Foundations of Algorithmic Decision Making: Probabilistic Reasoning & Representation

Foundations of Algorithmic Decision Making

Representation for Probabilistic Reasoning

Uncertainty is everywhere. Even systems governed by precise physical laws — like satellite trajectories — can drift because small imprecisions compound over time. Measurements contain noise. Sensors fail. Humans behave unpredictably.

A decision-making system that ignores uncertainty is fragile. One that models it explicitly becomes powerful. Probability gives us a structured language for representing uncertainty and reasoning under it. 

 

1. Degrees of Belief

Probability can be interpreted as a degree of belief — a numerical way of expressing how plausible we think a proposition is.

$$ A \succ B \quad \text{or} \quad P(A) > P(B) $$

To behave rationally, our beliefs must follow two core assumptions:

• Universal Comparability — Any two events can be ranked.
• Transitivity — If $$ A \succeq B $$ and $$ B \succeq C $$, then $$ A \succeq C $$.

Example Use: Medical Triage Prioritization

A clinician assigns degrees of belief to “Patient X will deteriorate within 2 hours” (e.g., P = 0.82) versus “Patient Y will deteriorate within 2 hours” (e.g., P = 0.47). This quantified belief directly informs who receives urgent attention — turning subjective judgment into auditable, consistent prioritization.

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2. Discrete Probability Distributions

A discrete distribution assigns probabilities to outcomes from a countable set — like dice rolls or categories. All probabilities must sum to 1:

$$ \sum_{i=1}^{n} P(X=i) = 1 $$

Bias toward face 6 (θ₆): 0.30

As you increase the bias toward face 6, the probabilities of the other faces decrease proportionally so that the total still sums to 1.

Example Use: Spam Email Classification

An email classifier outputs a discrete distribution over classes: P(spam) = 0.93, P(personal) = 0.04, P(work) = 0.02, P(newsletter) = 0.01. The sum is exactly 1. This allows not just binary “spam/ham” decisions, but calibrated risk assessment — e.g., routing high-confidence spam to quarantine, while low-confidence cases go to human review.

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3. Continuous Probability Distributions

When variables can take infinitely many values, we use probability density functions (PDFs). The Gaussian (Normal) distribution is the most fundamental example:

$$ p(x) = \mathcal{N}(x \mid \mu, \sigma^2) $$

Mean (μ): 0

Std Dev (σ): 1

Example Use: Sensor Fusion in Autonomous Vehicles

A self-driving car fuses lidar and radar distance readings into a single Gaussian distribution: 𝒩(μ = 12.4 m, σ² = 0.09 m²). This continuous representation captures both the best estimate (12.4 m) and its precision (±0.3 m), enabling safe path planning — e.g., braking only when P(distance < 10 m) > 0.99.

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4. Conditional Probability

Often, we want to update beliefs given evidence. Bayes’ Rule formalizes this transition from prior to posterior:

$$ P(x|y) = \frac{P(y|x)P(x)}{P(y)} $$

Example Use: Fraud Detection in Real-Time Payments

A bank computes P(fraud | transaction amount = $4,892, location = Tokyo, time = 2:17 AM) using Bayes’ rule. It combines the prior fraud rate (e.g., 0.001), the likelihood of that pattern given fraud (e.g., 0.21), and the overall probability of that pattern (e.g., 0.004). Result: P(fraud | evidence) ≈ 0.052 — triggering step-up authentication, not blocking.

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5. Sigmoid Model (Classification)

For binary decisions, we often use a smooth transition. Instead of a hard threshold, the sigmoid provides a confidence level:

$$ P(x=1 \mid y) = \frac{1}{1 + \exp\left(-\frac{2(y-\theta_1)}{\theta_2}\right)} $$

Threshold (θ₁): 0

Sharpness (θ₂): 1

Example Use: Credit Approval Scoring

A lending model maps a credit score y (e.g., FICO 680) through a sigmoid: P(approval | y=680) = 0.74. This isn’t just “yes/no” — it enables dynamic pricing (higher interest for lower P(approval)) and explains rejections (“Your score places you at 74% approval likelihood; raising it by 20 points would increase to 91%”).

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6. Bayesian Networks

Real systems involve many interacting variables. Bayesian networks represent these relationships using a directed acyclic graph.

$$ P(x_{1:n}) = \prod_{i=1}^{n} P(x_i \mid \text{pa}(x_i)) $$

This factorization works because of conditional independence—the key insight that makes large systems computationally tractable.

Example Use: Diagnostic Reasoning in Healthcare AI

A Bayesian network models symptoms (fever, cough), test results (PCR+, chest X-ray), and diseases (flu, pneumonia, COVID-19). Given fever + cough + PCR+, it computes P(COVID-19 | evidence) = 0.87 while automatically accounting for dependencies (e.g., fever is less specific if flu is present). Clinicians see not just the top diagnosis, but full posterior distributions across all conditions.

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Putting It All Together

Probabilistic reasoning provides the backbone of intelligent behavior. Whether you're building recommender systems or autonomous agents, these representations are the foundation of modern AI.