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

ExxonMobil bumps up 2030 target for Permian production

ExxonMobil Corp., Houston, is looking to grow production in the Permian basin to about 2.5 MMboe/d by 2030, an increase of 200,000 boe/d from executives’ previous forecasts and a jump of more than 45% from this year’s output. Helping drive that higher target is an expected 2030 cost profile that

Read More »

Antero adds to Marcellus portfolio, Infinity picks up divested Ohio Utica interests

Antero Resources Corp., Denver, Co., has signed deals to expand its Marcellus shale footprint in West Virginia and to divest its certain Ohio Utica shale assets. Adding the Marcellus assets expands Antero Resources’ core acreage position, enhancing its position “as the premier liquids developer in the Marcellus,” and provides the company “with further dry gas optionality for local demand from data centers and natural gas fired power plants,” said Michael Kennedy, president and chief executive officer, in a release Dec. 8. Marcellus acquisition from HG Energy Through a deal to acquire the upstream assets of HG Energy II LLC, Parkersburg, WV, Antero aims to add 850 MMcfed of expected Marcellus production in 2026. The deal, expected to close in second-quarter 2026, was signed for $2.8 billion in cash plus the assumption of HG Energy’s commodity hedge book. Antero said about 90% of HG natural gas production is hedged in 2026 and 2027 at average NYMEX prices of $4.00 and $3.88, respectively. The deal adds 385,000 net acres offsetting Antero’s existing 475,000 net core Marcellus acreage position and includes over 400 additional locations that immediately compete for capital (75% liquids), the company said in a related investor presentation.  Antero said it anticipates capital synergies of about $550 million inclusive of development planning optimization and drilling and completions savings. Another $400 in income-related synergies is expected. Separately, Antero Midstream agreed to acquire the midstream assets from HG Energy for $1.1 billion in cash. The deal includes about 50 miles of bi-directional dry and rich gas gathering pipelines and water assets in which Antero plans to invest about $25 million to integrate with its legacy gathering and water system. Utica sale to Infinity Natural Resources Infinity Natural Resources Inc., in a release Dec. 8, said subsidiary Infinity Natural Resources LLC will acquire upstream and

Read More »

Market Focus: Oversupply takes center stage, fundamentals catch up with the market

@import url(‘https://fonts.googleapis.com/css2?family=Inter:[email protected]&display=swap’); a { color: var(–color-primary-main); } .ebm-page__main h1, .ebm-page__main h2, .ebm-page__main h3, .ebm-page__main h4, .ebm-page__main h5, .ebm-page__main h6 { font-family: Inter; } body { line-height: 150%; letter-spacing: 0.025em; font-family: Inter; } button, .ebm-button-wrapper { font-family: Inter; } .label-style { text-transform: uppercase; color: var(–color-grey); font-weight: 600; font-size: 0.75rem; } .caption-style { font-size: 0.75rem; opacity: .6; } #onetrust-pc-sdk [id*=btn-handler], #onetrust-pc-sdk [class*=btn-handler] { background-color: #c19a06 !important; border-color: #c19a06 !important; } #onetrust-policy a, #onetrust-pc-sdk a, #ot-pc-content a { color: #c19a06 !important; } #onetrust-consent-sdk #onetrust-pc-sdk .ot-active-menu { border-color: #c19a06 !important; } #onetrust-consent-sdk #onetrust-accept-btn-handler, #onetrust-banner-sdk #onetrust-reject-all-handler, #onetrust-consent-sdk #onetrust-pc-btn-handler.cookie-setting-link { background-color: #c19a06 !important; border-color: #c19a06 !important; } #onetrust-consent-sdk .onetrust-pc-btn-handler { color: #c19a06 !important; border-color: #c19a06 !important; } <!–> In this Market Focus episode of the Oil & Gas Journal ReEnterprised podcast, Conglin Xu, managing editor, economics, takes a look at the growing oversupply in global crude markets and the shift now under way as fundamentals begin overtaking sentiment and geopolitics as the primary price driver. ]–>

Read More »

