenous, Poisson process. But what if reality begs for something different? In the next section, we’ll cover two extensions of the Poisson distribution when the constant λ assumption does not hold. These are not mutually exclusive, but neither they are the same: Time-varying λ: a single seller whose listing rate ramps up before holidays and slows down afterward Mixed Poisson distribution: multiple sellers listing items, each with their own λ can be seen as a mixture of various Poisson processes Time-varying λ The first extension allows λ to have its own value for each time t. The PMF then becomes Where the number of events 𝐾(𝑇) in an interval 𝑇 follows the Poisson distribution with a rate no longer equal to a fixed λ, but one equal to: More intuitively, integrating over the interval 𝑡 to 𝑡 + 𝑖 gives us a single number: the expected value of events over that interval. The integral will vary by each arbitrary interval, and that’s what makes λ change over time. To understand how that integration works, it was helpful for me to think of it like this: if the interval 𝑡 to 𝑡₁ integrates to 3, and 𝑡₁ to 𝑡₂ integrates to 5, then the interval 𝑡 to 𝑡₂ integrates to 8 = 3 + 5. That’s the two expectations summed up, and now the expectation of the entire interval. Practical implication One may want to modeling the expected value of the Poisson distribution as a function of time. For instance, to model an overall change in trend, or seasonality. In generative model notation: Time may be a continuous variable, or an arbitrary function of it. Process-varying λ: Mixed Poisson distribution But then there’s a gotcha. Remember when I said that λ has a dual role as the mean and variance? That still applies here. Looking at the “relaxed” PMF*, the only thing that changes is that λ can vary freely with time. But it’s still the one and only λ that orchestrates both the expected value and the dispersion of the PMF*. More precisely, 𝔼[𝑋] = Var(𝑋) still holds. There are various reasons for this constraint not to hold in reality. Model misspecification, event interdependence and unaccounted for heterogeneity could be the issues at hand. I’d like to focus on the latter case, as it justifies the Negative Binomial distribution — one of the topics I promised to open up. Heterogeneity and overdispersionImagine we are not dealing with one seller, but with 10 of them listing at different intensity levels, λᵢ, where 𝑖 = 1, 2, 3, …, 10 sellers. Then, essentially, we have 10 Poisson processes going on. If we unify the processes and estimate the grand λ, we simplify the mixture away. Meaning, we get a correct estimate of all sellers on average, but the resulting grand λ is naive and does not know about the original spread of λᵢ. It still assumes that the variance and mean are equal, as per the axioms of the distribution. This will lead to overdispersion and, in turn, to underestimated errors. Ultimately, it inflates the false positive rate and drives poor decision-making. We need a way to embrace the heterogeneity amongst sellers’ λᵢ. Negative binomial: Extending the Poisson distributionAmong the few ways one can look at the Negative Binomial distribution, one way is to see it as a compound Poisson process — 10 sellers, sounds familiar yet? That means multiple independent Poisson processes are summed up to a single one. Mathematically, first we draw λ from a Gamma distribution: λ ~ Γ(r, θ), then we draw the count 𝑋 | λ ~ Poisson(λ). In one image, it is as if we would sample from plenty Poisson distributions, corresponding to each seller. A negative Binomial distribution arises from many Poisson distributions. The more exposing alias of the Negative binomial distribution is Gamma-Poisson mixture distribution, and now we know why: the dictating λ comes from a continuous mixture. That’s what we needed to explain the heterogeneity amongst sellers. Let’s simulate this scenario to gain more intuition. Gamma mixture of lambda. First, we draw λᵢ from a Gamma distribution: λᵢ ~ Γ(r, θ). Intuitively, the Gamma distribution tells us about the variety in the intensity — listing rate — amongst the sellers. On a practical note, one can instill their assumptions about the degree of heterogeneity in this step of the model: how different are sellers? By varying the levels of heterogeneity, one can observe the impact on the final Poisson-like distribution. Doing this type of checks (i.e., posterior predictive check), is common in Bayesian modeling, where the assumptions are set explicitly. Gamma-Poisson mixture distribution versus homogenous Poisson distribution. Τhe dashed line reflects λ, which is 4 for both distributions. In the second step, we plug the obtained λ into the Poisson distribution: 𝑋 | λ ~ Poisson(λ), and obtain a Poisson-like distribution that represents the summed subprocesses. Notably, this unified process has a larger dispersion than expected from a homogeneous Poisson distribution, but it is in line with the Gamma mixture of λ. Heterogeneous λ and inference A practical consequence of introducing flexibility into your assumed distribution is that inference becomes more challenging. More parameters (i.e., the Gamma parameters) need to be estimated. Parameters act as flexible explainers of the data, tending to overfit and explain away variance in your variable. The more parameters you have, the better the explanation may seem, but the model also becomes more susceptible to noise in the data. Higher variance reduces the power to identify a difference in means, if one exists, because — well — it gets lost in the variance. Countering the loss of power Confirm whether you indeed need to extend the standard Poisson distribution. If not, simplify to the simplest, most fit model. A quick check on overdispersion may suffice for this. Pin down the estimates of the Gamma mixture distribution parameters using regularising, informative priors (think: Bayes). During my research process for writing this blog, I learned a great deal about the connective tissue underlying all of this: how the binomial distribution plays a fundamental role in the processes we’ve discussed. And while I’d love to ramble on about this, I’ll save it for another post, perhaps. In the meantime, feel free to share your understanding in the comments section below 👍. Conclusion The Poisson distribution is a simple distribution that can be highly suitable for modelling count data. However, when the assumptions do not hold, one can extend the distribution by allowing the rate parameter to vary as a function of time or other factors, or by assuming subprocesses that collectively make up the count data. This added flexibility can address the limitations, but it comes at a cost: increased flexibility in your modelling raises the variance and, consequently, undermines the statistical power of your model. If your end goal is inference, you may want to think twice and consider exploring simpler models for the data. Alternatively, switch to the Bayesian paradigm and leverage its built-in solution to regularise estimates: informative priors. I hope this has given you what you came for — a better intuition about the Poisson distribution. I’d love to hear your thoughts about this in the comments! Unless otherwise noted, all images are by the author.Originally published at https://aalvarezperez.github.io on January 5, 2025.

