Probabilistic

How Probabilistic Systems Work

In a deterministic method, a defined state under the same conditions leads to a defined result.

A simple example is a mathematical function:

f(x) = x + 2

For

x = 3

the result is always:

5

A probabilistic method, by contrast, can consider multiple possible outcomes and assign different probabilities to them.

Simplified, a distribution could, for example, look like this:

A → 70 %
B → 20 %
C → 10 %

The method therefore describes not only a single possible outcome, but a distribution of possible outcomes.

Probability Is Not Arbitrariness

A probabilistic system does not necessarily decide arbitrarily.
If one outcome is assigned a probability of 70 percent and another a probability of 1 percent, both are possible outcomes, but they are by no means equally likely.
The probabilities result from the respective model or method and the conditions and inputs relevant to it.
In addition, a method can define how a concrete outcome is selected from such a distribution.
Deterministic and random-based methods can be used or combined for this purpose.
The term probabilistic therefore initially describes the role of probabilities in the model or method.

It does not automatically mean:

“The system rolls dice.”

Language Models Work with Probabilities

In an autoregressive Language Model, the previous context is used to calculate which Tokens are likely continuations.

Simplified, after a given context a distribution such as the following could arise:

System       0.31
Application  0.18
Model        0.12
Method       0.07
...

The model therefore does not simply have a fixed next answer stored in it.
It calculates probabilities for possible continuations.
The subsequent output emerges step by step from the selected Tokens.
As a result, even small differences in the context or in Token selection can lead to different subsequent outputs.

Same Input, Different Output

With probabilistic systems, repeated processing of the same input does not necessarily have to lead to an identical output.

For a Language Model, for example, these answers could be produced:

OpenBSD is a Unix-like operating system.

and:

OpenBSD belongs to the family of Unix-like operating systems.

The character sequences differ.
In terms of content, however, both statements can still satisfy the same requirement.
A test that checks exclusively for character identity might evaluate one of the answers as incorrect even though there is no factual error.
This also changes the question of what is understood as a correct result.

Probabilistic Does Not Mean Untestable

The fact that multiple outcomes are possible does not prevent testing.
It must, however, be defined what is to be tested.

For a generated technical answer, for example, the following properties could be relevant:

• Are the statements it contains factually correct?
• Was the question asked answered?
• Is required information included?
• Were prohibited statements avoided?
• Are specified rules followed?
• Are output commands syntactically correct?
• Do these commands work on the intended system?
• Does quality remain within acceptable limits across multiple runs?

This shifts testing from:

Is output == expected output?

to questions such as:

Does the output meet the relevant requirements?

Individual Results and Behavior Across Multiple Runs

With a probabilistic system, a single successful test may have only limited significance.
A Language Model can solve a task correctly once and produce an error when run again.
Conversely, a single faulty run does not necessarily describe the entire behavior of a system.
It can therefore be useful to consider multiple runs.

For example, the following can be examined:

• frequency of certain errors
• stability of relevant statements
• compliance with rules
• differences between outputs
• behavior with changed inputs
• behavior with changed context
• boundary and special cases

The number of tests required depends on the particular application and the risk of incorrect results.

Reproducibility Requires Conditions

When examining probabilistic systems, it is important to know the conditions of an experiment as precisely as possible.

For a Language Model, these can include, for example:

• model used
• model version
• System Prompt
• User Prompt
• additional context
• generation parameters
• tools used
• external data sources
• time or version of external information

Two seemingly identical requests do not necessarily have to have been processed under actually identical conditions.
Missing information about these conditions makes the evaluation and reproducibility of results more difficult.

Deterministic Settings Do Not Completely Solve the Problem

In some probabilistic systems, selection of a concrete outcome can be made deterministic or configured with less random variation.
This can make outputs more reproducible.
The underlying model, however, remains a model that calculates probabilities over possible outcomes.
Even a reproducible incorrect answer remains incorrect.
More deterministic output therefore reduces certain forms of variation, but does not replace evaluation of content quality.

Why Probabilistic Systems Require Adapted Testing

• Multiple outputs can be correct: Character identity is often not a suitable quality criterion.
• One successful run is not always enough: Behavior can be examined across multiple executions.
• Quality criteria must be described: Correctness, completeness, or rule compliance must be made evaluable.
• Context influences outcomes: Changes in inputs can change the probability distribution and therefore the output.
• Reproducibility requires documented conditions: Model, context, and settings are part of the examination.
• Variation and errors are different things: A different answer is not automatically an incorrect answer.

Typical Example

A Language Model is to create installation instructions from a technical document.
Five runs produce five differently worded sets of instructions.
Four contain all required steps.
One set of instructions, however, omits a necessary configuration step.
Comparing the texts at character level is of little help in this evaluation.
Instead, it must be checked whether each output fulfills the relevant requirements.
For this purpose, a list of necessary steps could, for example, be defined.
In addition, technical statements could be checked against documentation or a real test system.
However, it may require adapted test criteria, repeated testing, or statistical evaluation methods.

Distinction

• Probabilistic describes systems or models in which probabilities are part of the description or calculation of possible outcomes.
• Deterministic, by contrast, describes behavior in which, under defined conditions, a particular state uniquely determines the following state or outcome.
• Random and probabilistic are not synonymous. Random selection can be part of a probabilistic method, but does not describe its entire operation.
• In Language Models, probability distributions over Tokens form an essential basis of text generation.
• For the Verification and Validation of probabilistic systems, suitable criteria and, where appropriate, statistical considerations must be used instead of exclusively expecting a single exact output.

Further Reading

Related Concepts

→ Language Model
→ Token
→ Temperature
→ Deterministic
→ Verification and Validation
→ Context Engineering
→ AI Evaluation
→ AI Assurance