SDT Foundations Presentation Slides Signal Detection Theory
Distinguishing Sensitivity from Strategy
Foundations Lesson 01
\( d' = z(H) - z(F) \)
The Diagnostic Dilemma
Consider two radiologists: Dr. A and Dr. B.
"Dr. A catches 95% of tumors but has a high false alarm rate. Dr. B catches 70% but rarely misidentifies healthy tissue."
The Critical Question:
Does Dr. A have better vision, or just a lower criterion for calling something a tumor?
Case Study Analysis
The SDT Framework
Internal Distributions
Perception occurs in the presence of noise . Every stimulus produces a distribution of evidence.
Noise Only: Random internal activity.Signal + Noise: The stimulus added to the background noise.
Criterion (\(\beta\))
Overlapping distributions create uncertainty.
Parameter 1: Sensitivity (\(d'\))
\(d'\) represents the distance between the means of the Noise and Signal+Noise distributions, normalized by their standard deviation.
\[ d' = \frac{\mu_{S+N} - \mu_N}{\sigma} \]
Higher \(d'\) = Better sensory discrimination.
\(d' = 0\) means no discrimination (chance performance).
Theoretical Note
\(d'\) is independent of the observer's strategy. It reflects the pure ability of the system to distinguish signal from noise.
The Receiver Operating Characteristic
False Alarm Rate Hit Rate Conservative Liberal
Key Properties of ROC
1 Plot of Hit Rate vs. False Alarm Rate .
2 The area under the curve (AUC) represents overall sensitivity.
3 Moving along the curve changes the criterion , not the sensitivity.
Transition to Workshop
We will now apply these formulas to empirical datasets. You will be calculating \(d'\) and \(c\) from a 2x2 contingency table.
Get Calculators
Open Dataset A
SDT Computation Workshop Worksheet Detection Analytics
Foundations of SDT: Quantitative Workshop
Name:
Date:
Part 1: The Contingency Table
"You are analyzing data from a sonar operator identifying submarines (Signal) versus underwater rock formations (Noise). In 200 trials, the submarine was present 100 times."
Stimulus Present (S) Stimulus Absent (N) "Yes" Response 82 (Hits) 24 (FAs) "No" Response 18 (Misses) 76 (CRs)
1.1 Calculate Rates:
Hit Rate (H) =
False Alarm Rate (F) =
Part 2: Parameter Estimation
Using the Z-table values provided below, calculate the observer's sensitivity (\(d'\)) and criterion (\(c\)).
z(0.82) ≈ 0.915
z(0.24) ≈ -0.706
z(0.76) ≈ 0.706
2.1 Calculate \(d'\):
Formula: \( d' = z(H) - z(F) \)
2.2 Calculate Criterion (\(c\)):
Formula: \( c = -0.5 \times [z(H) + z(F)] \)
2.3 Interpretation:
Based on your calculation of \(c\), is this observer Liberal, Conservative, or Neutral? Explain how you know by referencing the sign and magnitude of your result.
Part 3: The ROC Shift
Imagine the sonar operator is told that missing a submarine is 10x more costly than a false alarm. How would you expect their position on the ROC curve to shift? Sketch the original point and the new point below.
False Alarm Rate
Hit Rate
Explain the shift:
SDT Foundations Teacher Guide Instructor Resource
Lesson 01: SDT Foundations & Computation
Instructional Goals
This lesson bridges the gap between conceptual understanding of sensory "accuracy" and the mathematical reality of Signal Detection Theory. Students must leave this session understanding that performance is not a single number , but the interaction of sensory sensitivity (\(d'\)) and decision strategy (\(c\)).
Quick Stats
Duration: 90-120m
Format: Seminar/Workshop
Math Level: Intermediate
Facilitating the Radiological Hook
The Setup:
Present the Dr. A vs. Dr. B scenario. Ask students: "If you were the hospital administrator, which doctor would you hire?" Most will choose Dr. A (95% catch rate). Then reveal the False Alarm rates: Dr. A flags 40% of healthy patients for painful biopsies, while Dr. B flags only 5%.
Target Discussion Prompts
Is "accuracy" a useful metric here? Why not?
What costs are associated with a Miss vs. a False Alarm?
How can we prove they have the same eyesight?
