Mastering the Poisson Distribution: Intuition and Foundations

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.

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.

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:

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 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.

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.

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.

Stay Ahead

Explore More Insights

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.

SSHStalker botnet brute-forces its way onto 7,000 Linux machines

However, so far the botnet hasn’t done much other than maintaining persistence on infected machines. It has the ability to launch DDoS (distributed denial of service) attacks and conduct cryptomining, but hasn’t done anything yet to monetize its access. That, Flare says, suggests either the operator is still staging the

Cisco extends AgenticOps model across networking, security, observability products

At the Cisco Live EMEA event going on in Amsterdam this week, Cisco unveiled a range of updates across its networking and security portfolio that are aimed at helping customers tap agentic AI technologies to more effectively run their enterprise infrastructure. “Agentic AI orchestrates workflows, moves data, communicates with other

NetBrain’s new AI agents automate network diagnosis

In testing, the system handled the majority of real-world network issues. “90% of the real-world network issues that they had when they threw them at it, it handled it,” Nixon said. “[People] couldn’t quite believe that it was at the 90% mark. People went in thinking, ‘Well, if this gives me

IBM FlashSystems gain AI-assisted telemetry, analytics

For security, the systems include a new FlashCore Module all-flash drive, which brings hardware-accelerated, real-time ransomware detection, data reduction, analytics and operations. The devices can spot anomalies and patterns in data that need to be remediated, IBM noted. “The next-generation IBM FlashSystem elevates storage to an intelligent, always-available layer, where autonomous

USA Labor Market Report Underpins Energy Demand

In a market update sent to Rigzone late Wednesday, Rystad Energy outlined that the January U.S. labor market report “surprise[d]… to the upside, underpinning energy demand”. Rystad noted in the report that the latest U.S. jobs data “shows promise, with the unemployment rate falling by 4.3 percent, pointing to market stability”. “Non-farm payrolls increased by 130,000 jobs in January, while the rise for December was downwardly revised to 48,000,” it pointed out, adding that the unemployment rate in December was 4.4 percent. “The latest data compares with consensus expectations of job gains of around 70,000 and the unemployment rate holding steady at 4.4 percent,” Rystad said. In the update, Rystad Energy Chief Economist Claudio Galimberti noted that payroll growth exceeded expectations and that unemployment edged lower. “Following a series of weaker private indicators, the data suggests stabilization rather than strong acceleration,” Galimberti said. “Markets that had positioned for a rapid easing cycle responded by repricing yields higher and scaling back expectations for near-term rate cuts,” he added. “For energy markets, the implications are moderately supportive. A resilient labor market underpins demand for transport fuels, petrochemicals and power generation, reducing downside risks to U.S. consumption at a time when macro sentiment had turned cautious,” he continued. “While the U.S. is not the primary driver of incremental global oil demand, labor market stability reinforces the view that the demand picture is firming up,” he went on to state. Galimberti noted in the update that “revisions to prior data confirm that the cycle is mature, not accelerating”. “Still, in a market already balancing OPEC+ supply management against geopolitical risk, a firmer U.S. macro signal marginally strengthens the demand outlook,” he said. “The result is a modestly constructive backdrop for oil prices in the near term, without materially shifting the fundamentals,” Galimberti concluded. In

Energy Department Announces $175 Million to Modernize Coal Plants, Keeping Affordable Reliable Power Online for Americans

WASHINGTON—The U.S. Department of Energy (DOE) today announced $175 million in funding for six projects to modernize, retrofit, and extend the useful life of coal-fired power plants that serve rural and remote communities across the United States, keeping dependable energy sources online, strengthening grid reliability, and helping keep electricity costs low for American families and businesses. The projects are part of the Department’s $525 million effort to expand and reinvigorate America’s coal fleet through targeted upgrades that increase efficiency, extend plant life, and add dependable capacity using infrastructure that is already built and connected to the grid. Modernizing existing plants provides one of the fastest and most cost-effective ways to deliver reliable power while preserving high-wage energy jobs, particularly across Appalachian communities that have long powered the nation. “For years, previous administrations targeted America’s coal industry and the workers who power our country, forcing the premature closure of reliable plants, and driving up electricity costs,” said U.S. Secretary of Energy Chris Wright. “President Trump has ended the war on American coal and is restoring common-sense energy policy. These investments will keep America’s coal plants operating, keep costs low for Americans, and ensure we have the reliable power needed to keep the lights on and power our future.” These actions advance President Trump’s Executive Orders Reinvigorating America’s Beautiful Clean Coal Industry and Strengthening the Reliability and Security of the United States Electric Grid to restore common-sense energy policies that prioritize dependable power, affordability, and American workers. As electricity demand continues to grow, dependable, around-the-clock generation remains essential to maintaining a reliable and affordable power system. By upgrading existing coal facilities, DOE is strengthening the backbone of America’s power grid and ensuring communities have access to secure, reliable energy when they need it most. Selected projects include the following: Appalachian Power Company (Letart and Winfield, West Virginia) will upgrade two coal-fired plants in West

