# Uncertainty Resolution
## Resolution Process
```mermaid
graph TB
U[Uncertainty] --> |Measure| EN[Entropy]
U --> |Drive| IS[Information Seeking]
U --> |Guide| AR[Active Resolution]
EN --> |Quantifies| UV[Uncertainty Value]
IS --> |Generates| AQ[Action Query]
AR --> |Executes| A[Action]
AQ --> |Selects| A
UV --> |Prioritizes| AQ
A --> |Obtains| O[Observation]
O --> |Reduces| U
classDef state fill:#f9f,stroke:#333,stroke-width:2px
classDef process fill:#bbf,stroke:#333,stroke-width:2px
classDef action fill:#bfb,stroke:#333,stroke-width:2px
class U,O state
class EN,IS,AR process
class AQ,A action
```
## Information Flow
```mermaid
graph LR
subgraph Prior
B[Beliefs] --> |Entropy| H1[H(s)]
end
subgraph Posterior
O[Observation] --> |Update| BP[Beliefs|Obs]
BP --> |Entropy| H2[H(s|o)]
end
subgraph Reduction
H1 --> |Difference| IG[Information Gain]
H2 --> |Difference| IG
IG --> |Measures| UR[Uncertainty Resolution]
end
classDef state fill:#f9f,stroke:#333,stroke-width:2px
classDef measure fill:#bbf,stroke:#333,stroke-width:2px
classDef outcome fill:#bfb,stroke:#333,stroke-width:2px
class B,O,BP state
class H1,H2,IG measure
class UR outcome
```
## Resolution Dynamics
```mermaid
stateDiagram-v2
[*] --> HighUncertainty
state "Resolution Process" as RP {
HighUncertainty --> InformationGathering
InformationGathering --> BeliefUpdate
BeliefUpdate --> UncertaintyAssessment
state InformationGathering {
ActionSelection --> Observation
Observation --> InformationProcessing
}
state UncertaintyAssessment {
EntropyComputation --> ThresholdCheck
ThresholdCheck --> ResolutionDecision
}
}
RP --> [*]: Resolved
```
## Active Resolution Strategy
```mermaid
graph TD
subgraph Assessment
U[Uncertainty] --> |Measure| E[Entropy]
U --> |Identify| T[Target Areas]
end
subgraph Planning
E --> |Inform| S[Strategy]
T --> |Guide| S
S --> |Generate| P[Plan]
end
subgraph Execution
P --> |Select| A[Actions]
A --> |Execute| O[Observation]
O --> |Update| B[Beliefs]
end
subgraph Evaluation
B --> |Assess| R[Resolution]
R --> |Update| S
end
classDef state fill:#f9f,stroke:#333,stroke-width:2px
classDef process fill:#bbf,stroke:#333,stroke-width:2px
classDef action fill:#bfb,stroke:#333,stroke-width:2px
class U,B state
class E,T,S,R process
class A,O action
```
## Resolution Metrics
```mermaid
graph TD
subgraph Measures
H[Entropy] --> M[Metrics]
I[Information] --> M
D[Divergence] --> M
end
subgraph Analysis
M --> |Compute| S[Statistics]
M --> |Track| P[Progress]
M --> |Assess| E[Efficiency]
end
subgraph Decisions
S --> |Guide| A[Actions]
P --> |Update| ST[Strategy]
E --> |Optimize| R[Resources]
end
classDef measure fill:#f9f,stroke:#333,stroke-width:2px
classDef process fill:#bbf,stroke:#333,stroke-width:2px
classDef decision fill:#bfb,stroke:#333,stroke-width:2px
class H,I,D measure
class M,S,P,E process
class A,ST,R decision
```
## Mathematical Formulation
The uncertainty resolution can be quantified through various measures:
1. **Entropy Reduction**
$\Delta H = H(s) - H(s|o)$
- Initial entropy: $H(s) = -\sum_s P(s)\ln P(s)$
- Conditional entropy: $H(s|o) = -\sum_{s,o} P(s,o)\ln P(s|o)$
1. **Information Gain**
$IG = D_{KL}[P(s|o)\|P(s)]$
Links to:
- [[entropy]] - Uncertainty measure
- [[kl_divergence]] - Divergence measure
- [[mutual_information]] - Information measure
## Implementation
```python
def compute_uncertainty_resolution(
prior: np.ndarray, # Prior distribution P(s)
posterior: np.ndarray, # Posterior distribution P(s|o)
method: str = 'entropy' # Method to use
) -> float:
"""Compute uncertainty resolution.