The "Aha!" Moment
Direct students to the realization that Dr. A and Dr. B likely have the exact same sensory evidence but different internal decision thresholds .
Workshop Answer Key (Sonar Data)
1. Rate Calculations
Hit Rate (H) = 82/100 = 0.82
FA Rate (F) = 24/100 = 0.24
2. Sensitivity (\(d'\))
\(d' = z(0.82) - z(0.24)\)
\(d' = 0.915 - (-0.706) = \mathbf{1.621}\)
3. Criterion (\(c\))
\(c = -0.5 \times [z(0.82) + z(0.24)]\)
\(c = -0.5 \times [0.915 - 0.706] = \mathbf{-0.1045}\)
Interpretation
Since \(c\) is negative, the observer is Liberal . They are more likely to say "Yes" than the ideal observer (\(c=0\)).
Common Graduate Pitfalls
Confusing \(\beta\) and \(c\)
Students often use these interchangeably. Clarify that \(\beta\) is a likelihood ratio, while \(c\) is the distance from the ideal observer in units of SD. \(c\) is usually easier for students to intuit.
Z-score Directionality
Students often flip signs when FA rates are below 0.5. Emphasize that \(z(p)\) for \(p < 0.5\) is always negative.
Methodological Precision Slides Psychophysical Methods
From Classical Thresholds to Modern Adaptives
Methodology Lesson 02
The Silence Challenge
How do you measure the auditory threshold of an infant or a feline ?
"You cannot ask them 'Did you hear that?'. You must observe behavior, minimize trials, and maximize reliability."
Required Constraints:
Speed Precision Fatigue Management
Methods to Evaluate:
Method of Constant Stimuli
Method of Limits
Up-Down Adaptive Staircase
Method of Constant Stimuli
The Gold Standard
The experimenter presents a fixed set of stimuli in random order .
Pros: Highly precise, provides full psychometric curve.
Cons: Very inefficient; many trials far from threshold.
Stimulus Intensity P("Yes") 50% Threshold
Adaptive "Staircase" Procedures
The intensity of the next stimulus depends on the subject's previous response .
The Rule-Based Logic
Correct Response → Decrease Intensity
Incorrect Response → Increase Intensity
This "hones in" on the threshold rapidly.
Efficiency Wins
Most trials are placed right where they matter most: near the 50% detection point.
Reduces trial count by ~60% vs. Constant Stimuli.
Advanced: Bayesian Adaptive (QUEST)
Mathematical Prior
Starts with a probability distribution of where the threshold likely is.
Likelihood Update
Every response updates the "belief" about the threshold location.
The QUEST Convergence
"The model constantly recalculates the most informative intensity for the next trial."
Experimental Design Task
You are now tasked with designing a threshold estimation experiment for a challenging population. You must justify your methodological choice based on efficiency and data quality.
Mission: Hearing thresholds in the North Atlantic Right Whale
Experimental Design Protocol Worksheet Research Design Protocol
PROJECT ID: PSYCH-202-METHOD-A
Design Brief
Measuring sensory thresholds in "unreliable" or "non-communicative" populations requires creative methodological adaptation. You must design a threshold estimation protocol for Pre-Verbal Human Infants (6-12 months) to determine their absolute threshold for a specific 1000Hz tone.
Principal Investigator:
Protocol Date:
1 Methodological Selection
Constant Stimuli
Adaptive Staircase
Method of Adjustment
Justification for Selection (Considering infant attention span and movement noise):
2 Protocol Parameters
Starting Intensity & Step Size:
Response Metric (e.g., Head-turn, Sucking rate):
Criteria for Reversals (if using staircase):
Number of Catch Trials (to monitor bias):
3 Bias & Sensitivity Control
Infants often exhibit high false-alarm rates or rapid habituation. Explain how you will distinguish a lack of sensory sensitivity from a change in motivation or boredom.
AUTH: DEPT_PSYCH_GRAD_COMMITTEE STAMP: PROTOCOL_APPROVED_v1.2
Modeling Sensory Minds Slides Virtual Observers
Computational Modeling of Sensory Decisions
Modeling Lesson 03
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The Power of Simulation
Before we test humans, we build silicon subjects .
Computational models allow us to:
Generate precise performance predictions.
Systematically vary "internal noise".