Oil Gains As Middle East Tensions Rise

Oil gained as tensions in the Middle East outweighed concerns that there is a supply glut building in the market. West Texas Intermediate rose to settle above $64 after the Wall Street Journal said that the Pentagon has told a second aircraft carrier strike group to prepare to deploy to the Middle East, citing three US officials. That follows an earlier report from the news agency that the US is considering seizing tankers with Iranian crude. “Oil right now, and just the rest of the commodity complex, is really dominated by three things: geopolitics, trade and technology,” Francisco Blanch, head of commodities research at Bank of America Global Research, said in a Bloomberg Television interview. “Certainly, right now, geopolitics are the main driving force pushing oil close to the high end of this year’s range.” Iran is the fourth-largest OPEC producer, pumping an estimated 3.3 million barrels a day in January, according to a Bloomberg survey. Crude and condensate shipments totaled about 1.63 million barrels a day last month, vessel-tracking data show. The WSJ report meant crude erased earlier losses after President Donald Trump said in a social media post that he insisted that talks with Iran continue in a meeting with Israeli Prime Minister Benjamin Netanyahu. It was widely expected that Netanyahu would push for a broad curtailment of the Islamic Republic’s military activities in the region. The commodity has also received support earlier after strong US jobs data brightened the outlook for the world’s largest economy. “A resilient labor market underpins demand for transport fuels, petrochemicals and power generation, reducing downside risks to US consumption at a time when macro sentiment had turned cautious,” said Claudio Galimberti, chief economist at Rystad Energy. The strong numbers are a sign that the demand picture is firming up, he added. Crude

OPEC Says Oil Production Declined Last Month

OPEC+ oil production declined sharply last month amid losses in Kazakhstan, Venezuela and Iran, the group said. The 22 nations of the alliance produced an average of 42.448 million barrels a day in January, or 439,000 a day less than the previous month, according to a copy of the group’s monthly report obtained by Bloomberg. Kazakhstan accounted for more than half of the drop. While the report didn’t give a reason for the overall decline, Kazakhstan’s production fell as it suspended operations at the Tengiz oil field, the country’s largest. The Chevron-led venture started to restore output there at the end of last month. Separately, Venezuelan oil exports were disrupted by a US blockade during the ousting of former President Nicolas Maduro, while Iran continues to face American sanctions. Saudi Arabia and several other key nations held steady in January as the Organization of the Petroleum Exporting Countries and its allies began a three-month freeze to offset a seasonal lull in consumption. They’ll meet online on March 1 to review production levels for April and beyond. OPEC kept forecasts for global oil supply and demand unchanged for this year and next, according to the report. 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.

Ukraine Hits Lukoil Refinery

Ukraine attacked an oil refinery in Russia’s Volgograd region in the first major strike on Russia’s oil-processing industry this year. An overnight drone strike sparked a fire at the facility, Ukraine’s General Staff said on Telegram Wednesday. “The scope of the damage is being clarified,” it said, adding that the refinery helps supply the Russian army. Ukraine carried out multiple high-precision strikes on Russia’s energy assets last year, leading to refinery shutdowns, disruptions at oil terminals and the rerouting of some tankers. The attacks were designed to curb the Kremlin’s energy revenues and restrict fuel supplies to Russian front lines in the war, now nearing its fifth year. The Volgograd refinery, which was attacked several times last year, has a design capacity of about 300,000 barrels of crude a day. It mainly supplies oil products to southern Russia, with some volumes exported. The administration of the Volgograd region said in a Telegram statement that an an industrial plant caught fire after a drone attack but did not name the facility. Lukoil, Russia’s largest private oil producer, did not immediately respond to a request for comment. Satellite images from NASA’s Fire Information for Resource Management System show multiple fires at the refinery that began during the night of Feb. 10-11. The fires were not visible the previous day, according to the data. In January, Ukraine targeted three small independent Russian refineries, which together account for about 7% of Russia’s typical monthly crude throughput. The lull in drone strikes had offered temporary relief for Russia’s downstream sector, allowing refinery runs to gradually increase. Encouraged by the recovery, the government lifted its ban on most gasoline exports, permitting producers to resume shipments in February — a month earlier than planned. While Ukrainian attacks on Russia’s oil industry slowed in January, Moscow continued intense assaults on energy infrastructure

