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

Nutanix expands beyond HCI

The Pure Storage integration will also be supported within Cisco’s FlashStack offering, creating a “FlashStack with Nutanix” solution with storage provided by Pure, networking capabilities as well as UCS servers from Cisco, and then the common Nutanix Cloud Platform. Cloud Native AOS: Breaking free from hypervisors Another sharp departure from

Read More »

IBM introduces new generation of LinuxOne AI mainframe

In addition to generative AI applications, new multiple model AI approaches are engineered to enhance prediction and accuracy in many industry use cases like advanced fraud detection, image processing and retail automation, according to IBM. LinuxONE Emperor 5 also comes with advanced security features specifically designed for the AI threat

Read More »

Business leaders and SNP call on Starmer to visit Aberdeen amid North Sea job losses

Aberdeen business leaders and the SNP are calling on the Prime Minister to visit the north-east of Scotland as they blamed Labour policies for yet more job losses in the oil and gas sector. On Wednesday, Harbour Energy announced that it would cut 250 jobs from its onshore operations, accounting for a 25% reduction in headcount. The UK’s largest producer of oil and gas has claimed that the hostile fiscal policy facing oil and gas businesses prompted the decision as it slows investment in the country, opting to allocate funds overseas. On the day of this announcement, Aberdeen South MP and SNP Westminster leader Stephen Flynn brought the news to the attention of prime minister Sir Keir Starmer. © BloombergEmissions from chimneys at the British Steel Ltd. plant in Scunthorpe, UK. He asked Starmer to “explain to my constituents why he is willing to move heaven and earth to save jobs in Scunthorpe while destroying jobs in Scotland.” The SNP leader was referring to the government’s recent move to nationalise British Steel. The UK government took control of the British steel company from its Chinese owner, Jingye Group, after losses from its steelmaking operations forced it to the brink. Now the SNP MP, alongside his colleagues in Westminster and Holyrood, has written to the Labour Party leader, inviting him to see the impacts his government’s energy policy is having on Aberdeen and its people. “We are writing to you as the local MPs and MSPs for Aberdeen, to invite you to urgently visit Aberdeen to meet with local representatives, businesses, trade unions and workers to hear about the damaging impact that Labour government policies are having on Scottish energy jobs – and to discuss the urgent investment needed to protect jobs and deliver prosperity,” the letter reads. ‘Haemorrhaging investment in

Read More »

Oil Gains 3% as Trade Hopes Rise

Oil rose as President Donald Trump announced a trade framework with the UK, spurring some optimism about deals to come. West Texas Intermediate climbed 3.2% to approach $60 a barrel. Trump said the UK would fast-track US items through its customs process and reduce barriers on billions of dollars of agricultural, chemical, energy and industrial exports, including ethanol. Notably, the terms are limited in scope and a 10% baseline tariff remains. The British deal is raising investors’ confidence that agreements can be reached in the more complicated trade talks that lie ahead, specifically negotiations between US and Chinese officials kicking off this weekend. Trump said that the 145% levy against China, the world’s largest crude-importer, could be lowered if talks go well. “The real driver of risk assets today appears to be renewed optimism around progress in the US–China trade talks,” said Rebecca Babin, a senior energy trader at CIBC Private Wealth Group. “It’s also worth noting that sentiment toward crude remains overwhelmingly bearish.” Crude has slid since Trump took office on concerns that his global trade war will dent economic growth and slow energy demand. Adding to the bearishness, OPEC+ has decided to revive idled output faster than expected. Already, the drop in oil prices is spurring American shale producers to cut spending in the Permian Basin. Still, small pockets of bullishness are visible in the options market. There was active trading of Brent $95 September call options, which profit when futures rise. The US on Thursday sanctioned a third Chinese “teapot” oil refinery and various other entities associated with Iran, days ahead of a fourth round of nuclear talks between Washington and Tehran. The failure of the negotiations could push Brent up toward $70 a barrel, Citigroup analysts including Eric Lee said in a note. In the US,

Read More »