Simulate thousands of trials in seconds.
Test the optimality of decision rules.
Virtual Subject #001
Anatomy of the Model
1. Distributions
Generate random samples from two Gaussians (Noise and Signal+Noise).
2. The Rule
Set a mathematical criterion (e.g., "If sample > 1.5, say YES").
3. Comparison
Check response against ground truth to calculate Hits/FAs.
# Logic Flow
Manipulating Internal Noise
What happens to the ROC curve when we increase the standard deviation (\(\sigma\)) of our noise?
Prediction Task:
If \(\sigma\) increases while stimulus intensity remains constant, \(d'\) will [DECREASE / INCREASE] .
INTERNAL_NOISE_SIM_v4
The Optimality Challenge
We can code "motivation" by using a Payoff Matrix . Our virtual observer must now maximize its "total score".
S+N N "Yes" +10 -50 "No" -20 +10
Simulation Goal
Find the optimal criterion for this specific matrix. Does the model become more liberal or conservative when the penalty for a False Alarm is -50?
"The model will iterate through 100 possible criteria to find the one that yields the highest total payoff."
Simulation Workshop
Open your modeling environment. We will be building a virtual observer in Python/R to replicate human psychometric curves.
Load Notebook
Noise Params v1
Virtual Observer Simulation Log Worksheet DOC_TYPE: LOG_SIM_03
Virtual Observer Simulation
SENSORY MODELING LAB // DATA LOG
Simulation Lead:
Iterative Run #:
Architecture Logic
# Initialize Observer Parameters
# Run 1000 Trials
Experiment A: Scaling Variance
In your simulation, keep SIGNAL_MEAN constant at 2.0. Run three blocks of 500 trials each, varying the NOISE_SD. Record the resulting \(d'\).
Internal Noise (\(\sigma\)) Hit Rate FA Rate Calculated \(d'\) 0.5 (Low) 1.0 (Standard) 2.0 (High)
Computational Observation:
"How does the ratio of Signal to Noise impact the observer's ability to maintain a stable Hit Rate? Discuss the effect of 'masking' in computational terms."
Experiment B: Payoff Search
You are now modeling a "Virtual Doctor" where a Miss is 5x more expensive than a False Alarm. Use a loop to find the "Optimal Criterion" that maximizes profit across 5000 trials.
Search Results:
C = -1.0
C = -0.5
C = 0.0
C = 0.5
Model Inference:
Why did the model choose the specific criterion it did? Relate this back to the probability density of the Signal and Noise distributions at that point.
System Simulation Logic Approved
Temporal Dynamics Slides Deck Temporal Dynamics
Adaptation, Gain Control, and Criterion Shifts
Dynamics Lesson 04
The Disappearing Stimulus
Why does the steady hum of a refrigerator "vanish" until it suddenly switches off?
Sensory Adaptation: The reduction in sensitivity to a stimulus after prolonged exposure.
The Evolution of Efficiency
Why would our brains "stop listening" to constant input?
Gain Control
Sensitivity vs. Criterion Drift
1. Sensitivity Change
The physical response of the sensory neurons decreases (Gain Control).
\( d' \) decreases as \( \sigma \) effectively increases or \( \mu \) decreases.
2. Criterion Drift
The observer's decision threshold moves over time due to fatigue or boredom.
\( c \) shifts (usually becomes more conservative) over long sessions.
Mechanisms: Neural Gain Control
The brain maximizes its limited dynamic range by shifting its sensitivity to match the ambient stimulus level .
The Camera Analogy
Like an "Auto-Exposure" setting, your visual system lowers its sensitivity (Gain) in bright light to avoid saturation.
Response Intensity
System "slides" its sensitivity curve to prevent clipping.
Modeling Fatigue: Vigilance Decrement
In high-stakes detection (e.g., air traffic control), performance drops after ~30 minutes.
Research Question:
Is the operator losing sensitivity (\(d'\)) or just becoming too cautious (\(c\))?
Longitudinal SDT Results
d' c Time on Task
Many studies show stable sensitivity but a conservative criterion shift over time.
Adaptation Case Study
We will now analyze a longitudinal dataset from a 4-hour monitoring task to determine the exact mathematical nature of the performance decay.