TotalEnergies Cuts Buyback to Lower End of Range

(Update) February 11, 2026, 5:10 PM GMT: Article updated with comments on dividend growth, potential investment decisions and acquisitions from 14th paragraph. TotalEnergies SE trimmed its share buybacks to the lower end of its guidance range, aiming to keep debt in check as it adjusts to lower oil prices. The company plans to repurchase $750 million of stock in the first quarter, compared with $1.5 billion in the final three months of 2025, it said in an earnings statement Wednesday. For the year, its buyback target was kept at a range of $3 billion to $6 billion. TotalEnergies is the third and last of Europe’s top oil and gas producers to release earnings after Shell Plc and BP Plc published disappointing quarterly reports. The company has a lower ratio of debt to equity than its European peers and kept quarterly dividend unchanged. “This year we want to balance cash generation with cash expenditure,” Chief Executive Officer Patrick Pouyanne said during a press conference in Paris to discuss earnings. “We don’t know what will happen this year. We want to keep a healthy balance sheet.” Shares of Total closed 2.7% up, at their highest since July 2024. The company has a “solid balance sheet despite uncertain environment,“ Jefferies analysts led by Mark Wilson said in a note after the earnings release. While Big Oil is still churning out hefty profits, cash flows — particularly in Europe — have been undermined by last year’s 18% dive in crude prices. There are also widespread forecasts that the market will remain oversupplied this year as production swells both inside and outside the OPEC+ alliance. “Oil supply remains abundant, so the market is rather trending down,” Pouyanne said, adding that sanctions on Russia are causing a buildup of the nation’s crude at sea. Total’s adjusted

Energy providers seek flexible load strategies for data center operations

“In theory, yes, they’d have to wait a little bit longer while their queries are routed to a data center that has capacity,” said Lawrence. The one thing the industry cannot do is operate like it has in the past, where data center power was tuned and then forgotten for six months. Previously, data centers would test their power sources once or twice a year. They don’t have that luxury anymore. They need to check their power sources and loads far more regularly, according to Lawrence. “I think that for that for the data center industry to continue to survive like we all need it, there’s going to have to be some realignment on the incentives to why somebody would become flexible,” said Lawrence. The survey suggests that utilities and load operators expect to expand their demand response activities and budgets in the near term. Sixty-three percent of respondents anticipate DR program funding to grow by 50% or more over the next three years. While they remain a major source of load growth and system strain, 57% of respondents indicate that onsite power generation from data centers will be most important to improving grid stability over the next five years. One of the proposed fixes to the power shortage has been small modular nuclear reactors. These have gained a lot of traction in the marketplace even if they have nothing to sell yet. But Lawrence said that that’s not an ideal solution for existing power generators, ironically enough.

Nokia predicts huge WAN traffic growth, but experts question assumptions

Consumer, which includes both mobile access and fixed access, including fixed wireless access. Enterprise and industrial, which covers wide-area connectivity that supports knowledge work, automation, machine vision, robotics coordination, field support, and industrial IoT. AI, including applications that people directly invoke, such as assistants, copilots, and media generation, as well as autonomous use cases in which AI systems trigger other AI systems to perform functions and move data across networks. The report outlines three scenarios: conservative, moderate, and aggressive. “Our goal is to present scenarios that fall within a realistic range of possible outcomes, encouraging stakeholders to plan across the full spectrum of high-impact demand possibilities,” the report says. Nokia’s prediction for global WAN traffic growth ranges from a 13% CAGR for the conservative scenario to 16% CAGR for moderate and 22% CAGR for aggressive. Looking more closely at the moderate scenario, it’s clear that consumer traffic dominates. Enterprise and industrial traffic make up only about 14% to 17% of overall WAN traffic, although their share is expected to grow during the 10-year forecast period. “On the consumer side, the vast majority of traffic by volume is video,” says William Webb, CEO of the consulting firm Commcisive. Asked whether any of that consumer traffic is at some point served up by enterprises, the answer is a decisive “no.” It’s mostly YouTube and streaming services like Netflix, he says. In short, that doesn’t raise enterprise concerns. Nokia predicts AI traffic boom AI is a different story. “Consumer- and enterprise-generated AI traffic imposes a substantial impact on the wide-area network (WAN) by adding AI workloads processed by data centers across the WAN. AI traffic does not stay inside one data center; it moves across edge, metro, core, and cloud infrastructure, driving dense lateral flows and new capacity demands,” the report says. An

Cisco amps up Silicon One line, delivers new systems and optics for AI networking

