Agenda

<aside> 📌 Reminders

• A2 due November 26, 2024 </aside>

Priors

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A prior distribution: represents the initial beliefs or knowledge about a parameter before observing any data.

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• Key Points:
  • Denoted as $P(\theta)$, where $\theta$ is the parameter of interest.
  • Can be informative (based on prior knowledge) or uninformative (e.g., flat distribution).

Example: If you know a coin is fair, you might set $P(p)$ (the probability of heads) as a uniform prior between 0 and 1.

Posterior Distributions

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The posterior distribution: updates the prior distribution based on observed data using Bayes’ theorem:

$$ P(\theta \mid \text{data}) = \frac{P(\text{data} \mid \theta) P(\theta)}{P(\text{data})}  $$

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• $P(\theta \mid \text{data})$: Posterior (updated belief about $\theta$).
• $P(\text{data} \mid \theta)$: Likelihood (how likely the data is given $\theta$).
• $P(\theta)$: Prior.
• $P(\text{data})$: Normalizing constant.

Key Points:

• The posterior combines prior beliefs and evidence from the data.
• The shape of the posterior depends on both the prior and the likelihood.

Connection:

• Priors allow you to incorporate subjective beliefs.
• The posterior is central in Bayesian inference, enabling decisions based on both prior beliefs and observed data.
• A good choice of prior is crucial for meaningful results, especially with limited data.