Indian LNG Buyers Embrace USA Benchmark to Balance Volatility

Indian liquefied natural gas importers have signed a flurry of long-term purchase agreements linked to the US price benchmark, the latest effort by the nation’s buyers to protect themselves from volatile markets. State-owned companies have signed at least four contracts since December, totaling nearly 11 million tons per year, priced to the Henry Hub index, according to the executives familiar with the deals. Until now, most of India’s long-term contracts have been linked to crude oil, the traditional way to price LNG deals. Pricing the fuel to the Henry Hub index doesn’t necessarily mean that the fuel will come from the US, rather it is a move to hedge risk.  India’s consumers — from power plants to petrochemical facilities — are highly price-sensitive as gas competes head-to-head with cheaper and dirtier alternatives. Companies that relied on the spot market or oil-linked contracts have periodically been forced to cut back purchases due to price spikes. US gas futures have also been relatively less volatile and more liquid than the Asian spot benchmark, the Japan-Korea Marker. “The last ten year average shows that there have been periods during winter months JKM benchmark surged beyond imagination, while Henry Hub prices saw proportionally smaller growth,” Bharat Petroelum Corp Ltd’s Director Finance V.R.K. Gupta said. BPCL in February signed a deal with ADNOC Trading for 2.5 million tons of LNG for five years. The Mumbai-based refiner will evaluate the performance of the deal and may sign more such contracts, Gupta said.  Indian Oil Corp. last week signed a deal with Trafigura for 2.5 million tons, or 27 cargoes, spread over five years, with supplies starting the middle of this year. The recent deals have been signed at a 115% link to Henry Hub plus $5 to $6 per million British thermal units. The supply is

Read More »

PJM, utilities urge FERC to dismiss call for colocation settlement talks

The Federal Energy Regulatory Commission should reject a call for a 90-day pause in its deliberations over the PJM Interconnection’s rules for colocating data centers at power plants, according to PJM, major utilities and other organizations. “The national interest will be best served by a quick dismissal of this proceeding, and a ruling that the existing PJM Tariff remains just and reasonable,” PJM transmission owners said in a Wednesday filing urging FERC to dismiss a call for stakeholder settlement talks. “Rather than fighting about a wish list of new rules, the parties will then instead begin to focus on obtaining service under the rules in place today.” The transmission owners include utility companies such as American Electric Power, Dominion Energy, Duke Energy, Exelon, FirstEnergy and PPL Electric. “The record is clear — no matter how connected to the PJM transmission system, large loads pose both a safety and a reliability concern,” the utilities said. “It is unrealistic to ask the [transmission owners] to accede to these demands in the context of settlement procedures while those questions remain unresolved.” PJM also wants FERC to ignore the call for settlement discussions that was made in late April by the Electric Power Supply Association, the PJM Power Providers Group, Calpine, Cogentrix Energy Power Management, Constellation Energy Generation and LS Power Development. “The Commission should not pause its work on offering the industry guidance on a path forward for co-location arrangements,” PJM said in a Monday filing. The call for settlement talks lacks broad stakeholder support, PJM said, noting it is holding a workshop on “large load” issues on Friday. American Municipal Power, a wholesale power provider for public power utilities, and Northern Virginia Electric Cooperative and Northeastern Rural Electric Membership Corp. also oppose holding settlement talks. Beside the power generators and trade organizations,

Read More »

IRA’s fate unclear as Republicans look to finance megabill

Dive Brief: The Inflation Reduction Act – which passed in 2022 without any Republican support and is anticipated to cost taxpayers between $780 billion and $2 trillion over its first ten years – is likely to be targeted for cuts as the Republican-controlled Congress aims to cut spending by $2 trillion in order to cut taxes by $4.5 trillion. However, certain provisions of the IRA have won support with Republican lawmakers, setting up likely disagreements over cuts in the budget reconciliation process. That process is already expected to be “very contentious,” said Harry Godfrey, who leads Advanced Energy United’s federal investment and manufacturing working group. “[House Ways and Means Committee Chairman Jason Smith] and the Ways and Means Republicans will need revenue and will be seeking it, and have been saying all along that the IRA is an area they’re going to look at,” said Ryan Abraham, a principal with Ernst & Young’s Washington Council advisory practice. “But obviously there are some concerns among some members.” Dive Insight: Abraham noted the May 1 letter sent by 26 House Republicans to Chairman Smith, advocating for the preservation of the IRA’s 45U, 45Y, and 48E tax credits. The letter advocates on behalf of nuclear power specifically, “[urging Smith] to maintain federal investment in the existing nuclear energy fleet while accelerating deployment of the next generation of nuclear power technologies.” The 45U credit is the IRA’s zero-emission nuclear power production credit, while 48E and 45Y are technology-neutral credits, which were targeted in legislation introduced in April by Rep. Julie Fedorchak, R-N.D. “There’s a lot of concern that some of the bonus items that have been created in the IRA, like direct pay and transferability, which were also in that Fedorchak bill, could also get targeted,” Abraham said. However, he said, “Chairman Smith is aware

Read More »

Energy Department Aligns Award Criteria for For-profit, Non-profit Organizations, and State and Local Governments, Saving $935 Million Annually