You’ve probably used the normal distribution one or two times too many. We all have — It’s a true workhorse. But sometimes, we run into problems. For instance, when predicting or forecasting values, simulating data given a particular data-generating process, or when we try to visualise model output and explain them intuitively to non-technical stakeholders. Suddenly, things don’t make much sense: can a user really have made -8 clicks on the banner? Or even 4.3 clicks? Both are examples of how count data doesn’t behave.

I’ve found that better encapsulating the data generating process into my modelling has been key to having sensible model output. Using the Poisson distribution when it was appropriate has not only helped me convey more meaningful insights to stakeholders, but it has also enabled me to produce more accurate error estimates, better Inference, and sound decision-making.

In this post, my aim is to help you get a deep intuitive feel for the Poisson distribution by walking through example applications, and taking a dive into the foundations — the maths. I hope you learn not just how it works, but also why it works, and when to apply the distribution.

If you know of a resource that has helped you grasp the concepts in this blog particularly well, you’re invited to share it in the comments!

Outline

  1. Examples and use cases: Let’s walk through some use cases and sharpen the intuition I just mentioned. Along the way, the relevance of the Poisson Distribution will become clear.
  2. The foundations: Next, let’s break down the equation into its individual components. By studying each part, we’ll uncover why the distribution works the way it does.
  3. The assumptions: Equipped with some formality, it will be easier to understand the assumptions that power the distribution, and at the same time set the boundaries for when it works, and when not.
  4. When real life deviates from the model: Finally, let’s explore the special links that the Poisson distribution has with the Negative Binomial distribution. Understanding these relationships can deepen our understanding, and provide alternatives when the Poisson distribution is not suited for the job.

Example in an online marketplace

I chose to deep dive into the Poisson distribution because it frequently appears in my day-to-day work. Online marketplaces rely on binary user choices from two sides: a seller deciding to list an item and a buyer deciding to make a purchase. These micro-behaviours drive supply and demand, both in the short and long term. A marketplace is born.

Binary choices aggregate into counts — the sum of many such decisions as they occur. Attach a timeframe to this counting process, and you’ll start seeing Poisson distributions everywhere. Let’s explore a concrete example next.

Consider a seller on a platform. In a given month, the seller may or may not list an item for sale (a binary choice). We would only know if she did because then we’d have a measurable count of the event. Nothing stops her from listing another item in the same month. If she does, we count those events. The total could be zero for an inactive seller or, say, 120 for a highly engaged seller.

Over several months, we would observe a varying number of listed items by this seller — sometimes fewer, sometimes more — hovering around an average monthly listing rate. That is essentially a Poisson process. When we get to the assumptions section, you’ll see what we had to assume away to make this example work.

Other examples

Other phenomena that can be modelled with a Poisson distribution include:

  • Sports analytics: The number of goals scored in a match between two teams.
  • Queuing: Customers arriving at a help desk or customer support calls.
  • Insurance: The number of claims made within a given period.

Each of these examples warrants further inspection, but for the remainder of this post, we’ll use the marketplace example to illustrate the inner workings of the distribution.

The mathy bit

… or foundations.

I find opening up the probability mass function (PMF) of distributions helpful to understanding why things work as they do. The PMF of the Poisson distribution goes like:

Where λ is the rate parameter, and 𝑘 is the manifested count of the random variable (𝑘 = 0, 1, 2, 3, … events). Very neat and compact.

Graph: The probability mass function of the Poisson distribution, for a few different lambdas.
The probability mass function of the Poisson distribution, for a few different lambdas.

Contextualising λ and k: the marketplace example

In the context of our earlier example — a seller listing items on our platform — λ represents the seller’s average monthly listings. As the expected monthly value for this seller, λ orchestrates the number of items she would list in a month. Note that λ is a Greek letter, so read: λ is a parameter that we can estimate from data. On the other hand, 𝑘 does not hold any information about the seller’s idiosyncratic behaviour. It’s the target value we set for the number of events that may happen to learn about its probability.

The dual role of λ as the mean and variance

When I said that λ orchestrates the number of monthly listings for the seller, I meant it quite literally. Namely, λ is both the expected value and variance of the distribution, indifferently, for all values of λ. This means that the mean-to-variance ratio (index of dispersion) is always 1.

To put this into perspective, the normal distribution requires two parameters — 𝜇 and 𝜎², the average and variance respectively — to fully describe it. The Poisson distribution achieves the same with just one.

Having to estimate only one parameter can be beneficial for parametric inference. Specifically, by reducing the variance of the model and increasing the statistical power. On the other hand, it can be too limiting of an assumption. Alternatives like the Negative Binomial distribution can alleviate this limitation. We’ll explore that later.

Breaking down the probability mass function

Now that we know the smallest building blocks, let’s zoom out one step: what is λᵏ, 𝑒^⁻λ, and 𝑘!, and more importantly, what is each of these components’ function in the whole?

  • λᵏ is a weight that expresses how likely it is for 𝑘 events to happen, given that the expectation is λ. Note that “likely” here does not mean a probability, yet. It’s merely a signal strength.
  • 𝑘! is a combinatorial correction so that we can say that the order of the events is irrelevant. The events are interchangeable.
  • 𝑒^⁻λ normalises the integral of the PMF function to sum up to 1. It’s called the partition function of exponential-family distributions.

In more detail, λᵏ relates the observed value 𝑘 to the expected value of the random variable, λ. Intuitively, more probability mass lies around the expected value. Hence, if the observed value lies close to the expectation, the probability of occurring is larger than the probability of an observation far removed from the expectation. Before we can cross-check our intuition with the numerical behaviour of λᵏ, we need to consider what 𝑘! does.

Interchangeable events

Had we cared about the order of events, then each unique event could be ordered in 𝑘! ways. But because we don’t, and we deem each event interchangeable, we “divide out” 𝑘! from λᵏ to correct for the overcounting.