Aramco, ExxonMobil weigh new chemical complex for Samref refinery

Saudi Aramco and partner ExxonMobil Corp. subsidiary Mobil Yanbu Refining Co. Inc. are discussing the possibility of executing a major overhaul and expansion of 50-50 joint venture Saudi Aramco-Mobil Refinery Co. Ltd.’s (Samref) 400,000-b/d Samref refinery in Yanbu, Saudi Arabia. As part of a venture framework agreement (VFA) signed on Dec. 8, the partners will evaluate potential capital investments to expand and diversify the refinery’s existing production slate, including the addition of a grassroots petrochemical complex at the site, Aramco said in a statement. In addition to upgrading and diversifying Samref’s production to include lower-emission, high-quality distillates and high-performance chemicals, the project scope would involve works to improve the refinery’s energy efficiency and implement a sitewide integrated emissions reduction strategy, according to Aramco. With the VFA now signed, the companies said they will begin the project’s preliminary front-end engineering and design (pre-FEED) study, which will focus on opportunities to maximize the site’s operational advantage and enhance its competitiveness while meeting Saudi Arabia’s growing demand for high-quality petrochemical products. For Aramco, the proposed project—the design of which aims to increase the conversion of crude oil and other petroleum liquids into higher-value chemicals—further reinforces the company’s commitment to creating further value of its overall downstream business as well as its liquids-to-chemicals strategy, according to Mohammed Y. Al Qahtani, Aramco’s downstream president. “[The proposed expansion and integration project] will also position Samref as a key driver in the growth of [Saudi Arabia’s] petrochemical sector,” Al Qahtani added. Without disclosing a timeline as to when the partners expect to complete the pre-FEED study or reach final investment decision, Aramco confirmed existing plans for the potential project would remain subject to market conditions and necessary regulatory approvals. Samref previously completed modifications and renovations at the Yanbu refinery in 2014-15 related to a two-phased clean-fuels project

Read More »

Harbour Energy to add North Sea assets through Waldorf acquisition

@import url(‘https://fonts.googleapis.com/css2?family=Inter:[email protected]&display=swap’); a { color: var(–color-primary-main); } .ebm-page__main h1, .ebm-page__main h2, .ebm-page__main h3, .ebm-page__main h4, .ebm-page__main h5, .ebm-page__main h6 { font-family: Inter; } body { line-height: 150%; letter-spacing: 0.025em; font-family: Inter; } button, .ebm-button-wrapper { font-family: Inter; } .label-style { text-transform: uppercase; color: var(–color-grey); font-weight: 600; font-size: 0.75rem; } .caption-style { font-size: 0.75rem; opacity: .6; } #onetrust-pc-sdk [id*=btn-handler], #onetrust-pc-sdk [class*=btn-handler] { background-color: #c19a06 !important; border-color: #c19a06 !important; } #onetrust-policy a, #onetrust-pc-sdk a, #ot-pc-content a { color: #c19a06 !important; } #onetrust-consent-sdk #onetrust-pc-sdk .ot-active-menu { border-color: #c19a06 !important; } #onetrust-consent-sdk #onetrust-accept-btn-handler, #onetrust-banner-sdk #onetrust-reject-all-handler, #onetrust-consent-sdk #onetrust-pc-btn-handler.cookie-setting-link { background-color: #c19a06 !important; border-color: #c19a06 !important; } #onetrust-consent-sdk .onetrust-pc-btn-handler { color: #c19a06 !important; border-color: #c19a06 !important; } Harbour Energy plc has agreed to acquire substantially all the subsidiaries of Waldorf Energy Partners Ltd. and Waldorf Production Ltd., currently in administration, for $170 million. The company, in a release Dec. 12, said the deal would add oil-weighted production of 20,000 boe/d and 2P reserves of 35 MMboe. In addition, the deal would increase Harbour’s interest in its operated Catcher oil and gas field to 90% from 50% and provide a new production base  for Harbour in the northern North Sea with the addition of a 29.5% non-operated interest in the EnQuest plc-operated Kraken oil field. The deal is expected to close in second-quarter 2026, subject to regulatory approvals and full and final settlement of all creditor claims against Waldorf’s subsidiaries.