Those building blocks include the new G300 as well as the G200 51.2 Tbps chip, which is aimed at spine and aggregation applications, and the G100 25.6 Tbps chip, which is aimed at leaf operations. Expanded portfolio of Silicon One P200-powered systems Cisco in October rolled out the P200 Silicon One chip and the high-end, 51.2 Tbps 8223 router aimed at distributed AI workloads. That system supports Octal Small Form-Factor Pluggable (OSFP) and Quad Small Form-Factor Pluggable Double Density (QSFP-DD) optical form factors that help the box support geographically dispersed AI clusters. Cisco grew the G200 family this week with the addition of the 8122X-64EF-O, a 64x800G switch that will run the SONiC OS and includes support for Cisco 800G Linear Pluggable Optics (LPO) connectivity. LPO components typically set up direct links between fiber optic modules, eliminating the need for traditional components such as a digital signal processor. Cisco said its P200 systems running IOS XR software now better support core routing services to allow data-center-to-data-center links and data center interconnect applications. In addition, Cisco introduced a P200-powered 88-LC2-36EF-M line card, which delivers 28.8T of capacity. “Available for both our 8-slot and 18-slot modular systems, this line card enables up to an unprecedented 518.4T of total system bandwidth, the highest in the industry,” wrote Guru Shenoy, senior vice president of the Cisco provider connectivity group, in a blog post about the news. “When paired with Cisco 800G ZR/ZR+ coherent pluggable optics, these systems can easily connect sites over 1,000 kilometers apart, providing the high-density performance needed for modern data center interconnects and core routing.”

NetBox Labs ships AI copilot designed for network engineers, not developers

Natural language for network engineers Beevers explained that network operations teams face two fundamental barriers to automation. First, they lack accurate data about their infrastructure. Second, they aren’t software developers and shouldn’t have to become them. “These are not software developers. They are network engineers or IT infrastructure engineers,” Beevers said. “The big realization for us through the copilot journey is they will never be software developers. Let’s stop trying to make them be. Let’s let these computers that are really good at being software developers do that, and let’s let the network engineers or the data center engineers be really good at what they’re really good at.” That vision drove the development of NetBox Copilot’s natural language interface and its capabilities. Grounding AI in infrastructure reality The challenge with deploying AI in network operations is trust. Generic large language models hallucinate, produce inconsistent results, and lack the operational context to make reliable decisions. NetBox Copilot addresses this by grounding the AI agent in NetBox’s comprehensive infrastructure data model. NetBox serves as the system of record for network and infrastructure teams, maintaining a semantic map of devices, connections, IP addressing, rack layouts, power distribution and the relationships between these elements. Copilot has native awareness of this data structure and the context it provides. This enables queries that would be difficult or impossible with traditional interfaces. Network engineers can ask “Which devices are missing IP addresses?” to validate data completeness, “Who changed this prefix last week?” for change tracking and compliance, or “What depends on this switch?” for impact analysis before maintenance windows.

US pushes voluntary pact to curb AI data center energy impact

Others note that cost pressure isn’t limited to the server rack. Danish Faruqui, CEO of Fab Economics, said the AI ecosystem is layered from silicon to software services, creating multiple points where infrastructure expenses eventually resurface. “Cloud service providers are likely to gradually introduce more granular pricing models across cloud, AI, and SaaS offerings, tailored by customer type, as they work to absorb the costs associated with the White House energy and grid compact,” Faruqui said. This may not show up as explicit energy surcharges, but instead surface through reduced discounts, higher spending commitments, and premiums for guaranteed capacity or performance. “Smaller enterprises will feel the impact first, while large strategic customers remain insulated longer,” Rawat said. “Ultimately, the compact would delay and redistribute cost pressure; it does not eliminate it.” Implications for data center design The proposal is also likely to accelerate changes in how AI facilities are designed. “Data centers will evolve into localized microgrids that combine utility power with on-site generation and higher-level implementation of battery energy storage systems,” Faruqui said. “Designing for grid interaction will become imperative for AI data centers, requiring intelligent, high-speed switching gear, increased battery energy storage capacity for frequency regulation, and advanced control systems that can manage on-site resources.”

Intel teams with SoftBank to develop new memory type

However, don’t expect anything anytime soon. Intel’s Director of Global Strategic Partnerships Sanam Masroor outlined the plans in a blog post. Operations are expected to begin in Q1 2026, with prototypes due in 2027 and commercial products by 2030. While Intel has not come out and said it, that memory design is almost identical to HBM used in GPU accelerators and AI data centers. HBM sits right on the GPU die for immediate access to the GPU, unlike standard DRAM which resides on memory sticks plugged into the motherboard. HBM is much faster than DDR memory but is also much more expensive to produce. It’s also much more profitable than standard DRAM which is why the big three memory makers – Micron, Samsung, and SK Hynix – are favoring production of it.

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.

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

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

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