Since λᵏ is an exponential term, the output will always be larger as 𝑘 grows, holding λ constant. That is the opposite of our intuition that there is maximum probability when λ = 𝑘, as the output is larger when 𝑘 = λ + 1. But now that we know about the interchangeable events assumption — and the overcounting issue — we know that we have to factor in 𝑘! like so: λᵏ 𝑒^⁻λ / 𝑘!, to see the behaviour we expect.

Now let’s check the intuition of the relationship between λ and 𝑘 through λᵏ, corrected for 𝑘!. For the same λ, say λ = 4, we should see λᵏ 𝑒^⁻λ / 𝑘! to be smaller for values of 𝑘 that are far removed from 4, compared to values of 𝑘 that lie close to 4. Like so: inline code: 4²/2 = 8 is smaller than 4⁴/24 = 10.7. This is consistent with the intuition of a higher likelihood of 𝑘 when it’s near the expectation. The image below shows this relationship more generally, where you see that the output is larger as 𝑘 approaches λ.

Graph: The probability mass function without the normalising component e^-lambda.
The probability mass function without the normalising component e^-lambda.

The assumptions

First, let’s get one thing off the table: the difference between a Poisson process, and the Poisson distribution. The process is a stochastic continuous-time model of points happening in given interval: 1D, a line; 2D, an area, or higher dimensions. We, data scientists, most often deal with the one-dimensional case, where the “line” is time, and the points are the events of interest — I dare to say.

These are the assumptions of the Poisson process:

  1. The occurrence of one event does not affect the probability of a second event. Think of our seller going on to list another item tomorrow indifferently of having done so already today, or the one from five days ago for that matter. The point here is that there is no memory between events.
  2. The average rate at which events occur, is independent of any occurrence. In other words, no event that happened (or will happen) alters λ, which remains constant throughout the observed timeframe. In our seller example, this means that listing an item today does not increase or decrease the seller’s motivation or likelihood of listing another item tomorrow.
  3. Two events cannot occur at exactly the same instant. If we were to zoom at an infinite granular level on the timescale, no two listings could have been placed simultaneously; always sequentially.

From these assumptions — no memory, constant rate, events happening alone — it follows that 1) any interval’s number of events is Poisson-distributed with parameter λₜ and 2) that disjoint intervals are independent — two key properties of a Poisson process.

A Note on the distribution:
The distribution simply describes probabilities for various numbers of counts in an interval. Strictly speaking, one can use the distribution pragmatically whenever the data is nonnegative, can be unbounded on the right, has mean λ, and reasonably models the data. It would be just convenient if the underlying process is a Poisson one, and actually justifies using the distribution.

The marketplace example: Implications

So, can we justify using the Poisson distribution for our marketplace example? Let’s open up the assumptions of a Poisson process and take the test.

Constant λ

  • Why it may fail: The seller has patterned online activity; holidays; promotions; listings are seasonal goods.
  • Consequence: λ is not constant, leading to overdispersion (mean-to-variance ratio is larger than 1, or to temporal patterns.

Independence and memorylessness

  • Why it may fail: The propensity to list again is higher after a successful listing, or conversely, listing once depletes the stock and intervenes with the propensity of listing again.
  • Consequence: Two events are no longer independent, as the occurrence of one informs the occurrence of the other.

Simultaneous events

  • Why it may fail: Batch-listing, a new feature, was introduced to help the sellers.
  • Consequence: Multiple listings would come online at the same time, clumped together, and they would be counted simultaneously.

Balancing rigour and pragmatism

As Data Scientists on the job, we may feel trapped between rigour and pragmatism. The three steps below should give you a sound foundation to decide on which side to err, when the Poisson distribution falls short:

  1. Pinpoint your goal: is it inference, simulation or prediction, and is it about high-stakes output? List the worst thing that can happen, and the cost of it for the business.
  2. Identify the problem and solution: why does the Poisson distribution not fit, and what can you do about it? list 2-3 solutions, including changing nothing.
  3. Balance gains and costs: Will your workaround improve things, or make it worse? and at what cost: interpretability, new assumptions introduced and resources used. Does it help you in achieving your goal?

That said, here are some counters I use when needed.

When real life deviates from your model

Everything described so far pertains to the standard, or homogenous, Poisson process. But what if reality begs for something different?

In the next section, we’ll cover two extensions of the Poisson distribution when the constant λ assumption does not hold. These are not mutually exclusive, but neither they are the same:

  1. Time-varying λ: a single seller whose listing rate ramps up before holidays and slows down afterward
  2. Mixed Poisson distribution: multiple sellers listing items, each with their own λ can be seen as a mixture of various Poisson processes

Time-varying λ

The first extension allows λ to have its own value for each time t. The PMF then becomes

Where the number of events 𝐾(𝑇) in an interval 𝑇 follows the Poisson distribution with a rate no longer equal to a fixed λ, but one equal to:

More intuitively, integrating over the interval 𝑡 to 𝑡 + 𝑖 gives us a single number: the expected value of events over that interval. The integral will vary by each arbitrary interval, and that’s what makes λ change over time. To understand how that integration works, it was helpful for me to think of it like this: if the interval 𝑡 to 𝑡₁ integrates to 3, and 𝑡₁ to 𝑡₂ integrates to 5, then the interval 𝑡 to 𝑡₂ integrates to 8 = 3 + 5. That’s the two expectations summed up, and now the expectation of the entire interval.

Practical implication 
One may want to modeling the expected value of the Poisson distribution as a function of time. For instance, to model an overall change in trend, or seasonality. In generative model notation:

Time may be a continuous variable, or an arbitrary function of it.

Process-varying λ: Mixed Poisson distribution

But then there’s a gotcha. Remember when I said that λ has a dual role as the mean and variance? That still applies here. Looking at the “relaxed” PMF*, the only thing that changes is that λ can vary freely with time. But it’s still the one and only λ that orchestrates both the expected value and the dispersion of the PMF*. More precisely, 𝔼[𝑋] = Var(𝑋) still holds.

There are various reasons for this constraint not to hold in reality. Model misspecification, event interdependence and unaccounted for heterogeneity could be the issues at hand. I’d like to focus on the latter case, as it justifies the Negative Binomial distribution — one of the topics I promised to open up.

