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Highest Density Interval (HDI)

Source: R/hdi.R
hdi.Rd

Compute the Highest Density Interval (HDI) of posterior distributions. All points within this interval have a higher probability density than points outside the interval. The HDI can be used in the context of uncertainty characterisation of posterior distributions as Credible Interval (CI).

Usage

hdi(x, ...)

# S3 method for class 'numeric'
hdi(x, ci = 0.95, verbose = TRUE, ...)

# S3 method for class 'data.frame'
hdi(x, ci = 0.95, rvar_col = NULL, verbose = TRUE, ...)

# S3 method for class 'stanreg'
hdi(
  x,
  ci = 0.95,
  effects = c("fixed", "random", "all"),
  component = c("location", "all", "conditional", "smooth_terms", "sigma",
    "distributional", "auxiliary"),
  parameters = NULL,
  verbose = TRUE,
  ...
)

# S3 method for class 'brmsfit'
hdi(
  x,
  ci = 0.95,
  effects = c("fixed", "random", "all"),
  component = c("conditional", "zi", "zero_inflated", "all"),
  parameters = NULL,
  verbose = TRUE,
  ...
)

# S3 method for class 'get_predicted'
hdi(x, ci = 0.95, use_iterations = FALSE, verbose = TRUE, ...)

Arguments

x

Vector representing a posterior distribution, or a data frame of such vectors. Can also be a Bayesian model. bayestestR supports a wide range of models (see, for example, methods("hdi")) and not all of those are documented in the 'Usage' section, because methods for other classes mostly resemble the arguments of the .numeric or .data.framemethods.

...

Currently not used.

ci

Value or vector of probability of the (credible) interval - CI (between 0 and 1) to be estimated. Default to .95 (95%).

verbose

Toggle off warnings.

rvar_col

A single character - the name of an rvar column in the data frame to be processed. See example in p_direction().

effects

Should results for fixed effects, random effects or both be returned? Only applies to mixed models. May be abbreviated.

component

Should results for all parameters, parameters for the conditional model or the zero-inflated part of the model be returned? May be abbreviated. Only applies to brms-models.

parameters

Regular expression pattern that describes the parameters that should be returned. Meta-parameters (like lp__ or prior_) are filtered by default, so only parameters that typically appear in the summary() are returned. Use parameters to select specific parameters for the output.

use_iterations

Logical, if TRUE and x is a get_predicted object, (returned by insight::get_predicted()), the function is applied to the iterations instead of the predictions. This only applies to models that return iterations for predicted values (e.g., brmsfit models).

Value

A data frame with following columns:

  • Parameter The model parameter(s), if x is a model-object. If x is a vector, this column is missing.

  • CI The probability of the credible interval.

  • CI_low, CI_high The lower and upper credible interval limits for the parameters.

Details

Unlike equal-tailed intervals (see eti()) that typically exclude 2.5% from each tail of the distribution and always include the median, the HDI is not equal-tailed and therefore always includes the mode(s) of posterior distributions. While this can be useful to better represent the credibility mass of a distribution, the HDI also has some limitations. See spi() for details.

The 95% or 89% Credible Intervals (CI) are two reasonable ranges to characterize the uncertainty related to the estimation (see here for a discussion about the differences between these two values).

The 89% intervals (ci = 0.89) are deemed to be more stable than, for instance, 95% intervals (Kruschke, 2014). An effective sample size of at least 10.000 is recommended if one wants to estimate 95% intervals with high precision (Kruschke, 2014, p. 183ff). Unfortunately, the default number of posterior samples for most Bayes packages (e.g., rstanarm or brms) is only 4.000 (thus, you might want to increase it when fitting your model). Moreover, 89 indicates the arbitrariness of interval limits - its only remarkable property is being the highest prime number that does not exceed the already unstable 95% threshold (McElreath, 2015).

However, 95% has some advantages too. For instance, it shares (in the case of a normal posterior distribution) an intuitive relationship with the standard deviation and it conveys a more accurate image of the (artificial) bounds of the distribution. Also, because it is wider, it makes analyses more conservative (i.e., the probability of covering 0 is larger for the 95% CI than for lower ranges such as 89%), which is a good thing in the context of the reproducibility crisis.

A 95% equal-tailed interval (ETI) has 2.5% of the distribution on either side of its limits. It indicates the 2.5th percentile and the 97.5h percentile. In symmetric distributions, the two methods of computing credible intervals, the ETI and the HDI, return similar results.

This is not the case for skewed distributions. Indeed, it is possible that parameter values in the ETI have lower credibility (are less probable) than parameter values outside the ETI. This property seems undesirable as a summary of the credible values in a distribution.

On the other hand, the ETI range does change when transformations are applied to the distribution (for instance, for a log odds scale to probabilities): the lower and higher bounds of the transformed distribution will correspond to the transformed lower and higher bounds of the original distribution. On the contrary, applying transformations to the distribution will change the resulting HDI.

Note

There is also a plot()-method implemented in the see-package.

References

  • Kruschke, J. (2014). Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press.

  • McElreath, R. (2015). Statistical rethinking: A Bayesian course with examples in R and Stan. Chapman and Hall/CRC.

See also

Other interval functions, such as hdi(), eti(), bci(), spi(), si().

Other ci: bci(), ci(), eti(), si(), spi()

Author

Credits go to ggdistribute and HDInterval.

