ML | Dempster Shafer Theory

Dempster-Shafer Theory (DST) is an evidence theory for managing uncertainty and making decisions when information is incomplete or conflicting. Unlike traditional methods, it combines evidence from multiple sources and assigns belief, plausibility and doubt to possible outcomes. This helps us assess the likelihood of different scenarios even when we don’t have all the facts. It is useful in situations where relying on a single piece of evidence isn’t enough to make an informed decision.

Need for Dempster-Shafer Theory(DST)

Traditional methods like Bayesian probability are valuable but struggle with uncertainty, especially in real-world situations where information is incomplete or conflicting. Dempster-Shafer Theory (DST) is needed for the following reasons:

1. Combining Conflicting Evidence: While Bayesian probability handles one piece of evidence at a time, it allows us to combine conflicting or incomplete evidence to offer a more holistic understanding.
2. Handling Uncertainty and Ignorance: Bayesian models assume we have complete knowledge but DST can handle situations where we lack complete information, making it more flexible in uncertain environments.
3. Greater Flexibility: Unlike Bayesian methods that work with fixed probabilities, it allows us to assess belief, giving us a clearer view of the range of possible outcomes.
4. Better Decision-Making: It provides a structured approach to decision-making when faced with incomplete or uncertain information, helping us make more informed choices.

Key Concepts of Dempster-Shafer Theory

DST operates based on a few key concepts that help us manage uncertainty and conflicting information. Let’s see them:

1. Frame of Discernment: A frame of discernment is simply the set of all possible outcomes in a given situation.

2. Basic Probability Assignment (BPA): It assigns belief to subsets of the frame of discernment, not individual elements. BPAs express the degree of confidence in a set of possibilities, given the available evidence. Importantly, the total of all BPAs in the frame must sum to 1.

3. Belief, Plausibility and Doubt:

• Belief: Represents the confidence in a hypothesis based on the evidence.
• Plausibility: Measures how likely a hypothesis might still be true even if the evidence isn’t fully conclusive.
• Doubt: Reflects uncertainty about a hypothesis showing how likely it is that the hypothesis is false.

4. Dempster’s Rule of Combination: It is used to combine multiple sources of evidence into a single belief. This is useful when dealing with conflicting or incomplete evidence.

Example of Applying Dempster-Shafer Theory

Let’s take a mathematical approach to applying Dempster-Shafer Theory (DST) with an example.

1. Problem Setup

In a room, four people A, B, C and D are present. The lights go out and when they come back on, B has been stabbed in the back. No one entered or left the room and B didn’t commit suicide. Our goal is to find who the murderer is.

The set of possible hypotheses (Frame of Discernment) is:

• Either A or C or D has killed him.
• Either {A, C} or {C, D} or {A, D} have killed him.
• Or the three of them have killed him i.e {A, C, D}
• None of them have killed him {o} (let's say).

2. Power Set and Mass Function

The Power Set of P consists of all possible subsets of P including the empty set and the full set.

For P = {(A, B, C, D)} , the power set 2^P is:

2^P = {∅, {A}, {B}, {C}, {D}, {A, B}-----{B, C, D}, {A, B, C, D}}

Given a power set, we assign mass functions to these subsets. The mass function m(K) refers to the belief assigned to a particular subset, representing evidence that supports a particular hypothesis or combination of hypotheses.

3. Belief and Plausibility

Using the mass function, we calculate belief and plausibility for various hypotheses. For example, if we want to find the belief and plausibility for the hypothesis that {A, D} are involved, we proceed as follows:

• Belief in K: The belief for a hypothesis K (e.g {A, D}) is the sum of the masses of all subsets of K. If K= {A, D} then: Bel({A, D}) = m(A) + m(D) + m(A, D)
• Plausibility in K: Plausibility for K is the sum of masses for all sets that intersect with K. For K= {A, D} we would calculate: Pl({A, D}) = m(A) + m(D) + m(A, D) + m(A, B) + m(D, C) and so on for all subsets that intersect {A, D}.

4. Combining Evidence

DST good at combining evidence from multiple sources. Suppose one witness claims A is guilty and another suggests that A and C are involved. It can combine these pieces of evidence into one belief:

• If the evidence for A is strong but unclear about whether C was involved, we combine the masses accordingly to get a unified belief and plausibility for different combinations of suspects.

Through this process, it provides a structured way of updating our beliefs as we gather more evidence even when the evidence is conflicting or incomplete.

Applications of Dempster Shafer Theory

1. Medical Diagnosis: It helps combine multiple test results or symptoms to make a more accurate diagnosis by handling uncertainty in medical data.
2. Sensor Fusion: It integrates data from different sensors to create a cohesive and accurate understanding even when individual sensors provide conflicting information.
3. Machine Learning: It improves classification tasks by managing uncertainty in data allowing for more flexible and reliable predictions.
4. Robotics: It is used in robotics to combine uncertain data from various sources, helping robots make better decisions in dynamic environments.

Challenges of Dempster Shafer Theory

1. Computational Complexity: It can be resource-intensive especially with large sets of data or numerous evidence sources, making it harder to scale.
2. Conflict Handling: It struggles with extreme conflicts between evidence sources which can lead to unreliable results if not properly managed.
3. Interpretability: The results from DST may be difficult to interpret, particularly when combining evidence with high conflict or uncertainty.