Heterogeneity and overdispersion
Imagine we are not dealing with one seller, but with 10 of them listing at different intensity levels, λᵢ, where 𝑖 = 1, 2, 3, …, 10 sellers. Then, essentially, we have 10 Poisson processes going on. If we unify the processes and estimate the grand λ, we simplify the mixture away. Meaning, we get a correct estimate of all sellers on average, but the resulting grand λ is naive and does not know about the original spread of λᵢ. It still assumes that the variance and mean are equal, as per the axioms of the distribution. This will lead to overdispersion and, in turn, to underestimated errors. Ultimately, it inflates the false positive rate and drives poor decision-making. We need a way to embrace the heterogeneity amongst sellers’ λᵢ.

Negative binomial: Extending the Poisson distribution
Among the few ways one can look at the Negative Binomial distribution, one way is to see it as a compound Poisson process — 10 sellers, sounds familiar yet? That means multiple independent Poisson processes are summed up to a single one. Mathematically, first we draw λ from a Gamma distribution: λ ~ Γ(r, θ), then we draw the count 𝑋 | λ ~ Poisson(λ).

In one image, it is as if we would sample from plenty Poisson distributions, corresponding to each seller.

A negative Binomial distribution arises from many Poisson distributions.
A negative Binomial distribution arises from many Poisson distributions.

The more exposing alias of the Negative binomial distribution is Gamma-Poisson mixture distribution, and now we know why: the dictating λ comes from a continuous mixture. That’s what we needed to explain the heterogeneity amongst sellers.

Let’s simulate this scenario to gain more intuition.

Gamma mixture of lambda.
Gamma mixture of lambda.

First, we draw λᵢ from a Gamma distribution: λᵢ ~ Γ(r, θ). Intuitively, the Gamma distribution tells us about the variety in the intensity — listing rate — amongst the sellers.

On a practical note, one can instill their assumptions about the degree of heterogeneity in this step of the model: how different are sellers? By varying the levels of heterogeneity, one can observe the impact on the final Poisson-like distribution. Doing this type of checks (i.e., posterior predictive check), is common in Bayesian modeling, where the assumptions are set explicitly.

Gamma-Poisson mixture distribution versus homogenous Poisson distribution. Τhe dashed line reflects λ, which is 4 for both distributions.
Gamma-Poisson mixture distribution versus homogenous Poisson distribution. Τhe dashed line reflects λ, which is 4 for both distributions.

In the second step, we plug the obtained λ into the Poisson distribution: 𝑋 | λ ~ Poisson(λ), and obtain a Poisson-like distribution that represents the summed subprocesses. Notably, this unified process has a larger dispersion than expected from a homogeneous Poisson distribution, but it is in line with the Gamma mixture of λ.

Heterogeneous λ and inference

A practical consequence of introducing flexibility into your assumed distribution is that inference becomes more challenging. More parameters (i.e., the Gamma parameters) need to be estimated. Parameters act as flexible explainers of the data, tending to overfit and explain away variance in your variable. The more parameters you have, the better the explanation may seem, but the model also becomes more susceptible to noise in the data. Higher variance reduces the power to identify a difference in means, if one exists, because — well — it gets lost in the variance.

Countering the loss of power

  1. Confirm whether you indeed need to extend the standard Poisson distribution. If not, simplify to the simplest, most fit model. A quick check on overdispersion may suffice for this.
  2. Pin down the estimates of the Gamma mixture distribution parameters using regularising, informative priors (think: Bayes).

During my research process for writing this blog, I learned a great deal about the connective tissue underlying all of this: how the binomial distribution plays a fundamental role in the processes we’ve discussed. And while I’d love to ramble on about this, I’ll save it for another post, perhaps. In the meantime, feel free to share your understanding in the comments section below 👍.

Conclusion

The Poisson distribution is a simple distribution that can be highly suitable for modelling count data. However, when the assumptions do not hold, one can extend the distribution by allowing the rate parameter to vary as a function of time or other factors, or by assuming subprocesses that collectively make up the count data. This added flexibility can address the limitations, but it comes at a cost: increased flexibility in your modelling raises the variance and, consequently, undermines the statistical power of your model.

If your end goal is inference, you may want to think twice and consider exploring simpler models for the data. Alternatively, switch to the Bayesian paradigm and leverage its built-in solution to regularise estimates: informative priors.

I hope this has given you what you came for — a better intuition about the Poisson distribution. I’d love to hear your thoughts about this in the comments!

Unless otherwise noted, all images are by the author.
Originally published at https://aalvarezperez.github.io on January 5, 2025.

Shape
Shape
Stay Ahead

Explore More Insights

More Articles

Stay ahead with more perspectives on cutting-edge power, infrastructure, energy,  bitcoin and AI solutions. Explore these articles to uncover strategies and insights shaping the future of industries.

Shape

RFID boosts Amazon’s autonomous retail tech

The new RFID lanes are built for merchandise and apparel. These items are much harder to track with camera-based systems since they can be folded, stacked, or carried out of a store in bulk. RFID tags solve that problem by identifying every item. The lanes combine several systems working together

Read More »

Cisco extends Nexus 9000 support to Intel Gaudi 3 AI accelerators

Partnerships, validated designs strengthen Cisco offerings Cisco’s AI offerings also include Nvidia technologies, such as Spectrum-X-based switches that are part of Cisco Secure AI Factory with Nvidia.  Cisco also works with AMD and its Instinct AI GPUs for networking and compute stack in large AI clusters. In addition, Cisco integrates

Read More »

