Posterior belief

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.

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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)

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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).

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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.

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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: $P(p > 0.5 \mid \text{data}) = 0.9$.

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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.

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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.

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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.