Why Can Probability Functions Represent Uncertainty? — How AI Calculates “Plausibility”

books : What Kind of Mathematical Worldview Is AI Built Upon?: From Finite Rules to Generative Models

Last week, we considered transcendental functions and AI.

Transcendental functions such as exponential functions, logarithmic functions, and trigonometric functions describe change, growth, periodicity, and continuous transformation—phenomena that cannot be expressed through simple arithmetic operations alone. The ability of AI to learn complex relationships and generate new representations from inputs also rests on such nonlinear mathematical structures.

But complexity is not the only thing AI must handle.

The real world is always uncertain.

How much demand will there be tomorrow?
Is an equipment anomaly actually occurring?
Will a customer leave?
Which word is likely to come next in this sentence?

AI does not know a single correct answer in advance. It compares multiple possibilities and assigns each a degree of plausibility as it makes predictions and generates outputs.

The fundamental structure that enables this is the probability function.

A Probability Function Does Not Decide the Future in a Single Way

When we hear the word “function,” we often imagine something that returns a single answer for a given input.

[
y=f(x)
]

For example, a function may take temperature as an input and return electricity demand. Or it may take the price of a product and return its sales volume. Such functions describe a correspondence between inputs and outputs.

But real-world events are not determined by a single value.

Even with the same temperature, the same day of the week, and the same location, tomorrow’s sales may differ. It may rain, or it may not. Human behavior also includes unobservable circumstances and chance.

A probability function therefore does not return just one answer. Instead, it assigns weights to multiple possible outcomes.

[
P(Y=y \mid X=x)
]

This represents the likelihood that outcome (Y=y) will occur under the condition (X=x).

What matters is that a probability function is not magic that predicts the future perfectly.

It does not eliminate uncertainty in the future. It expresses multiple possibilities in a form that can be compared.

Why Do Numbers Become Probabilities?

Simply assigning a number to each candidate does not make those numbers probabilities.

For values to be treated as probabilities, they must satisfy at least two important conditions.

First, the value assigned to each outcome must be between 0 and 1.

[
0 \leq P(y) \leq 1
]

Second, the probabilities of all possible outcomes must add up to 1.

[
\sum_y P(y)=1
]

This is easier to understand if we think of probability as distributing a total amount of possibility—fixed at 1—across multiple candidates.

Suppose an AI produces the following forecast for tomorrow’s weather.

[
P(\text{sunny})=0.65
] [
P(\text{cloudy})=0.25
] [
P(\text{rainy})=0.10
]

The total is 1.

The AI is not declaring, “Tomorrow will be sunny.” It presents several possible futures—sun, clouds, and rain—and shows how consistent each one is with the available information, as a single distribution.

Only through this structure can we say, “Sunny weather is the most likely outcome, but the possibility of rain should not be ignored.”

Probability is not simply another name for ambiguity.

It is a mathematical structure for expressing multiple possibilities in an ordered way.

How Does AI Create Probabilities?

Inside an AI model, probabilities are not usually calculated from the beginning.

In many cases, the model first calculates a raw score for each candidate. These values are called logits.

[
z_1, z_2, \ldots, z_n
]

For example, an AI examining an image and determining whether it is closer to a cat, a dog, or a bird may assign scores such as:

• Cat: 2.8
• Dog: 1.4
• Bird: -0.7

At this stage, these values are not probabilities. They can be negative, and their sum is not necessarily 1.

A representative mechanism used to convert them into probabilities is the Softmax function.

[
P(y_i \mid x)=\frac{e^{z_i}}{\sum_{j=1}^{n}e^{z_j}}
]

This function transforms each candidate’s score into a positive value through the exponential function and then divides it by the total across all candidates.

This achieves two things at once:

• Every value falls between 0 and 1.
• The values for all candidates add up to 1.

In other words, Softmax converts the relatively unconstrained internal scores of an AI model into a representation constrained as a probability distribution.