Args:
prior: Prior probability distribution
posterior: Posterior probability distribution
method: Method to use ('entropy' or 'kl')
Returns:
Amount of uncertainty resolved
"""
if method == 'entropy':
prior_entropy = compute_entropy(prior)
post_entropy = compute_entropy(posterior)
return prior_entropy - post_entropy
elif method == 'kl':
return compute_kl_divergence(posterior, prior)
else:
raise ValueError(f"Unknown method: {method}")
```
## Properties
1. **Non-negativity**
- Always reduces uncertainty
- Zero for no information gain
- Links to [[information_theory_axioms]]
1. **Bounded Reduction**
- Limited by initial uncertainty
- Asymptotic behavior
- Links to [[information_bounds]]
1. **Active Nature**
- Requires strategic action selection
- Path-dependent resolution
- Links to [[active_learning]]
## Related Concepts
- [[active_inference]] - Framework
- [[information_seeking]] - Strategy
- [[optimal_experiment_design]] - Design
- [[uncertainty_quantification]] - Measurement
- [[information_geometry]] - Geometry
## References
- [[friston_2017]] - Active Inference
- [[mackay_2003]] - Information Theory
- [[lindley_1956]] - Information Measures
- [[chaloner_1995]] - Optimal Design
## Visualization Methods
```mermaid
graph TD
subgraph Uncertainty Landscape
U[Uncertainty] --> |Map| L[Landscape]
L --> |Identify| H[Hotspots]
H --> |Target| A[Actions]
end
subgraph Resolution Progress
A --> |Execute| O[Observation]
O --> |Update| B[Beliefs]
B --> |Reduce| U
U --> |Track| P[Progress]
P --> |Visualize| V[Visualization]
end
subgraph Visualization Types
V --> M1[Heatmaps]
V --> M2[Trajectories]
V --> M3[State Space]
V --> M4[Time Series]
M1 --> |Show| D1[Distribution]
M2 --> |Show| D2[Evolution]
M3 --> |Show| D3[Structure]
M4 --> |Show| D4[Dynamics]
end
classDef state fill:#f9f,stroke:#333,stroke-width:2px
classDef process fill:#bbf,stroke:#333,stroke-width:2px
classDef viz fill:#bfb,stroke:#333,stroke-width:2px
class U,B,L state
class A,O,P process
class M1,M2,M3,M4,V viz
```
## Visualization Components
```mermaid
graph TD
subgraph Data Processing
D[Data] --> |Extract| F[Features]
F --> |Process| M[Metrics]
M --> |Prepare| V[Visualization]
end
subgraph Visual Elements
V --> P1[Primary Views]
V --> P2[Secondary Views]
V --> P3[Auxiliary Views]
P1 --> |Main| E1[Entropy Plot]
P1 --> |Main| E2[Belief Plot]
P2 --> |Support| E3[Action Plot]
P2 --> |Support| E4[Progress Plot]
P3 --> |Context| E5[History]
P3 --> |Context| E6[Statistics]
end
subgraph Interaction
E1 --> |Select| I[Interaction]
E2 --> |Select| I
E3 --> |Select| I
E4 --> |Select| I
I --> |Update| V
end
classDef data fill:#f9f,stroke:#333,stroke-width:2px
classDef viz fill:#bbf,stroke:#333,stroke-width:2px
classDef interact fill:#bfb,stroke:#333,stroke-width:2px
class D,F,M data
class V,P1,P2,P3 viz
class I,E1,E2,E3,E4,E5,E6 interact
```
## Implementation Example
```python
class UncertaintyVisualizer:
"""Visualize uncertainty resolution process."""
def __init__(self, resolution_data: Dict[str, np.ndarray]):
self.data = resolution_data
self.fig = plt.figure(figsize=(15, 10))
self.gs = self.fig.add_gridspec(3, 2)
def plot_entropy_evolution(self, ax=None):
"""Plot entropy reduction over time."""
if ax is None:
ax = self.fig.add_subplot(self.gs[0, 0])
time = np.arange(len(self.data['entropy']))
ax.plot(time, self.data['entropy'], 'b-', label='Entropy')
ax.set_title('Uncertainty Evolution')
ax.set_xlabel('Time Step')
ax.set_ylabel('Entropy')
ax.grid(True)
def plot_belief_landscape(self, ax=None):
"""Plot belief landscape with uncertainty."""
if ax is None:
ax = self.fig.add_subplot(self.gs[0, 1])
belief_grid = self.data['belief_landscape']
im = ax.imshow(belief_grid, cmap='viridis')
ax.set_title('Belief Landscape')
plt.colorbar(im, ax=ax)
def plot_resolution_progress(self, ax=None):
"""Plot resolution progress metrics."""