Dataset: Vigilance_Delta_09
Vigilance Analytics Case Study Worksheet Vigilance Analytics
CASE STUDY: LONGITUDINAL SDT ANALYSIS // REF_ID_909
Archive Date: 01.17.2026
Researcher:
Session ID:
Subject Context: Radar Monitoring
"Subject #14 participated in a 120-minute simulated radar task. Signals were sparse (p=0.1). Below is the performance data segmented into four 30-minute blocks."
Time Block Hits (out of 50) False Alarms (out of 450) Calculated \(d'\) Calculated \(c\) 00 - 30 min 44 22 30 - 60 min 38 15 60 - 90 min 28 8 90 - 120 min 22 3
Trend Analysis
1. Sensitivity Decay Test:
Does the calculated \(d'\) remain stable or decrease? What does this suggest about the subject's primary sensory system?
2. The Criterion Shift:
Quantify the shift in \(c\) from Block 1 to Block 4. Is the observer becoming more Liberal or Conservative?
3. Theoretical Synthesis
A common "Vigilance Decrement" is observed in these tasks. Given your findings above, propose a practical intervention (e.g., changes to the interface, break schedules, or payoff changes) that could correct for the criterion shift without necessarily changing the observer's sensitivity.
Seminar Discussion Point
"If adaptation is an evolutionary advantage (conserving metabolic energy), why does it create life-threatening risks in modern technological tasks like air traffic control? Is this a failure of the 'Bayesian Brain' to account for modern priors?"
The Bayesian Brain Presentation Deck Slides The Bayesian Brain
Perception as Optimal Statistical Inference
Inference Lesson 05
P(H|E) = \frac{P(E|H)P(H)}{P(E)}
Illusion or Inference?
Most people see visual illusions as "failures" of the brain.
The Bayesian View: Illusions are actually optimal predictions based on noisy data and strong prior beliefs about how the world works.
Example: Shadow Casting
The brain "knows" light usually comes from above. This is a prior .
Perceptual Priors
The Calculus of Perception
Prior
Existing knowledge or probability of an event (e.g., "Faces are usually convex").
×
Likelihood
Current sensory evidence from the eyes/ears (often noisy).
Posterior
The resulting percept: A weighted average of what we expect and what we see.
Precision-Weighted Fusion
The brain doesn't just combine expectations and evidence; it weights them by their certainty (precision) .
Noisy Evidence (Fog): Brain relies more on Priors.Clear Evidence: Brain relies more on Likelihood.
Prior Likelihood Posterior
Critiquing the Theory
Is the brain actually doing Bayesian math?
Where do the "Priors" come from? (Innate vs. Learned)
Is "Optimal" always "Adaptive"?
Seminar Discussion
"If our brain is an optimal inference engine, why do we suffer from chronic pain or phantom limbs? Can these be modeled as 'maladaptive priors'?"
Seminar Synthesis
We will now break into seminar groups to critique the 'Bayesian Brain' literature and map visual illusions to formal Bayesian models.
The Optimal Inference Seminar
Bayesian Brain Seminar Analysis Worksheet Optimal Inference Seminar
Topic: The Bayesian Brain Hypothesis
Graduate Seminar // Lesson 05
Part 1: Perceptual Mapping
Select a classic visual illusion (e.g., The Hollow Face, Ames Room, or Adelson's Checker-shadow). Deconstruct the illusion using the Bayesian framework by identifying the Prior, the Likelihood, and the resulting Posterior.
The Prior (Expectation)
What does the brain "assume" about the world based on lifetime statistics?
The Likelihood (Evidence)
What noisy or ambiguous sensory signals are currently hitting the receptors?
The Posterior (Percept)
How does the brain resolve the conflict? Why is this solution "optimal"?
Part 2: Critical Discourse
Seminar Question 1: The Curse of Precision
In Bayesian models of schizophrenia and autism, it is hypothesized that the brain assigns too much precision to either the likelihood or the prior. How would an "over-precise" likelihood impact the perception of random noise?
Seminar Question 2: Falsifiability
Critics argue that "Bayesian Brain" models are "just-so stories" because any behavior can be explained by postulating the correct prior. How can we mathematically test if a brain is truly being Bayesian vs. using a simple heuristic?
Closing Reflection
"If the brain is an inference engine, our reality is a controlled hallucination."
Researcher Signature:
