# IAT: Integrated Autocorrelation Time In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

The `IAT` function estimates integrated autocorrelation time, which is the computational inefficiency of a continuous chain or MCMC sampler. IAT is also called the IACT, ACT, autocorrelation time, autocovariance time, correlation time, or inefficiency factor. A lower value of `IAT` is better. `IAT` is a MCMC diagnostic that is an estimate of the number of iterations, on average, for an independent sample to be drawn, given a continuous chain or Markov chain. Put another way, `IAT` is the number of correlated samples with the same variance-reducing power as one independent sample.

IAT is a univariate function. A multivariate form is not included.

## Usage

 `1` ```IAT(x) ```

## Arguments

 `x` This requried argument is a vector of samples from a chain.

## Details

`IAT` is a MCMC diagnostic that is often used to compare continuous chains of MCMC samplers for computational inefficiency, where the sampler with the lowest `IAT`s is the most efficient sampler. Otherwise, chains may be compared within a model, such as with the output of `LaplacesDemon` to learn about the inefficiency of the continuous chain. For more information on comparing MCMC algorithmic inefficiency, see the `Juxtapose` function.

`IAT` is also estimated in the `PosteriorChecks` function. `IAT` is usually applied to a stationary, continuous chain after discarding burn-in iterations (see `burnin` for more information). The `IAT` of a continuous chain correlates with the variability of the mean of the chain, and relates to Effective Sample Size (`ESS`) and Monte Carlo Standard Error (`MCSE`).

`IAT` and `ESS` are inversely related, though not perfectly, because each is estimated a little differently. Given N samples and taking autocorrelation into account, `ESS` estimates a reduced number of M samples. Conversely, `IAT` estimates the number of autocorrelated samples, on average, required to produce one independently drawn sample.

The `IAT` function is similar to the `IAT` function in the `Rtwalk` package of Christen and Fox (2010), which is currently unavailabe on CRAN.

## Value

The `IAT` function returns the integrated autocorrelation time of a chain.

## Author(s)

Statisticat, LLC. software@bayesian-inference.com

## References

Christen, J.A. and Fox, C. (2010). "A General Purpose Sampling Algorithm for Continuous Distributions (the t-walk)". Bayesian Analysis, 5(2), p. 263–282.

`burnin`, `Compare`, `ESS`, `LaplacesDemon`, `MCSE`, and `PosteriorChecks`.
 ```1 2 3``` ```library(LaplacesDemon) theta <- rnorm(100) IAT(theta) ```