Candidates with larger scores stand out more strongly because of the exponential function. At the same time, the other candidates do not disappear completely.

Before AI decides, “This is the answer,” it constructs a probability distribution that represents the relationship among the alternatives.

LLMs Create a Probability Distribution for the Next Word

Large language models (LLMs) operate on the same basic structure.

For example, given the context:

“AI handles uncertainty…”

an LLM calculates scores for many possible next tokens.

“by handling it”
“by understanding it”
“by reducing it”
“by representing it”
“by making decisions under it”

It then converts these scores into a probability distribution through the Softmax function.

[
P(\text{next token} \mid \text{previous context})
]

Text generation is the process of repeatedly selecting the most probable next word—or, in some cases, selecting among candidates according to the probability distribution.

This is where the setting called temperature becomes meaningful.

When the temperature is lower, the highest-probability candidates are more likely to be selected. Outputs tend to be more stable, but may converge on similar expressions.

When the temperature is higher, lower-probability candidates are more likely to be selected as well. Outputs become more diverse, but unexpected expressions and errors may also increase.

This means that the trade-off between creativity and reliability in AI can be adjusted through the shape of a probability distribution.

A High Probability Is Not the Same as Truth

This is the most important point.

Even if an AI assigns a probability of 0.95 to an answer, that does not guarantee that the answer is “95 percent likely to be true.”

It means that, given the model’s training data, the provided context, and its internal representations, the model has assigned a high weight to that candidate.

If the training data is biased, the probabilities may also be biased. If input information is missing, the model may create a plausible but incorrect distribution. If the real world has changed since the model was trained, probabilities based on the past may no longer work well.

Moreover, the token probabilities produced by an LLM are not probabilities of facts in the external world.

They are probabilities of which expression is likely to follow in a given linguistic context.

That is why AI cannot be entrusted with real-world decisions and actions on the basis of probability alone.

Connecting Probability to Decisions and Actions

Probability is an important input to decision-making.

But probability itself is not a decision.

For example, even if an AI predicts a high probability of equipment failure, that does not necessarily mean operations should be stopped immediately. The impact on safety, the cost of stopping operations, available alternatives, sensor reliability, and the ability to verify the situation on site must all be considered.

Likewise, even if the probability of customer churn is high, an AI should not necessarily be allowed to offer a discount automatically. The customer relationship, contractual terms, brand policy, the knowledge of the account manager, and accountability all matter.

To connect AI probability outputs to action, at least the following layers are needed.

[
\text{Probability}
\rightarrow
\text{Context}
\rightarrow
\text{Constraint}
\rightarrow
\text{Human Gate}
\rightarrow
\text{Decision Trace}
]

Probability indicates possibilities.
Context places those possibilities back into the actual situation.
Constraint defines the boundaries that must not be crossed: safety, legal, ethical, and authority-related boundaries.
Human Gate preserves decisions that require human intervention.
Decision Trace records what was decided, by whom, and on what grounds.

This is the design that moves AI closer to being not merely a prediction engine, but a participant capable of collaborating responsibly within organizations and society.

Not Eliminating Uncertainty, but Taking Responsibility for It

Probability functions are not merely tools for simplifying an ambiguous world into certain answers.

They are structures that make multiple possibilities comparable and enable decisions under uncertainty.

It is important that AI can handle probability. But what truly matters in the AI era is not simply increasing probabilities.

What is that probability based on?
How much uncertainty does it contain?
Who is allowed to use the result?
Which actions require human approval?
Can the outcome of a decision later be explained and improved?

Knowledge Flow connects the evidence for decisions. Trust Infrastructure defines the boundaries of action. Decision Trace supports accountability for decisions.

Probability is the mathematics that enables AI to handle uncertainty.

And how people, AI, and organizations take responsibility for that uncertainty—and turn it into better action—is where the essence of future AI infrastructure lies.