Turkey in Talks With Chevron for Joint Oil and Gas Exploration

Turkey’s state energy company is in talks with Chevron Corp to jointly explore for oil and gas, according to a Turkish official familiar with the discussions. Turkish Petroleum Corp, known as TPAO, would work with Chevron on seismic studies and drilling, the official said, asking not to be identified because the talks are private. The potential accord is the latest move by Ankara to boost energy production and comes amid a broader warming of US-Turkey ties. It would follow a January agreement with Exxon Mobil Corp on joint exploration in the Black Sea and Mediterranean.  Turkey’s Energy Ministry didn’t respond to a request for comment. TPAO couldn’t be reached for comment.  “Chevron has a diverse exploration and production portfolio globally and continues to assess potential opportunities,” a spokesperson for Chevron told Bloomberg. “As a matter of policy, we do not comment on commercial matters.” It’s not clear which projects the companies could collaborate on. TPAO is already active in the Black Sea, Iraq, Russia and Somalia, and previously drilled in the eastern Mediterranean, where Chevron operates fields in Israeli and Cypriot waters.  Ankara has sought to cut its almost total dependency on imported oil and gas in recent years by increasing domestic production and expanding TPAO’s operations abroad. The firm has expanded its fleet of specialist vessels for offshore energy exploration and recently announced plans to raise as much as $4 billion in its debut Islamic debt sale. One of TPAO’s newest ships, called Cagri Bey, is expected to begin drilling work in Somalia in April or May, Energy Minister Alparslan Bayraktar told state-run broadcaster TRT Haber last week. WHAT DO YOU THINK? Generated by readers, the comments included herein do not reflect the views and opinions of Rigzone. All comments are subject to editorial review. Off-topic, inappropriate or insulting comments will be removed.

Read More »

Crude Gains as IEA Lifts Demand Outlook

Oil edged higher as traders assessed US President Donald Trump’s statement that the US had reached a “framework” for a deal on Greenland. West Texas Intermediate rose less than 1% to settle below $61 a barrel, following equities and the dollar higher. Trump said he would refrain from previously threatened tariffs on European nations opposing his effort to take possession of Greenland. The US’s renewed push to acquire the sovereign Danish territory threatened to trigger a trade war that could curb economic growth and bring down oil prices. Markets also digested renewed concerns over military action in Iran, adding to a geopolitical risk premium that further supported prices. Trump is still pressing aides on “decisive” military options in Iran, the Wall Street Journal reported Tuesday. Reaction in the oil markets has appeared relatively muted, said Frank Monkam, head of macro trading at Buffalo Bayou Commodities. “The price action today is basically range trade with an upside bias where dips are being bought until we get clarity on Iran, which has taken a backseat to Greenland/Davos talks for now,” Monkam said. Helping add a floor to the market, the IEA increased its forecast for global oil demand growth in 2026, slightly trimming a projected supply glut that has weighed on prices. The projections, however, still leave oil markets facing a significant excess. Stockpiles are on track to swell by 3.7 million barrels a day this year, according to the IEA’s latest assessment, though the agency has cautioned that the actual overhang may not reach these levels in practice. The excess supply has cushioned against uncertainties in Iran, Russia, Kazakhstan, and Venezuela, the agency added. In Kazakhstan, the operator of the Tengiz oilfield has declared force majeure on crude deliveries into the Caspian Pipeline Consortium, adding a hurdle that has offered some

Read More »

Russia Says Ukraine Attacked Afipsky Refinery Overnight

Russia said Ukrainian drones targeted the Afipsky refinery in the Krasnodar region overnight, in the latest attack on the nation’s energy infrastructure. Debris of the unmanned aerial vehicle fell on the territory of the facility and caused a fire, which “was quickly extinguished,” regional emergencies authorities said in a Telegram post. “There were no casualties or damage to infrastructure.” Bloomberg couldn’t independently verify the claim. ForteInvest, which operates the Afipsky refinery, didn’t immediately respond to a request for comment on the potential impact on processing rates.  Ukraine and Russia have been trading attacks on energy infrastructure as Kremlin’s full-scale invasion of its neighbor is about to enter a fifth year, with Kyiv and Moscow remaining at an impasse over a proposed peace plan. While Kyiv has reduced the intensity of attacks on Russian refineries and oil export facilities so far this month compared with the end of last year, Moscow has stepped up strikes on Ukraine’s power sector, leaving millions of people without heating and water amid freezing temperatures. The Afipsky refinery has a processing capacity of as much as 9.1 million tons of crude oil annually, or some 180,000 barrels per day. The facility, which has been a target for repeated Ukrainian attacks, was last hit in December.  WHAT DO YOU THINK? Generated by readers, the comments included herein do not reflect the views and opinions of Rigzone. All comments are subject to editorial review. Off-topic, inappropriate or insulting comments will be removed.

Read More »

EIA Sees USA Crude Oil Production Dropping in 2026, 2027

In its latest short term energy outlook (STEO), which was released on January 13, the U.S. Energy Information Administration (EIA) projected that total U.S. crude oil production will drop in 2026 and 2027. According to this STEO, the EIA sees U.S. crude oil output, including lease condensate, averaging 13.59 million barrels per day in 2026 and 13.25 million barrels per day in 2027. This production averaged 13.61 million barrels per day in 2025, the STEO showed. The STEO projected that Lower 48 States production, excluding the Gulf of America, will come in at 11.11 million barrels per day in 2026 and 10.87 million barrels per day in 2027. In the report, Federal Gulf of America production is forecast to average 2.00 million barrels per day in 2026 and 1.89 million barrels per day in 2027 and Alaska output is projected to come in at 0.47 million barrels per day in 2026 and 0.50 million barrels per day in 2027. In 2025, Lower 48 States production averaged 11.28 million barrels per day, Federal Gulf of America production came in at 1.91 million barrels per day, and Alaska output averaged 0.42 million barrels per day, the STEO showed. A quarterly breakdown included in the EIA’s latest STEO projected that U.S. crude oil production will come in at 13.73 million barrels per day in the first quarter of this year, 13.65 million barrels per day in the second quarter, 13.47 million barrels per day in the third quarter, 13.50 million barrels per day in the fourth quarter, 13.43 million barrels per day in the first quarter of 2027, 13.31 million barrels per day in the second quarter, and 13.13 million barrels per day across the third and fourth quarter of next year. “We forecast U.S. crude oil production will remain close to the

Read More »

