# inference

##### statistical inference on greta models

Carry out statistical inference on greta models by MCMC or likelihood/posterior optimisation.

##### Usage

```
mcmc(model, sampler = hmc(), n_samples = 1000, thin = 1,
warmup = 1000, chains = 4, n_cores = NULL, verbose = TRUE,
pb_update = 50, one_by_one = FALSE, initial_values = initials())
```stashed_samples()

extra_samples(draws, n_samples = 1000, thin = 1, n_cores = NULL,
verbose = TRUE, pb_update = 50, one_by_one = FALSE)

initials(...)

opt(model, optimiser = bfgs(), max_iterations = 100,
tolerance = 1e-06, initial_values = initials(), adjust = TRUE,
hessian = FALSE)

##### Arguments

- model
greta_model object

- sampler
sampler used to draw values in MCMC. See

`samplers`

for options.- n_samples
number of MCMC samples to draw per chain (after any warm-up, but before thinning)

- thin
MCMC thinning rate; every

`thin`

samples is retained, the rest are discarded- warmup
number of samples to spend warming up the mcmc sampler (moving chains toward the highest density area and tuning sampler hyperparameters).

- chains
number of MCMC chains to run

- n_cores
the maximum number of CPU cores used by each sampler (see details).

- verbose
whether to print progress information to the console

- pb_update
how regularly to update the progress bar (in iterations)

- one_by_one
whether to run TensorFlow MCMC code one iteration at a time, so that greta can handle numerical errors as 'bad' proposals (see below).

- initial_values
an optional

`initials`

object (or list of`initials`

objects of length`chains`

) giving initial values for some or all of the variables in the model. These will be used as the starting point for sampling/optimisation.- draws
an mcmc.list object returned by

`mcmc`

or`stashed_samples`

- ...
named numeric values, giving initial values of some or all of the variables in the model (unnamed variables will be automatically initialised)

- optimiser
an

`optimiser`

object giving the optimisation algorithm and parameters See`optimisers`

.- max_iterations
the maximum number of iterations before giving up

- tolerance
the numerical tolerance for the solution, the optimiser stops when the (absolute) difference in the joint density between successive iterations drops below this level

- adjust
whether to account for Jacobian adjustments in the joint density. Set to

`FALSE`

(and do not use priors) for maximum likelihood estimates, or`TRUE`

for maximum*a posteriori*estimates.- hessian
whether to return a list of

*analytically*differentiated Hessian arrays for the parameters

##### Details

For `mcmc()`

if `verbose = TRUE`

, the progress bar shows
the number of iterations so far and the expected time to complete the phase
of model fitting (warmup or sampling). Occasionally, a proposed set of
parameters can cause numerical instability (I.e. the log density or its
gradient is `NA`

, `Inf`

or `-Inf`

); normally because the log
joint density is so low that it can't be represented as a floating point
number. When this happens, the progress bar will also display the
proportion of proposals so far that were 'bad' (numerically unstable) and
therefore rejected. Some numerical instability during the warmup phase is
normal, but 'bad' samples during the sampling phase can lead to bias in
your posterior sample. If you only have a few bad samples (<10\
usually resolve this with a longer warmup period or by manually defining
starting values to move the sampler into a more reasonable part of the
parameter space. If you have more samples than that, it may be that your
model is misspecified. You can often diagnose this by using
`calculate()`

to evaluate the values of greta arrays, given
fixed values of model parameters, and checking the results are what you
expect.

greta runs multiple chains simultaneously with a single sampler,
vectorising all operations across the chains. E.g. a scalar addition in
your model is computed as an elementwise vector addition (with vectors
having length `chains`

), a vector addition is computed as a matrix
addition etc. TensorFlow is able to parallelise these operations, and this
approach reduced computational overheads, so this is the most efficient of
computing on multiple chains.

Multiple mcmc samplers (each of which can simultaneously run multiple
chains) can also be run in parallel by setting the execution plan with the
`future`

package. Only `plan(multisession)`

futures or
`plan(cluster)`

futures that don't use fork clusters are allowed,
since forked processes conflict with TensorFlow's parallelism. Explicitly
parallelising chains on a local machine with `plan(multisession)`

will
probably be slower than running multiple chains simultaneously in a single
sampler (with `plan(sequential)`

, the default) because of the overhead
required to start new sessions. However, `plan(cluster)`

can be used
to run chains on a cluster of machines on a local or remote network. See
`future::cluster`

for details, and the
`future.batchtools`

package to set up plans on clusters with job
schedulers.

