# Elicitation: Prior Elicitation In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

Prior elicitation is the act of inducing personal opinion to be expressed by the probabilities the person associates with an event (Savage, 1971). The `elicit` function elicits personal opinion and the `delicit` function estimates probability density to be used with model specification in the `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `LaplacesDemon.hpc`, `PMC`, or `VariationalBayes` functions.

## Usage

 ```1 2``` ```delicit(theta, x, a=-Inf, b=Inf, log=FALSE) elicit(n, cats, cat.names, show.plot=FALSE) ```

## Arguments

 `theta` This is a scalar or vector of parameters for which the density is estimated with respect to the kernel density estimate of `x`. `x` This is the elicited vector. `a` This is an optional lower bound for support. `b` This is an optional upper bound for support. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned. `n` This is the number of chips. `cats` This is a vector of k categories, bins, or intervals. When the variable is continuous, the mid-point of each category is used. For example, if the continuous interval [0,1] has 5 equal-sized categories, then `cats=c(0.1,0.3,0.5,0.7,0.9)`. `cat.names` This is a vector of category names. For example, if the continuous interval [0,1] has 5 equal-sized categories, then one way or naming the categories may be ```cat.names=c("0:<.2", ".2:<.4", ".4:<.6", ".6:<.8", ".8:1")```. `show.plot` Logical. If `show.plot=TRUE`, then a barplot is shown after each allocation of chips.

## Details

The `elicit` function elicits a univariate, discrete, non-conjugate, informative, prior probability distribution by offering a number of chips (specified as `n` by the statistician) for the user to allocate into categories specified by the statistician. The results of multiple elicitations (meaning, with multiple people), each the output of `elicit`, may be combined with the `c` function in base R.

This discrete distribution is included with the data for a model and supplied to a model specification function, where in turn it is supplied to the `delicit` function, which estimates the density at the current value of the prior distribution, p(theta). The prior distribution may be either continuous or discrete, will be proper, and may have bounded support (constrained to an interval).

For a minimal example, a statistician elicits the prior probability distribution for a regression effect, β. Non-statisticians would not be asked about expected parameters, but could be asked about how much y would be expected to change given a one-unit change in x. After consulting with others who have prior knowledge, the support does not need to be bounded, and their guesses at the range result in the statistician creating 5 catgories from the interval [-1,4], where each interval has a width of one. The statistician schedules time with 3 people, and each person participates when the statistician runs the following R code:

```x <- elicit(n=10, cats=c(-0.5, 0.5, 1.5, 2.5, 3.5), cat.names=c("-1:<0", "0:<1", "1:<2", "2:<3", "3:4"), show.plot=TRUE)```

Each of the 3 participants receives 10 chips to allocate among the 5 categories according to personal beliefs in the probability of the regression effect. When the statistician and each participant accept their elicited distribution, all 3 vectors are combined into one vector. In the model form, the prior is expressed as

p(beta) ~ EL

and the code for the model specification is

`elicit.prior <- delicit(beta, x, log=TRUE)`

This method is easily extended to priors that are multivariate, correlated, or conditional.

As an alternative, Hahn (2006) also used a categorical approach, eliciting judgements about the relative likelihood of each category, and then minimizes the KLD (for more information on KLD, see the `KLD` function).

## Author(s)

Statisticat, LLC. [email protected]

## References

Hahn, E.D. (2006). "Re-examining Informative Prior Elicitation Through the Lens of Markov chain Monte Carlo Methods". Journal of the Royal Statistical Society, A 169 (1), p. 37–48.

Savage, L.J. (1971). "Elicitation of Personal Probabilities and Expectations". Journal of the American Statistical Association, 66(336), p. 783–801.

`de.Finetti.Game`, `KLD`, `IterativeQuadrature`, `LaplaceApproximation`, `LaplacesDemon`, `LaplacesDemon.hpc`, `PMC`, and `VariationalBayes`.
 ```1 2 3 4 5``` ```library(LaplacesDemon) x <- c(1,2,2,3,3,3,4,7,8,8,9,10) #Elicited with elicit function theta <- seq(from=-5,to=15,by=.1) plot(theta, delicit(theta,x), type="l", xlab=expression(theta), ylab=expression("p(" * theta * ")")) ```