Western Midstream Secures New Deals with Occidental,ConocoPhillips

Western Midstream Partners LP (WES) announced Tuesday it has amended its natural gas gathering and processing contracts in the Delaware Basin with Occidental Petroleum Corp and expanded its partnership with ConocoPhillips in the same basin. The new agreements with ConocoPhillips and Occidental advance WES’ transition to fixed-fee arrangements in the maturing basin, The Woodlands, Texas-based company said in a statement on its website. The previous agreement with Occidental already provided for a transition to a fixed-fee structure; the new agreement speeds up that transition, according to WES. Houston, Texas-based oil, gas and chemicals producer Occidental agreed to reduce its ownership in WES from about 42 percent to around 40 percent under the renegotiated gathering and processing contracts, “further positioning WES as a standalone midstream enterprise”, WES said. “Following this amendment, approximately nine percent of WES’ total revenue will remain subject to cost-of-service rates, with approximately one percent of total revenue subject to cost-of-service rates expiring in the late 2020s”, WES said. “The remaining cost-of-service rate provisions extend into the mid-to-late 2030s and include provisions to convert to fixed-fee structures at that time. “All significant fixed-fee contracts with Occidental, including the contracts being amended, are effective through the mid-to-late 2030s”. The new gas gathering contract provides “volumetric protection via substantial minimum volume commitments (MVCs) through the original cost-of-service term, and from that point forward, the existing acreage dedication and fixed-fee structure continues through the duration of the contract”, WES added. The new gas processing contract “continues to provide volumetric protection via MVCs through 2035”, WES said. As part of the renegotiated contracts, Occidental will surrender to WES 15.3 million common units currently owned by Occidental. The volume represents around $610 million of limited partnership interests, according to WES. “This transfer was structured on terms intended to represent a value-neutral exchange for the economic concessions reflected in

Read More »

Why Are USA NatGas Prices Rising Today?

U.S. natural gas prices are rising today due to a combination of weather risk, production softness, and positioning, rather than any single structural shift. That’s what Ole R. Hvalbye, a commodities analyst at Skandinaviska Enskilda Banken AB (SEB), told Rigzone in an exclusive interview on Wednesday. “First, the weather premium has kicked in hard,” Hvalbye said. “Forecasts now show temperatures in the Lower-48 turning well below normal from around January 23 and extending into early February, particularly across the eastern half of the United States,” he added. “That directly lifts heating demand expectations at a time of year when the market is already sensitive(!). As a result, Henry Hub has surged from … [around] $3 per MMBtu [million British thermal units] last week to nearly $5 per MMBtu intraday today!” he noted. “Second, supply has tightened at the margin. Lower-48 dry gas production has dipped to around 110.5 Bcfpd [billion cubic feet per day], down from over 112 Bcfpd earlier this week, partly reflecting cold-weather disruptions,” Hvalbye continued. “At the same time, LNG feedgas demand remains elevated at just over 18 Bcfpd, even though flows at Sabine Pass eased slightly today, partly offset by higher intake at Elba Island,” he stated. “Third, positioning and short covering are amplifying the move,” Hvalbye highlighted. The SEB commodities analyst told Rigzone that trading volumes in Henry Hub futures hit a record high earlier this week and added that today’s rally has been pushed by hedge funds covering short positions built up during the recent sell-off. “That adds momentum once prices start moving,” he pointed out. Looking at the demand side, Hvalbye told Rigzone that U.S. gas consumption “has eased back toward ~108 Bcfpd from very high cold-weather levels earlier this week” but added that “that hasn’t been enough to offset the weather risk

Read More »

CyrusOne Hones AI-Era Data Center Strategy for Power, Pace, and Reliability

In the second half of 2025, CyrusOne was racing to secure buildable power and faster time-to-market capacity for AI-era customers. At the same time, its reputation for mission-critical reliability took a very public hit when a disruption at a CyrusOne facility helped knock CME trading offline. The incident forced the company into an unusually open conversation about redundancy, cooling systems, and operational discipline: systems that are meant to disappear in normal operation, and dominate the story when they malfunction. From Projects to a Playbook Which projects, missteps, and strategic moves from 2025 are now shaping how CyrusOne enters 2026? Nowhere is that view clearer than in Texas. There, CyrusOne has been leaning hard into a “power + land + interconnect” model: treating deliverable power and grid position as part of the product, not just a prerequisite. If you map the company’s announcements since late July, Texas reveals the playbook. Secure power, secure substations and grid position, then build multi-phase campuses designed to scale quickly as demand materializes. The Calpine “Powered Land” Deal: From 190 MW to 400 MW in Three Months On July 30, 2025, CyrusOne and Calpine announced a 190-MW agreement tied to a hyperscale campus (DFW10) adjacent to Calpine’s Thad Hill Energy Center in Bosque County, Texas. The structure bundled power, grid connection, and land into a single development package, with CyrusOne saying the site was already under construction and targeting operation by Q4 2026. Just three months later, on November 3–4, the partners announced a second phase, adding 210 MW and taking the campus to 400 MW. The update emphasized coordination to support grid reliability during scarcity; such curtailment and operational-coordination concepts are becoming table stakes for ERCOT-scale megaprojects. Together, the two announcements show CyrusOne placing a large bet on an emerging model: power-ready campuses, or “powered

Read More »

Forrester study quantifies benefits of Cisco Intersight

If IT groups are to be the strategic business partners their companies need, they require solutions that can improve infrastructure life cycle management in the age of artificial intelligence (AI) and heightened security threats. To quantify the value of such solutions, Cisco recently commissioned Forrester Consulting to conduct a Total Economic Impact™ analysis of Cisco Intersight. The comprehensive study found that for a composite organization, Intersight delivered 192% return on investment (ROI) and a payback period of less than six months, along with significant tangible benefits to IT and businesses. Cisco Intersight overview Cisco Intersight is a cloud-native IT operations platform for infrastructure life cycle management. It provides IT teams with comprehensive visibility, control, and automation capabilities for Cisco’s portfolio of compute solutions for data centers, colocation facilities, and edge environments based on the Cisco Unified Computing System (Cisco UCS). Intersight also integrates with leading operating systems, storage providers, hypervisors, and third-party IT service management and security tools. Intersight’s unified, policy-driven approach to infrastructure management helps IT groups automate numerous tasks and, as Forrester found, free up time to dedicate to strategic projects. Forrester study quantifies the benefits of Cisco Intersight  A composite organization using Cisco Intersight achieved:192% ROI and payback in less than six months$3.3M net present value over three years$2.7M from improved uptime and resilience 50% reduction in mean time to resolution $1.7M from increased IT productivity$267K benefit from decreased time to value due to faster project execution and earlier return on infrastructure investments Forrester Total Economic Impact study findings The analyst firm conducted detailed interviews with IT decision-makers and Intersight users at six organizations, from which it created one composite organization: a multinational technology-driven company with $10 billion in annual revenue, 120 branch locations, and a team of six engineers managing its 1,000 servers deployed in several

