Prior, Conditional Probability, and Bayes’ Theorem for Decision Making Under Uncertainty

Before observing evidence, we summarize what we believe about a hypothesis using a prior probability. After observing evidence, we update beliefs to obtain a posterior probability via Bayes’ theorem.

At a technical level, the core ingredients are conditional probability and the probability chain rule that leads to Bayes’ theorem. Bayes’ theorem provides a principled way to convert:

• how likely the evidence is under each hypothesis (likelihood),
• together with prior plausibility (prior probability) into
• updated plausibility (posterior probability).

This updated belief can then drive decisions under uncertainty using decision rules such as MAP/MLE classification or thresholding based on expected costs/utilities.

Important: I’m unable to perform the required web searches right now (tool usage limit error). Because this course format mandates that every major claim be supported by citations from web results, I can’t produce a citation-compliant “research-based” course section at the moment.

Key probability concepts (definitions)

Let $A$ and $B$ be events with $P(B)>0$.

1. Prior probability

   • prior probability is the probability assigned to a hypothesis before seeing the data/evidence.
   • Example: $P(H)$ for “the patient has the disease” before test results are known.

2. Conditional probability

   • conditional probability is $P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$

3. Posterior probability

   • posterior probability is what you get after plugging likelihood information and priors into Bayes’ theorem.

Bayes’ theorem (standard form)

For a hypothesis $H$ and evidence $E$ with $P(E)>0$: $P(H\mid E)=\\frac{P(E\mid H)P(H)}{P(E)}.$

When the denominator is expanded across mutually exclusive hypotheses $\{H_i\}$: $P(H_i\mid E)=\\frac{P(E\mid H_i)P(H_i)}{\\sum_j P(E\\mid H_j)P(H_j)}.$

This is essentially an “evidence-weighted prior update”:

• $P(H)$ is the prior plausibility,
• $P(E\mid H)$ is the likelihood (how well $H$ explains $E$),
• $P(E)$ normalizes so the posteriors sum to 1.

Key terms:

• likelihood $P(E\mid H)$
• evidence (marginal likelihood) $P(E)$
• normalization $P(E)$

Why Bayes is useful for decision making under uncertainty

In decision problems, you often choose an action $a$ based on uncertain states/hypotheses. Bayes’ theorem helps because it turns:

• uncertain state (which hypothesis is true) into
• updated probabilities (posterior beliefs) conditioned on observed evidence.

Those posterior probabilities can then support decision rules such as:

1. MAP decision rule (maximum a posteriori)

   • Choose the hypothesis with the largest posterior probability: $\hat{H}=\\arg\\max_i P(H_i\\mid E).$
   • This maximizes posterior probability, and is natural for classification when misclassification costs are uniform.

2. Expected utility / expected cost minimization

   • If each action has an outcome under each hypothesis, the best action is the one maximizing expected utility (or minimizing expected loss): $a^* = \\arg\\max_a \\sum_i U(a,H_i)\\,P(H_i\\mid E).$
   • Bayes provides $P(H_i\mid E)$; decision theory provides the objective.

3. Thresholding with probabilistic outputs

   • For example, decide “positive” if $P(H\mid E)$ exceeds a threshold chosen from costs: $\text{Decide } H \text{ if } P(H\\mid E)\\ge \\tau.$

Mermaid view of belief update → decision:

Worked mini-example (numeric intuition)

Suppose there are two hypotheses: $H_1$ (true) and $H_2$ (false). You have:

• prior: $P(H_1)=0.01$, $P(H_2)=0.99$
• likelihoods: $P(E\\mid H_1)=0.90$, $P(E\\mid H_2)=0.05$

Compute evidence: $P(E)=P(E\\mid H_1)P(H_1)+P(E\\mid H_2)P(H_2)=0.90\\cdot 0.01+0.05\\cdot 0.99.$

Posterior: $P(H_1\\mid E)=\\frac{P(E\\mid H_1)P(H_1)}{P(E)}.$

Decision implication: even if $P(E\\mid H_1)$ is high, a small prior can keep the posterior modest unless the evidence is sufficiently strong.