The `Acceptance.Rate`

function calculates the acceptance rate per
chain from a matrix of posterior MCMC samples.

`AcceptanceRate(x)`

x

This required argument accepts a \(S \times J\)
numeric matrix of \(S\) posterior samples for \(J\) variables,
such as `Posterior1`

or `Posterior2`

from an object of
class `demonoid`

.

The `AcceptanceRate`

function returns a vector of acceptance
rates, one for each chain.

The acceptance rate of an MCMC algorithm is the percentage of iterations in which the proposals were accepted.

Optimal Acceptance Rates
The optimal acceptance rate varies with the number of parameters and
by algorithm. Algorithms with componentwise Gaussian proposals have an
optimal acceptance rate of 0.44, regardless of the number of
parameters. Algorithms that update with multivariate Gaussian
proposals tend to have an optimal acceptance rate that ranges from
0.44 for one parameter (one IID Gaussian target distribution) to 0.234
for an infinite number of parameters (IID Gaussian target
distributions), and 0.234 is approached quickly as the number of
parameters increases. The AHMC, HMC, and THMC algorithms have an
optimal acceptance rate of 0.67, except with the algorithm
specification `L=1`

, where the optimal acceptance rate is
0.574. The target acceptance rate is specified in HMCDA and NUTS, and
the recommended rate is 0.65 and 0.60 respectively. Some algorithms
have an acceptance rate of 1, such as AGG, ESS, GG, GS (MISS only),
SGLD, or Slice.

Global and Local Acceptance Rates
`LaplacesDemon`

reports the global acceptance rate for the
un-thinned chains. However, componentwise algorithms make a proposal
per parameter, and therefore have a local acceptance rate for each
parameter. Since only the global acceptance rate is reported, the
`AcceptanceRate`

function may be used to calculate the local
acceptance rates from a matrix of un-thinned posterior samples.

Thinning
Thinned samples tend to have higher local acceptance rates than
un-thinned samples. With enough samples and enough thinning, local
acceptance rates approach 1. Local acceptance rates do not need to
approach the optimal acceptance rates above. Conversely, local
acceptance rates do not need to approach 1, because too much
information may possibly be discarded by thinning. For more
information on thinning, see the `Thin`

function.

Diagnostics
The `AcceptanceRate`

function may be used to calculate local
acceptance rates on a matrix of thinned or un-thinned samples. Any
chain with a local acceptance rate that is an outlier may be studied
for reasons that may cause the outlier. A local acceptance rate
outlier does not violate theory and is often acceptable, but may
indicate a potential problem. Only some of the many potential problems
include: identifiability, model misspecification, multicollinearity,
multimodality, choice of prior distributions, or becoming trapped in a
low-probability space. The solution to local acceptance rate outliers
tends to be either changing the MCMC algorithm or re-specifying the
model or priors. For example, an MCMC algorithm that makes
multivariate Gaussian proposals for a large number of parameters may
have low global and local acceptance rates when far from the target
distributions.

`LaplacesDemon`

,
`MISS`

,
`PosteriorChecks`

, and
`Thin`

.

```
# NOT RUN {
library(LaplacesDemon)
AcceptanceRate(matrix(rnorm(5000),1000,5))
# }
```

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