Read More »

SoftBank launches software stack for AI data center operations

Addressing enterprise challenges The software provides two main services, according to SoftBank. The Kubernetes-as-a-Service component automates the stack from BIOS and RAID settings through the OS, GPU drivers, networking, Kubernetes controllers, and storage, the company said. It reconfigures physical connectivity using Nvidia NVLink and memory allocation as users create, update, or delete clusters, according to the announcement. The system allocates nodes based on GPU proximity and NVLink domain configuration to reduce latency, SoftBank said. Enterprises currently face complex GPU cluster provisioning, Kubernetes lifecycle management, inference scaling, and infrastructure tuning challenges that require deep expertise, according to Dai. SoftBank’s automated approach addresses these pain points by handling BIOS-to-Kubernetes configuration, optimizing GPU interconnects, and abstracting inference into API-based services, he said. This allows teams to focus on model development rather than infrastructure maintenance, Dai said. The Inference-as-a-Service component lets users deploy inference services by selecting large language models without configuring Kubernetes or underlying infrastructure, according to the company. It provides OpenAI-compatible APIs and scales across multiple nodes on platforms including the GB200 NVL72, SoftBank said. The software includes tenant isolation through encrypted communications, automated system monitoring and failover, and APIs for connecting to portal, customer management, and billing systems, according to the announcement.

Read More »

OpenAI shifts AI data center strategy toward power-first design

The shift to ‘energy sovereignty’  Analysts say the move reflects a fundamental shift in data center strategy, moving from “fiber-first” to “power-first” site selection. “Historically, data centers were built near internet exchange points and urban centers to minimize latency,” said Ashish Banerjee, senior principal analyst at Gartner. “However, as AI training requirements reach the gigawatt scale, OpenAI is signaling that they will prioritize regions with ‘energy sovereignty’, places where they can build proprietary generation and transmission, rather than fighting for scraps on an overtaxed public grid.” For network architecture, this means a massive expansion of the “middle mile.” By placing these behemoth data centers in energy-rich but remote locations, the industry will have to invest heavily in long-haul, high-capacity dark fiber to connect these “power islands” back to the edge. “We should expect a bifurcated network: a massive, centralized core for ‘cold’ model training located in the wilderness, and a highly distributed edge for ‘hot’ real-time inference located near the users,” Banerjee added. Manish Rawat, a semiconductor analyst at TechInsights, also noted that the benefits may come at the cost of greater architectural complexity. “On the network side, this pushes architectures toward fewer mega-hubs and more regionally distributed inference and training clusters, connected via high-capacity backbone links,” Rawat said. “The trade-off is higher upfront capex but greater control over scalability timelines, reducing dependence on slow-moving utility upgrades.”

Read More »

CleanArc’s Virginia Hyperscale Bet Meets the Era of Pay-Your-Way Power

What CleanArc’s Project Really Signals About Scaling in Virginia The more important story is what the project signals about how developers believe they can still scale in Virginia at hyperscale magnitude. To wit: 1) The campus is sized like a grid project, not a real estate project At 900 MW, CleanArc is not simply building a few facilities. It is effectively planning a utility-interface program that will require staged substation, transmission, and interconnection work over many years. The company describes the campus as a “flagship” designed for scalable demand and sustainability-focused procurement. Power delivery is planned in three 300 MW phases: the first targeted for 2027, the second for 2030, and the final block sometime between 2033 and 2035. That scale changes what “site selection” really means. For projects of this magnitude, the differentiator is no longer “Can we entitle buildings?” but “Can we secure a credible path for large power blocks, with predictable commercial terms, while regulators are rewriting the rules?” 2) It’s being marketed as sustainability-forward in a market that increasingly requires it CleanArc frames the campus as aligned with sustainability-focused infrastructure: a posture that is no longer optional for hyperscale procurement teams. That does not mean the grid power itself is automatically carbon-free. It means the campus is being positioned to support the modern contracting stack, involving renewables, clean-energy attributes, and related structures, while still delivering what hyperscalers buy first: capacity, reliability, and delivery certainty. 3) The timing is strategic as Virginia tightens around very large load CleanArc is launching its flagship in the nation’s premier data center corridor at the same moment Virginia has moved to formalize a large-customer category that explicitly includes data centers. The implication is not that Virginia has become anti-data center. It is that the state is entering a phase where it

Read More »

xAI’s AI Factories: From Colossus to MACROHARDRR in the Gigawatt Era

Colossus: The Prototype For much of the past year, xAI’s infrastructure story did not unfold across a portfolio of sites. It unfolded inside a single building in Memphis, where the company first tested what an “AI factory” actually looks like in physical form. That building had a name that matched the ambition: Colossus. The Memphis-area facility, carved out of a vacant Electrolux factory, became shorthand for a new kind of AI build: fast, dense, liquid-cooled, and powered on a schedule that often ran ahead of the grid. It was an “AI factory” in the literal sense: not a cathedral of architecture, but a machine for turning electricity into tokens. Colossus began as an exercise in speed. xAI took over a dormant industrial building in Southwest Memphis and turned it into an AI training plant in months, not years. The company has said the first major system was built in about 122 days, and then doubled in roughly 92 more, reaching around 200,000 GPUs. Those numbers matter less for their bravado than for what they reveal about method. Colossus was never meant to be bespoke. It was meant to be repeatable. High-density GPU servers, liquid cooling at the rack, integrated CDUs, and large-scale Ethernet networking formed a standardized building block. The rack, not the room, became the unit of design. Liquid cooling was not treated as a novelty. It was treated as a prerequisite. By pushing heat removal down to the rack, xAI avoided having to reinvent the data hall every time density rose. The building became a container; the rack became the machine. That design logic, e.g. industrial shell plus standardized AI rack, has quietly become the template for everything that followed. Power: Where Speed Met Reality What slowed the story was not compute, cooling, or networking. It was power.