Read More »

EIA: US oil inventories drop 1.8 million bbl

US commercial crude inventories for the week ended Dec. 5, excluding those in the Strategic Petroleum Reserve, dropped 1.8 million bbl from the previous week to 425.7 million bbl, which is about 4% below the average range for this time of year, according to the US Energy Information Administration’s (EIA) Weekly Petroleum Status Report. Total motor gasoline inventories gained 6.4 million bbl last week and are about 1% below the 5-year average range for this time of year. Finished gasoline inventories and blending components inventories rose. Distillate fuel inventories increased by 2.5 million bbl but are 7% below the 5-year average for this time of year. EIA reported that US crude refinery inputs last week averaged 16.9 million b/d, down 17,000 b/d from the previous week’s average. Refineries operated at 94.5% of their operable capacity. Gasoline production decreased to 9.6 million b/d, while distillate fuel production increased by 380,000 b/d, averaging 5.4 million b/d. US crude imports averaged 6.6 million b/d, up 609,000 b/d from the previous week’s average. Over the last 4 weeks, crude imports averaged 6.2 million b/d, down 7.7% from the same 4-week period last year. Total motor gasoline imports, including both finished gasoline and gasoline blending components, averaged 659,000 b/d. Distillate fuel imports averaged 181,000 b/d last week.

Read More »

Macquarie: Oil oversupply deepens as inventory growth accelerates

The global crude market remains firmly oversupplied, and current trends point to an increasingly imbalanced supply–demand environment through late-2025 and into early-2026. According to a note from Macquarie Group, a peak surplus of more than 4 million b/d is expected by first-quarter 2026. Signs of the surplus are showing with continued offshore builds, increasing onshore builds, and extremely strong freight rates, said Vikas Dwivedi, global energy strategist, Macquarie. Offshore crude inventories have surged by roughly 250 million bbl since late August, while onshore storage has risen by about 30 million bbl over the same period. Over the past month alone, combined offshore and onshore builds have accelerated to nearly 3 million b/d. Despite these rapid stock increases, the forward curve remains in backwardation—supported by continued Atlantic-to-Asia flows. Roughly one-third of offshore builds are linked to long-haul shipments originating from the Americas and moving toward Asia, tying up floating storage and delaying the impact on onshore tanks. However, as these volumes eventually discharge, the market expects visible onshore builds to accelerate—particularly in Asia first, followed by the US and Europe, the report said. “We expect onshore builds to accelerate through year 2025 and into early 2026, a process which should drive Brent towards the low $50 range with a possibility of reaching $45/bbl,” Macquarie reported.   Market signals Crude tanker freight rates are exceptionally high amid limited vessel availability. If rates climb further, key arbitrage pathways could shut, forcing more crude into onshore storage. Conversely, once cargoes discharge and vessels are freed up, freight rates could correct lower—another step that typically precedes visible inventory builds. “Either way, a freight sell off should lead rising storage statistics,” Macquarie said.  Meantime, price pressure is now spreading across core benchmarks. North Sea markers have softened, while key Brent-linked grades such as West African barrels and

Read More »

Executive Roundtable: Converging Disciplines in the AI Buildout

At Data Center Frontier, we rely on industry leaders to help us understand the most urgent challenges facing digital infrastructure. And in the fourth quarter of 2025, the data center industry is adjusting to a new kind of complexity.  AI-scale infrastructure is redefining what “mission critical” means, from megawatt density and modular delivery to the chemistry of cooling fluids and the automation of energy systems. Every project has arguably in effect now become an ecosystem challenge, demanding that electrical, mechanical, construction, and environmental disciplines act as one.  For this quarter’s Executive Roundtable, DCF convened subject matter experts from Ecolab, EdgeConneX, Rehlko and Schneider Electric – leaders spanning the full chain of facilities design, deployment, and operation. Their insights illuminate how liquid cooling, energy management, and sustainable process design in data centers are now converging to set the pace for the AI era. Our distinguished executive panelists for this quarter include: Rob Lowe, Director RD&E – Global High Tech, Ecolab Phillip Marangella, Chief Marketing and Product Officer, EdgeConneX Ben Rapp, Manager, Strategic Project Development, Rehlko Joe Reele, Vice President, Datacenter Solution Architects, Schneider Electric Today: Engineering the New Normal – Liquid Cooling at Scale  Today’s kickoff article grapples with how, as liquid cooling technology transitions to default hyperscale design, the challenge is no longer if, but how to scale builds safely, repeatably, and globally.  Cold plates, immersion, dielectric fluids, and liquid-to-chip loops are converging into factory-integrated building blocks, yet variability in chemistry, serviceability, materials, commissioning practices, and long-term maintenance threatens to fragment adoption just as demand accelerates.  Success now hinges on shared standards and tighter collaboration across OEMs, builders, and process specialists worldwide. So how do developers coordinate across the ecosystem to make liquid cooling a safe, maintainable global default? What’s Ahead in the Roundtable Over the coming days, our panel