If `n_cores = NULL`

and mcmc samplers are being run sequentially, each
sampler will be allowed to use all CPU cores (possibly to compute multiple
chains sequentially). If samplers are being run in parallel with the
`future`

package, `n_cores`

will be set so that ```
n_cores *
future::nbrOfWorkers
```

is less than the number
of CPU cores.

If the sampler is aborted before finishing (and `future`

parallelism isn't being used), the samples collected so far can be
retrieved with `stashed_samples()`

. Only samples from the sampling
phase will be returned.

Samples returned by `mcmc()`

and `stashed_samples()`

can
be added to with `extra_samples()`

. This continues the chain from the
last value of the previous chain and uses the same sampler and model as was
used to generate the previous samples. It is not possible to change the
sampler or extend the warmup period.

Because `opt()`

acts on a list of greta arrays with possibly
varying dimension, the `par`

and `hessian`

objects returned by
`opt()`

are named lists, rather than a vector (`par`

) and a
matrix (`hessian`

), as returned by `optim()`

.
Because greta arrays may not be vectors, the hessians may not be matrices,
but could be higher-dimensional arrays. To return a hessian matrix covering
multiple model parameters, you can construct your model so that all those
parameters are in a vector, then split the vector up to define the model.
The parameter vector can then be passed to model. See example.

##### Value

`mcmc`

, `stashed_samples`

& `extra_samples`

- an
`mcmc.list`

object that can be analysed using functions from the coda
package. This will contain mcmc samples of the greta arrays used to create
`model`

.

`opt`

- a list containing the following named elements:

`par`

a named list of the optimal values for the greta arrays specified in`model`

`value`

the (unadjusted) negative log joint density of the model at the parameters 'par'`iterations`

the number of iterations taken by the optimiser`convergence`

an integer code, 0 indicates successful completion, 1 indicates the iteration limit`max_iterations`

had been reached`hessian`

(if`hessian = TRUE`

) a named list of hessian matrices/arrays for the parameters (w.r.t.`value`

)

##### Examples

```
# NOT RUN {
# define a simple Bayesian model
x <- rnorm(10)
mu <- normal(0, 5)
sigma <- lognormal(1, 0.1)
distribution(x) <- normal(mu, sigma)
m <- model(mu, sigma)
# carry out mcmc on the model
draws <- mcmc(m, n_samples = 100)
# add some more samples
draws <- extra_samples(draws, 200)
#' # initial values can be passed for some or all model variables
draws <- mcmc(m, chains = 1, initial_values = initials(mu = -1))
# if there are multiple chains, a list of initial values should be passed,
# othewise the same initial values will be used for all chains
inits <- list(initials(sigma = 0.5), initials(sigma = 1))
draws <- mcmc(m, chains = 2, initial_values = inits)
# you can auto-generate a list of initials with something like this:
inits <- replicate(4,
initials(mu = rnorm(1), sigma = runif(1)),
simplify = FALSE)
draws <- mcmc(m, chains = 4, initial_values = inits)
# or find the MAP estimate
opt_res <- opt(m)
# get the MLE of the normal variance
mu <- variable()
variance <- variable(lower = 0)
distribution(x) <- normal(mu, sqrt(variance))
m2 <- model(variance)
# adjust = FALSE skips the jacobian adjustments used in MAP estimation, to
# give the true maximum likelihood estimates
o <- opt(m2, adjust = FALSE)
# the MLE corresponds to the *unadjusted* sample variance, but differs
# from the sample variance
o$par
mean((x - mean(x)) ^ 2) # same
var(x) # different
# initial values can also be passed to optimisers:
o <- opt(m2, initial_values = initials(variance = 1))
# and you can return a list of the hessians for each of these parameters
o <- opt(m2, hessians = TRUE)
o$hessians
# to get a hessian matrix across multiple greta arrays, you must first
# combine them and then split them up for use in the model (so that the
# combined vector is part of the model) and pass that vector to model:
params <- c(variable(), variable(lower = 0))
mu <- params[1]
variance <- params[2]
distribution(x) <- normal(mu, sqrt(variance))
m3 <- model(params)
o <- opt(m3, hessians = TRUE)
o$hessians
# }
```

*Documentation reproduced from package greta, version 0.3.0, License: Apache License 2.0*