Read More »

Microsoft will invest $80B in AI data centers in fiscal 2025

And Microsoft isn’t the only one that is ramping up its investments into AI-enabled data centers. Rival cloud service providers are all investing in either upgrading or opening new data centers to capture a larger chunk of business from developers and users of large language models (LLMs).  In a report published in October 2024, Bloomberg Intelligence estimated that demand for generative AI would push Microsoft, AWS, Google, Oracle, Meta, and Apple would between them devote $200 billion to capex in 2025, up from $110 billion in 2023. Microsoft is one of the biggest spenders, followed closely by Google and AWS, Bloomberg Intelligence said. Its estimate of Microsoft’s capital spending on AI, at $62.4 billion for calendar 2025, is lower than Smith’s claim that the company will invest $80 billion in the fiscal year to June 30, 2025. Both figures, though, are way higher than Microsoft’s 2020 capital expenditure of “just” $17.6 billion. The majority of the increased spending is tied to cloud services and the expansion of AI infrastructure needed to provide compute capacity for OpenAI workloads. Separately, last October Amazon CEO Andy Jassy said his company planned total capex spend of $75 billion in 2024 and even more in 2025, with much of it going to AWS, its cloud computing division.

Read More »

John Deere unveils more autonomous farm machines to address skill labor shortage

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More Self-driving tractors might be the path to self-driving cars. John Deere has revealed a new line of autonomous machines and tech across agriculture, construction and commercial landscaping. The Moline, Illinois-based John Deere has been in business for 187 years, yet it’s been a regular as a non-tech company showing off technology at the big tech trade show in Las Vegas and is back at CES 2025 with more autonomous tractors and other vehicles. This is not something we usually cover, but John Deere has a lot of data that is interesting in the big picture of tech. The message from the company is that there aren’t enough skilled farm laborers to do the work that its customers need. It’s been a challenge for most of the last two decades, said Jahmy Hindman, CTO at John Deere, in a briefing. Much of the tech will come this fall and after that. He noted that the average farmer in the U.S. is over 58 and works 12 to 18 hours a day to grow food for us. And he said the American Farm Bureau Federation estimates there are roughly 2.4 million farm jobs that need to be filled annually; and the agricultural work force continues to shrink. (This is my hint to the anti-immigration crowd). John Deere’s autonomous 9RX Tractor. Farmers can oversee it using an app. While each of these industries experiences their own set of challenges, a commonality across all is skilled labor availability. In construction, about 80% percent of contractors struggle to find skilled labor. And in commercial landscaping, 86% of landscaping business owners can’t find labor to fill open positions, he said. “They have to figure out how to do

Read More »

2025 playbook for enterprise AI success, from agents to evals

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More 2025 is poised to be a pivotal year for enterprise AI. The past year has seen rapid innovation, and this year will see the same. This has made it more critical than ever to revisit your AI strategy to stay competitive and create value for your customers. From scaling AI agents to optimizing costs, here are the five critical areas enterprises should prioritize for their AI strategy this year. 1. Agents: the next generation of automation AI agents are no longer theoretical. In 2025, they’re indispensable tools for enterprises looking to streamline operations and enhance customer interactions. Unlike traditional software, agents powered by large language models (LLMs) can make nuanced decisions, navigate complex multi-step tasks, and integrate seamlessly with tools and APIs. At the start of 2024, agents were not ready for prime time, making frustrating mistakes like hallucinating URLs. They started getting better as frontier large language models themselves improved. “Let me put it this way,” said Sam Witteveen, cofounder of Red Dragon, a company that develops agents for companies, and that recently reviewed the 48 agents it built last year. “Interestingly, the ones that we built at the start of the year, a lot of those worked way better at the end of the year just because the models got better.” Witteveen shared this in the video podcast we filmed to discuss these five big trends in detail. Models are getting better and hallucinating less, and they’re also being trained to do agentic tasks. Another feature that the model providers are researching is a way to use the LLM as a judge, and as models get cheaper (something we’ll cover below), companies can use three or more models to

Read More »

OpenAI’s red teaming innovations define new essentials for security leaders in the AI era

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More OpenAI has taken a more aggressive approach to red teaming than its AI competitors, demonstrating its security teams’ advanced capabilities in two areas: multi-step reinforcement and external red teaming. OpenAI recently released two papers that set a new competitive standard for improving the quality, reliability and safety of AI models in these two techniques and more. The first paper, “OpenAI’s Approach to External Red Teaming for AI Models and Systems,” reports that specialized teams outside the company have proven effective in uncovering vulnerabilities that might otherwise have made it into a released model because in-house testing techniques may have missed them. In the second paper, “Diverse and Effective Red Teaming with Auto-Generated Rewards and Multi-Step Reinforcement Learning,” OpenAI introduces an automated framework that relies on iterative reinforcement learning to generate a broad spectrum of novel, wide-ranging attacks. Going all-in on red teaming pays practical, competitive dividends It’s encouraging to see competitive intensity in red teaming growing among AI companies. When Anthropic released its AI red team guidelines in June of last year, it joined AI providers including Google, Microsoft, Nvidia, OpenAI, and even the U.S.’s National Institute of Standards and Technology (NIST), which all had released red teaming frameworks. Investing heavily in red teaming yields tangible benefits for security leaders in any organization. OpenAI’s paper on external red teaming provides a detailed analysis of how the company strives to create specialized external teams that include cybersecurity and subject matter experts. The goal is to see if knowledgeable external teams can defeat models’ security perimeters and find gaps in their security, biases and controls that prompt-based testing couldn’t find. What makes OpenAI’s recent papers noteworthy is how well they define using human-in-the-middle

Read More »
More Articles