When to use: When we want to find the posterior probability mass given a specific boundary – or set of boundaries. 

- Example: What is the posterior probability that the proportion of water (p) is < 0.5? 

- Calculate using samples from the posterior: 
-  draw our 10,000 sample from our posterior distribution, w/ replacement
- Samples <- sample(p_grid, prob=posterior, size=1e4, replacement=TRUE)
- add  up the samples < 0.5 
- Sum(samples < 0.5) / 1e4
- 0.1726





Intervals of Define Boundaries

- Commonly seen in scientific journals as confidence intervals 
- Best approach = use samples from the posterior 

Example - Find the boundaries of the lower 80% posterior probability:

quantile(samples, 0.8)
      80%
0.7607608  # 80% of the posterior prob lies  between 0 and 0.76

- The result reports two parameter values that contain between them a specified amount of posterior probability 
- This type of interval  assigns equal probability mass to each tail is known as a percentile interval (PI) 
- Highest Posterior Density Interval (HPDI) – finds the boundaries by using the narrowest interval containing the specified probability mass. 


Intervals of Defined Mass

Important note: Choosing a point estimate to describe the posterior distribution is hardly ever necessary and is often harmful as it discards information  

Point estimates: is a single-value summary of the posterior distribution  
- mean
- median
- mode

Which point estimate to use?

- Loss function - a rule that tells you the cost associated with using any particular point estimate.  

- Different loss functions nominate different point estimates
- Goal: find the parameter value (or a decision) that will minimize our expected loss; do this by using the posterior to average over our uncertainty in the true value


Point Estimates

Goal: to summarize and interpret from the posterior distribution

Example questions to answer:
- How much posterior probability lies below some parameter value?
- How much posterior probability lies between two parameter values?
- Which parameter value marks the lower 5% of the posterior probability?
- Which range of parameter values contains 90% of the posterior probability?
- Which parameter value has highest posterior probability?

We can answer these question by the below 3 categories:
1. Intervals of defined boundaries
2. Intervals of defined mass
3. Point estimates

New York University

- Sampling from a grid-approximate posterior
- Sampling to summarize
- Sampling to simulate prediction

Sampling the Imaginary

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

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