Read More »

DCF Trends Summit 2025: AI for Good – How Operators, Vendors and Cooling Specialists See the Next Phase of AI Data Centers

At the 2025 Data Center Frontier Trends Summit (Aug. 26-28) in Reston, Va., the conversation around AI and infrastructure moved well past the hype. In a panel sponsored by Schneider Electric—“AI for Good: Building for AI Workloads and Using AI for Smarter Data Centers”—three industry leaders explored what it really means to design, cool and operate the new class of AI “factories,” while also turning AI inward to run those facilities more intelligently. Moderated by Data Center Frontier Editor in Chief Matt Vincent, the session brought together: Steve Carlini, VP, Innovation and Data Center Energy Management Business, Schneider Electric Sudhir Kalra, Chief Data Center Operations Officer, Compass Datacenters Andrew Whitmore, VP of Sales, Motivair Together, they traced both sides of the “AI for Good” equation: building for AI workloads at densities that would have sounded impossible just a few years ago, and using AI itself to reduce risk, improve efficiency and minimize environmental impact. From Bubble Talk to “AI Factories” Carlini opened by acknowledging the volatility surrounding AI investments, citing recent headlines and even Sam Altman’s public use of the word “bubble” to describe the current phase of exuberance. “It’s moving at an incredible pace,” Carlini noted, pointing out that roughly half of all VC money this year has flowed into AI, with more already spent than in all of the previous year. Not every investor will win, he said, and some companies pouring in hundreds of billions may not recoup their capital. But for infrastructure, the signal is clear: the trajectory is up and to the right. GPU generations are cycling faster than ever. Densities are climbing from high double-digits per rack toward hundreds of kilowatts. The hyperscale “AI factories,” as NVIDIA calls them, are scaling to campus capacities measured in gigawatts. Carlini reminded the audience that in 2024,

Read More »

FinOps Foundation sharpens FOCUS to reduce cloud cost chaos

“The big change that’s really started to happen in late 2024 early 2025 is that the FinOps practice started to expand past the cloud,” Storment said. “A lot of organizations got really good at using FinOps to manage the value of cloud, and then their organizations went, ‘oh, hey, we’re living in this happily hybrid state now where we’ve got cloud, SaaS, data center. Can you also apply the FinOps practice to our SaaS? Or can you apply it to our Snowflake? Can you apply it to our data center?’” The FinOps Foundation’s community has grown to approximately 100,000 practitioners. The organization now includes major cloud vendors, hardware providers like Nvidia and AMD, data center operators and data cloud platforms like Snowflake and Databricks. Some 96 of the Fortune 100 now participate in FinOps Foundation programs. The practice itself has shifted in two directions. It has moved left into earlier architectural and design processes, becoming more proactive rather than reactive. It has also moved up organizationally, from director-level cloud management roles to SVP and COO positions managing converged technology portfolios spanning multiple infrastructure types. This expansion has driven the evolution of FOCUS beyond its original cloud billing focus. Enterprises are implementing FOCUS as an internal standard for chargeback reporting even when their providers don’t generate native FOCUS data. Some newer cloud providers, particularly those focused on AI infrastructure, are using the FOCUS specification to define their billing data structures from the ground up rather than retrofitting existing systems. The FOCUS 1.3 release reflects this maturation, addressing technical gaps that have emerged as organizations apply cost management practices across increasingly complex hybrid environments. FOCUS 1.3 exposes cost allocation logic for shared infrastructure The most significant technical enhancement in FOCUS 1.3 addresses a gap in how shared infrastructure costs are allocated and

Read More »