WASHINGTON — The U.S. Department of Energy (DOE) today announced three new policy actions that are projected to save more than $935 million annually for the American taxpayer, while expanding American innovation and scientific research. In three new policy memorandums, the DOE announced that it will follow best practices used by fellow grant providers and limit “indirect costs” of DOE funding to 10% for state and local governments, 15% for non-profit organizations, and 15% for for-profit companies. The Energy Department expects to generate over $935 million in annual cost savings for the American people, delivering on President Trump’s commitment to bring greater transparency and efficiency to federal government spending. Estimated savings are based on applying the new policies to 2024 fiscal year spending. “This action ensures that Department of Energy funds are supporting state, local, for-profit and non-profit initiatives that make energy more affordable and secure for Americans, not funding administrative costs,” U.S. Secretary of Energy Chris Wright said. “By aligning our policy on indirect costs with industry standards, we are increasing accountability of taxpayer dollars and ensuring the American people are getting the greatest value possible from these DOE programs.” These policy actions follow an announcement made in April to limit financial support of “indirect costs” of DOE research funding at colleges and universities to 15%, saving an estimated additional $405 million annually. By enacting indirect cost limits, the Department aligns its practices with those common for other grant providers. The full three memorandums are available below: POLICY FLASH SUBJECT: Adjusting Department of Energy Financial Assistance Policy for State and Local Governments’ Financial Assistance Awards BACKGROUND: Pursuant to 5 U.S.C. 553(a)(2), the Department of Energy (“Department”) is updating its policy with respect to Department financial assistance funding awarded to state and local governments. Through its financial assistance programs (which include grants and cooperative agreements),

Read More »

Tech CEOs warn Senate: Outdated US power grid threatens AI ambitions

The implications are clear: without dramatic improvements to the US energy infrastructure, the nation’s AI ambitions could be significantly constrained by simple physical limitations – the inability to power the massive computing clusters necessary for advanced AI development and deployment. Streamlining permitting processes The tech executives have offered specific recommendations to address these challenges, with several focusing on the need to dramatically accelerate permitting processes for both energy generation and the transmission infrastructure needed to deliver that power to AI facilities, the report added. Intrator specifically called for efforts “to streamline the permitting process to enable the addition of new sources of generation and the transmission infrastructure to deliver it,” noting that current regulatory frameworks were not designed with the urgent timelines of the AI race in mind. This acceleration would help technology companies build and power the massive data centers needed for AI training and inference, which require enormous amounts of electricity delivered reliably and consistently. Beyond the cloud: bringing AI to everyday devices While much of the testimony focused on large-scale infrastructure needs, AMD CEO Lisa Su emphasized that true AI leadership requires “rapidly building data centers at scale and powering them with reliable, affordable, and clean energy sources.” Su also highlighted the importance of democratizing access to AI technologies: “Moving faster also means moving AI beyond the cloud. To ensure every American benefits, AI must be built into the devices we use every day and made as accessible and dependable as electricity.”

Read More »

Networking errors pose threat to data center reliability

Still, IT and networking issues increased in 2024, according to Uptime Institute. The analysis attributed the rise in outages due to increased IT and network complexity, specifically, change management and misconfigurations. “Particularly with distributed services, cloud services, we find that cascading failures often occur when networking equipment is replicated across an entire network,” Lawrence explained. “Sometimes the failure of one forces traffic to move in one direction, overloading capacity at another data center.” The most common causes of major network-related outages were cited as: Configuration/change management failure: 50% Third-party network provider failure: 34% Hardware failure: 31% Firmware/software error: 26% Line breakages: 17% Malicious cyberattack: 17% Network overload/congestion failure: 13% Corrupted firewall/routing tables issues: 8% Weather-related incident: 7% Configuration/change management issues also attributed for 62% of the most common causes of major IT system-/software-related outages. Change-related disruptions consistently are responsible for software-related outages. Human error continues to be one of the “most persistent challenges in data center operations,” according to Uptime’s analysis. The report found that the biggest cause of these failures is data center staff failing to follow established procedures, which has increased by about 10 percentage points compared to 2023. “These are things that were 100% under our control. I mean, we can’t control when the UPS module fails because it was either poorly manufactured, it had a flaw, or something else. This is 100% under our control,” Brown said. The most common causes of major human error-related outages were reported as:

Read More »

Liquid cooling technologies: reducing data center environmental impact

“Highly optimized cold-plate or one-phase immersion cooling technologies can perform on par with two-phase immersion, making all three liquid-cooling technologies desirable options,” the researchers wrote. Factors to consider There are numerous factors to consider when adopting liquid cooling technologies, according to Microsoft’s researchers. First, they advise performing a full environmental, health, and safety analysis, and end-to-end life cycle impact analysis. “Analyzing the full data center ecosystem to include systems interactions across software, chip, server, rack, tank, and cooling fluids allows decision makers to understand where savings in environmental impacts can be made,” they wrote. It is also important to engage with fluid vendors and regulators early, to understand chemical composition, disposal methods, and compliance risks. And associated socioeconomic, community, and business impacts are equally critical to assess. More specific environmental considerations include ozone depletion and global warming potential; the researchers emphasized that operators should only use fluids with low to zero ozone depletion potential (ODP) values, and not hydrofluorocarbons or carbon dioxide. It is also critical to analyze a fluid’s viscosity (thickness or stickiness), flammability, and overall volatility. And operators should only use fluids with minimal bioaccumulation (the buildup of chemicals in lifeforms, typically in fish) and terrestrial and aquatic toxicity. Finally, once up and running, data center operators should monitor server lifespan and failure rates, tracking performance uptime and adjusting IT refresh rates accordingly.