if ax is None:
ax = self.fig.add_subplot(self.gs[1, :])
metrics = self.data['resolution_metrics']
time = np.arange(len(metrics))
ax.plot(time, metrics, 'r-', label='Resolution')
ax.set_title('Resolution Progress')
ax.set_xlabel('Time Step')
ax.set_ylabel('Resolution Metric')
ax.grid(True)
def plot_all(self):
"""Generate complete visualization suite."""
self.plot_entropy_evolution()
self.plot_belief_landscape()
self.plot_resolution_progress()
plt.tight_layout()
return self.fig
```
Links to:
- [[visualization_guide]] - Visualization standards
- [[plotting_utilities]] - Plotting tools
- [[interactive_visualization]] - Interactive features
## Visualization Types
1. **Uncertainty Maps**
- Heatmaps of belief distribution
- Entropy landscapes
- Information gain fields
- Links to [[spatial_visualization]]
1. **Temporal Evolution**
- Resolution trajectories
- Convergence plots
- Learning curves
- Links to [[temporal_visualization]]
1. **State Space Views**
- Belief manifolds
- Value landscapes
- Action trajectories
- Links to [[state_space_visualization]]
1. **Interactive Elements**
- Selection tools
- Filtering options
- Dynamic updates
- Links to [[interactive_tools]]
## Learning Dynamics
```mermaid
graph TD
subgraph Resolution Phases
P1[Initial Phase] --> |High Uncertainty| L1[Active Learning]
L1 --> |Information Gain| U1[Uncertainty Reduction]
U1 --> |Progress| P2[Middle Phase]
P2 --> |Refined Search| L2[Targeted Learning]
L2 --> |Specific Information| U2[Focused Resolution]
U2 --> |Convergence| P3[Final Phase]
P3 --> |Low Uncertainty| L3[Model Application]
L3 --> |Validation| U3[Stability Check]
U3 --> |Maintenance| P4[Steady State]
end
classDef phase fill:#f9f,stroke:#333,stroke-width:4px
classDef learning fill:#bbf,stroke:#333,stroke-width:2px
classDef state fill:#bfb,stroke:#333,stroke-width:2px
class P1,P2,P3,P4 phase
class L1,L2,L3 learning
class U1,U2,U3 state
```
## Resolution Framework
```mermaid
graph LR
subgraph Components
U[Uncertainty] --> |Measure| E[Entropy]
U --> |Drive| IS[Information Seeking]
U --> |Guide| AR[Active Resolution]
end
subgraph Process
E --> |Quantify| UV[Uncertainty Value]
IS --> |Generate| AQ[Action Query]
AR --> |Execute| A[Action]
UV --> |Prioritize| AQ
AQ --> |Select| A
end
subgraph Outcome
A --> |Obtain| O[Observation]
O --> |Update| B[Beliefs]
B --> |Reduce| U
end
classDef component fill:#f9f,stroke:#333,stroke-width:2px
classDef process fill:#bbf,stroke:#333,stroke-width:2px
classDef outcome fill:#bfb,stroke:#333,stroke-width:2px
class U,E,IS,AR component
class UV,AQ,A process
class O,B outcome
```
## Enhanced Relationships
### Core Components
- [[information_theory]] - Theoretical foundation
- [[active_inference]] - Framework context
- [[belief_updating]] - State estimation
- [[action_selection]] - Decision making
- [[entropy]] - Uncertainty measure
### Resolution Methods
- [[information_gain]] - Knowledge acquisition
- [[active_learning]] - Strategic learning
- [[optimal_experiment_design]] - Query design
- [[exploration_strategies]] - Search methods
- [[convergence_control]] - Learning stability
### Analysis Tools
- [[information_metrics]] - Measurement tools
- [[convergence_analysis]] - Learning progress
- [[visualization_methods]] - Visual analysis
- [[performance_evaluation]] - System assessment
- [[statistical_tests]] - Validation methods
### Implementation Aspects
- [[numerical_methods]] - Computation tools
- [[optimization_algorithms]] - Search techniques
- [[probability_distributions]] - Distribution handling
- [[computational_efficiency]] - Performance
- [[numerical_stability]] - Robustness
## Theoretical Foundations
- [[information_geometry]] - Geometric view
- [[statistical_inference]] - Learning framework
- [[decision_theory]] - Choice framework
- [[optimization_theory]] - Search principles
- [[learning_dynamics]] - System behavior