Examples

library(bayestestR)

posterior <- rnorm(1000)
hdi(posterior, ci = 0.89)
#> 89% HDI: [-1.46, 1.79]
hdi(posterior, ci = c(0.80, 0.90, 0.95))
#> Highest Density Interval
#> 
#> 80% HDI       |       90% HDI |       95% HDI
#> ---------------------------------------------
#> [-1.39, 1.24] | [-1.59, 1.77] | [-2.10, 1.80]

bayestestR::hdi(iris[1:4])
#> Identical densities found along different segments of the distribution,
#>   choosing rightmost.
#> Highest Density Interval
#> 
#> Parameter    |      95% HDI
#> ---------------------------
#> Sepal.Length | [4.60, 7.70]
#> Sepal.Width  | [2.20, 3.90]
#> Petal.Length | [1.00, 6.10]
#> Petal.Width  | [0.10, 2.30]
bayestestR::hdi(iris[1:4], ci = c(0.80, 0.90, 0.95))
#> Identical densities found along different segments of the distribution,
#>   choosing rightmost.
#> Identical densities found along different segments of the distribution,
#>   choosing rightmost.
#> Identical densities found along different segments of the distribution,
#>   choosing rightmost.
#> Highest Density Interval
#> 
#> Parameter    |      80% HDI |      90% HDI |      95% HDI
#> ---------------------------------------------------------
#> Sepal.Length | [4.90, 6.90] | [4.40, 6.90] | [4.60, 7.70]
#> Sepal.Width  | [2.50, 3.60] | [2.40, 3.80] | [2.20, 3.90]
#> Petal.Length | [1.30, 5.50] | [1.10, 5.80] | [1.00, 6.10]
#> Petal.Width  | [0.10, 1.90] | [0.20, 2.30] | [0.10, 2.30]
# \donttest{
model <- suppressWarnings(
  rstanarm::stan_glm(mpg ~ wt + gear, data = mtcars, chains = 2, iter = 200, refresh = 0)
)
bayestestR::hdi(model)
#> Highest Density Interval 
#> 
#> Parameter   |        95% HDI
#> ----------------------------
#> (Intercept) | [29.21, 49.50]
#> wt          | [-6.99, -4.05]
#> gear        | [-2.18,  1.68]
bayestestR::hdi(model, ci = c(0.80, 0.90, 0.95))
#> Highest Density Interval 
#> 
#> Parameter   |        80% HDI |        90% HDI |        95% HDI
#> --------------------------------------------------------------
#> (Intercept) | [31.68, 46.67] | [29.21, 47.18] | [29.21, 49.50]
#> wt          | [-6.30, -4.23] | [-6.70, -4.11] | [-6.99, -4.05]
#> gear        | [-1.53,  1.08] | [-1.89,  1.41] | [-2.18,  1.68]

bayestestR::hdi(emmeans::emtrends(model, ~1, "wt", data = mtcars))
#> Highest Density Interval
#> 
#> X1      |        95% HDI
#> ------------------------
#> overall | [-6.99, -4.05]

model <- brms::brm(mpg ~ wt + cyl, data = mtcars)
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 7e-06 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.07 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
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#> Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.023 seconds (Warm-up)
#> Chain 1:                0.018 seconds (Sampling)
#> Chain 1:                0.041 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 3e-06 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2: 
#> Chain 2: 
#> Chain 2: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 2: 
#> Chain 2:  Elapsed Time: 0.023 seconds (Warm-up)
#> Chain 2:                0.019 seconds (Sampling)
#> Chain 2:                0.042 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 3e-06 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.03 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3: 
#> Chain 3: 
#> Chain 3: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 3: 
#> Chain 3:  Elapsed Time: 0.024 seconds (Warm-up)
#> Chain 3:                0.019 seconds (Sampling)
#> Chain 3:                0.043 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 4e-06 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.04 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4: 
#> Chain 4: 
#> Chain 4: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 4: 
#> Chain 4:  Elapsed Time: 0.023 seconds (Warm-up)
#> Chain 4:                0.02 seconds (Sampling)
#> Chain 4:                0.043 seconds (Total)
#> Chain 4: 
bayestestR::hdi(model)
#> Highest Density Interval 
#> 
#> Parameter   |        95% HDI
#> ----------------------------
#> (Intercept) | [36.15, 43.24]
#> wt          | [-4.81, -1.60]
#> cyl         | [-2.37, -0.66]
bayestestR::hdi(model, ci = c(0.80, 0.90, 0.95))
#> Highest Density Interval 
#> 
#> Parameter   |        80% HDI |        90% HDI |        95% HDI
#> --------------------------------------------------------------
#> (Intercept) | [37.05, 41.78] | [36.79, 42.75] | [36.15, 43.24]
#> wt          | [-4.22, -2.23] | [-4.53, -1.88] | [-4.81, -1.60]
#> cyl         | [-2.07, -0.97] | [-2.20, -0.78] | [-2.37, -0.66]

bf <- BayesFactor::ttestBF(x = rnorm(100, 1, 1))
bayestestR::hdi(bf)
#> Highest Density Interval
#> 
#> Parameter  |      95% HDI
#> -------------------------
#> Difference | [0.77, 1.19]
bayestestR::hdi(bf, ci = c(0.80, 0.90, 0.95))
#> Highest Density Interval
#> 
#> Parameter  |      80% HDI |      90% HDI |      95% HDI
#> -------------------------------------------------------
#> Difference | [0.85, 1.12] | [0.82, 1.17] | [0.78, 1.20]
# }