1. Definition

  • Posterior belief is the updated belief about a parameter or hypothesis after observing data.
  • In Bayesian statistics, it is mathematically represented as the posterior distribution.
  • It combines:
    • Prior belief (what we thought before seeing data), and
    • Likelihood (evidence from data).

In short:

Posterior belief = prior belief updated by evidence.

2. Bayes’ Theorem and Posterior Belief

$P(\theta \mid D) = \frac{P(D \mid \theta) \cdot P(\theta)}{P(D)}$

  • $P(\theta)$ = Prior belief (before data)
  • $P(D \mid \theta)$ = Likelihood (compatibility of data with parameter)
  • $P(\theta \mid D)$ = Posterior belief (updated belief after data)

3. Interpretation

  • The posterior belief is usually a distribution over possible parameter values, not just a single number.
  • It reflects how plausible different parameter values are after seeing the data.
  • From the posterior distribution, we can compute posterior probabilities (probability of specific events).

4. Example – Coin Toss

  • Parameter of interest: probability of heads (ppp).
  • Prior belief: uniform prior → Beta(1,1).
  • Data: 10 tosses, 7 heads.
  • Posterior belief: Beta(8,4).
    • Centered around ~0.67 → meaning our updated belief is that ppp is most likely around 0.67.

5. Posterior Belief vs Posterior Probability

  • Posterior belief → the whole distribution after updating with data.
  • Posterior probability → a specific probability derived from that distribution.

Example:

  • Posterior belief: $p \sim Beta(8,4)$.
  • Posterior probability: $P(p > 0.5 \mid \text{data}) = 0.9$.

6. Applications

  • A/B testing: posterior belief about conversion rate difference.
  • Clinical trials: posterior belief about treatment effectiveness.
  • Machine learning: Bayesian models updating parameter distributions.

7. Key Takeaway

  • Posterior belief = the updated distribution of parameter values after seeing data.
  • It’s broader than posterior probability: the latter is just one number extracted from the former.

In short:
Posterior belief is your updated belief after observing data, represented by the posterior distribution. Posterior probability is a specific probability value derived from that distribution.