Aetherflux joins the race to launch orbital data centers by 2027

Enterprises will connect to and manage orbital workloads “the same way they manage cloud workloads today,” using optical links, the spokesperson added. The company’s approach is to “continuously launch new hardware and quickly integrate the latest architectures,” with older systems running lower-priority tasks to serve out the full useful lifetime of their high-end GPUs. The company declined to disclose pricing. Aetherflux plans to launch about 30 satellites at a time on SpaceX Falcon 9 rockets. Before the data center launch, the company will launch a power-beaming demonstration satellite in 2026 to test transmission of one kilowatt of energy from orbit to ground stations, using infrared lasers. Competition in the sector has intensified in recent months. In November, Starcloud launched its Starcloud-1 satellite carrying an Nvidia H100 GPU, which is 100 times more powerful than any previous GPU flown in space, according to the company, and demonstrated running Google’s Gemma AI model in orbit. In the same month, Google announced Project Suncatcher, with a 2027 demonstration mission planned. Analysts see limited near-term applications Despite the competitive activity, orbital data centers won’t replace terrestrial cloud regions for general hosting through 2030, said Ashish Banerjee, senior principal analyst at Gartner. Instead, they suit specific workloads, including meeting data sovereignty requirements for jurisdictionally complex scenarios, offering disaster recovery immune to terrestrial risks, and providing asynchronous high-performance computing, he said. “Orbital centers are ideal for high-compute, low-I/O batch jobs,” Banerjee said. “Think molecular folding simulations for pharma, massive Monte Carlo financial simulations, or training specific AI model weights. If the job takes 48 hours, the 500ms latency penalty of LEO is irrelevant.” One immediate application involves processing satellite-generated data in orbit, he said. Earth observation satellites using synthetic aperture radar generate roughly 10 gigabytes per second, but limited downlink bandwidth creates bottlenecks. Processing data in

Read More »

Here’s what Oracle’s soaring infrastructure spend could mean for enterprises

He said he had earlier told analysts in a separate call that margins for AI workloads in these data centers would be in the 30% to 40% range over the life of a customer contract. Kehring reassured that there would be demand for the data centers when they were completed, pointing to Oracle’s increasing remaining performance obligations, or services contracted but not yet delivered, up $68 billion on the previous quarter, saying that Oracle has been seeing unprecedented demand for AI workloads driven by the likes of Meta and Nvidia. Rising debt and margin risks raise flags for CIOs For analysts, though, the swelling debt load is hard to dismiss, even with Oracle’s attempts to de-risk its spend and squeeze more efficiency out of its buildouts. Gogia sees Oracle already under pressure, with the financial ecosystem around the company pricing the risk — one of the largest debts in corporate history, crossing $100 billion even before the capex spend this quarter — evident in the rising cost of insuring the debt and the shift in credit outlook. “The combination of heavy capex, negative free cash flow, increasing financing cost and long-dated revenue commitments forms a structural pressure that will invariably finds its way into the commercial posture of the vendor,” Gogia said, hinting at an “eventual” increase in pricing of the company’s offerings. He was equally unconvinced by Magouyrk’s assurances about the margin profile of AI workloads as he believes that AI infrastructure, particularly GPU-heavy clusters, delivers significantly lower margins in the early years because utilisation takes time to ramp.

Read More »

New Nvidia software gives data centers deeper visibility into GPU thermals and reliability

Addressing the challenge Modern AI accelerators now draw more than 700W per GPU, and multi-GPU nodes can reach 6kW, creating concentrated heat zones, rapid power swings, and a higher risk of interconnect degradation in dense racks, according to Manish Rawat, semiconductor analyst at TechInsights. Traditional cooling methods and static power planning increasingly struggle to keep pace with these loads. “Rich vendor telemetry covering real-time power draw, bandwidth behavior, interconnect health, and airflow patterns shifts operators from reactive monitoring to proactive design,” Rawat said. “It enables thermally aware workload placement, faster adoption of liquid or hybrid cooling, and smarter network layouts that reduce heat-dense traffic clusters.” Rawat added that the software’s fleet-level configuration insights can also help operators catch silent errors caused by mismatched firmware or driver versions. This can improve training reproducibility and strengthen overall fleet stability. “Real-time error and interconnect health data also significantly accelerates root-cause analysis, reducing MTTR and minimizing cluster fragmentation,” Rawat said. These operational pressures can shape budget decisions and infrastructure strategy at the enterprise level.

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