Read More »

Cisco unveils prototype quantum networking chip

Clock synchronization allows for coordinated time-dependent communications between end points that might be cloud databases or in large global databases that could be sitting across the country or across the world, he said. “We saw recently when we were visiting Lawrence Berkeley Labs where they have all of these data sources such as radio telescopes, optical telescopes, satellites, the James Webb platform. All of these end points are taking snapshots of a piece of space, and they need to synchronize those snapshots to the picosecond level, because you want to detect things like meteorites, something that is moving faster than the rotational speed of planet Earth. So the only way you can detect that quickly is if you synchronize these snapshots at the picosecond level,” Pandey said. For security use cases, the chip can ensure that if an eavesdropper tries to intercept the quantum signals carrying the key, they will likely disturb the state of the qubits, and this disturbance can be detected by the legitimate communicating parties and the link will be dropped, protecting the sender’s data. This feature is typically implemented in a Quantum Key Distribution system. Location information can serve as a critical credential for systems to authenticate control access, Pandey said. The prototype quantum entanglement chip is just part of the research Cisco is doing to accelerate practical quantum computing and the development of future quantum data centers.  The quantum data center that Cisco envisions would have the capability to execute numerous quantum circuits, feature dynamic network interconnection, and utilize various entanglement generation protocols. The idea is to build a network connecting a large number of smaller processors in a controlled environment, the data center warehouse, and provide them as a service to a larger user base, according to Cisco.  The challenges for quantum data center network fabric

Read More »

Zyxel launches 100GbE switch for enterprise networks

Port specifications include: 48 SFP28 ports supporting dual-rate 10GbE/25GbE connectivity 8 QSFP28 ports supporting 100GbE connections Console port for direct management access Layer 3 routing capabilities include static routing with support for access control lists (ACLs) and VLAN segmentation. The switch implements IEEE 802.1Q VLAN tagging, port isolation, and port mirroring for traffic analysis. For link aggregation, the switch supports IEEE 802.3ad for increased throughput and redundancy between switches or servers. Target applications and use cases The CX4800-56F targets multiple deployment scenarios where high-capacity backbone connectivity and flexible port configurations are required. “This will be for service providers initially or large deployments where they need a high capacity backbone to deliver a primarily 10G access layer to the end point,” explains Nguyen. “Now with Wi-Fi 7, more 10G/25G capable POE switches are being powered up and need interconnectivity without the bottleneck. We see this for data centers, campus, MDU (Multi-Dwelling Unit) buildings or community deployments.” Management is handled through Zyxel’s NebulaFlex Pro technology, which supports both standalone configuration and cloud management via the Nebula Control Center (NCC). The switch includes a one-year professional pack license providing IGMP technology and network analytics features. The SFP28 ports maintain backward compatibility between 10G and 25G standards, enabling phased migration paths for organizations transitioning between these speeds.

Read More »

Engineers rush to master new skills for AI-driven data centers

According to the Uptime Institute survey, 57% of data centers are increasing salary spending. Data center job roles that saw the highest increases were in operations management – 49% of data center operators said they saw highest increases in this category – followed by junior and mid-level operations staff at 45%, and senior management and strategy at 35%. Other job categories that saw salary growth were electrical, at 32% and mechanical, at 23%. Organizations are also paying premiums on top of salaries for particular skills and certifications. Foote Partners tracks pay premiums for more than 1,300 certified and non-certified skills for IT jobs in general. The company doesn’t segment the data based on whether the jobs themselves are data center jobs, but it does track 60 skills and certifications related to data center management, including skills such as storage area networking, LAN, and AIOps, and 24 data center-related certificates from Cisco, Juniper, VMware and other organizations. “Five of the eight data center-related skills recording market value gains in cash pay premiums in the last twelve months are all AI-related skills,” says David Foote, chief analyst at Foote Partners. “In fact, they are all among the highest-paying skills for all 723 non-certified skills we report.” These skills bring in 16% to 22% of base salary, he says. AIOps, for example, saw an 11% increase in market value over the past year, now bringing in a premium of 20% over base salary, according to Foote data. MLOps now brings in a 22% premium. “Again, these AI skills have many uses of which the data center is only one,” Foote adds. The percentage increase in the specific subset of these skills in data centers jobs may vary. The Uptime Institute survey suggests that the higher pay is motivating workers to stay in the

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