BaytesPMCMC.jl is a library to perform particle MCMC proposal steps for parameter in a `ModelWrapper`

struct, see ModelWrappers.jl.

BaytesPMCMC.jl implements a Particle Gibbs as well as an Particle Metropolis sampler. Note that the latter does not need a specified log objective function for the acceptance rate, but instead uses an approximation from a particle filter. It is recommended to use a fixed stepsize for this sampler, or, if possible, use Particle Gibbs instead.

Let us start with creating a univariate normal Mixture model with two states via ModelWrappers.jl:

```
using ModelWrappers, BaytesMCMC, BaytesFilters, BaytesPMCMC
using Distributions, Random, UnPack
_rng = Random.MersenneTwister(1)
N = 10^3
# Parameter
μ = [-2., 2.]
σ = [1., 1.]
p = [.05, .95]
# Latent data
latent = rand(_rng, Categorical(p), N)
data = [rand(_rng, Normal(μ[iter], σ[iter])) for iter in latent]
# Create ModelWrapper struct, assuming we do not know latent
latent_init = rand(_rng, Categorical(p), N)
myparameter = (;
μ = Param([Normal(-2., 5), Normal(2., 5)], μ, ),
σ = Param([Gamma(2.,2.), Gamma(2.,2.)], σ, ),
p = Param(Dirichlet(2, 2), p, ),
latent = Param([Categorical(p) for _ in Base.OneTo(N)], latent_init, ),
)
mymodel = ModelWrapper(myparameter)
myobjective = Objective(mymodel, data)
```

Particle Metropolis uses a particle filter to estimate the parameter latent, and an MCMC kernel to estimate all other parameter iteratively. This method is likelihood-free and uses an estimate from the particle filter for the acceptance ratio. As such, one does not need to state the log objective function at all, but gradient based mcmc kernels cannot be used either in this case. To assign a Particle Metropolis sampler, we only have to assign the particle filter dynamics as in BaytesFilters.jl:

```
# Assign Model dynamics
function BaytesFilters.dynamics(objective::Objective{<:ModelWrapper{BaseModel}})
@unpack model, data = objective
@unpack μ, σ, p = model.val
initial_latent = Categorical(p)
transition_latent(particles, iter) = initial_latent
transition_data(particles, iter) = Normal(μ[particles[iter]], σ[particles[iter]])
return Markov(initial_latent, transition_latent, transition_data)
end
dynamics(myobjective)
# Assign an objective for both a particle filter and an mcmc kernel:
myobjective_pf = Objective(mymodel, data, :latent)
myobjective_mcmc = Objective(mymodel, data, (:μ, :σ, :p))
# Assign Particle Metropolis algorithm
mcmcdefault = MCMCDefault(;
stepsize = ConfigStepsize(; ϵ = 1.0, stepsizeadaption = UpdateFalse()),
)
pmetropolis = ParticleMetropolis(
#Particle filter
ParticleFilter(_rng, myobjective_pf),
#MCMC kernel
MCMC(_rng, Metropolis, myobjective_mcmc, mcmcdefault)
)
# Proposal steps work exactly as in BaytesFilters.jl and BaytesMCMC.jl
_val, _diagnostics = propose!(_rng, pmetropolis, mymodel, data)
```

Particle Gibbs uses a conditional particle filter along with an MCMC kernel. In order to use this sampler, one has to define an objective function. However, we can condition the target function on the latent sequence, which results usually in a much easier and faster form than the (marginal) likelihood, where latent variables have to be integrated out. Once defined, we can also use more advanced mcmc kernels for estimation.

```
function (objective::Objective{<:ModelWrapper{BaseModel}})(θ::NamedTuple)
@unpack model, data, tagged = objective
@unpack μ, σ, p, latent = θ
## Prior -> a faster shortcut without initializing the priors again
lprior = log_prior(tagged.info.transform.constraint, ModelWrappers.subset(θ, tagged.parameter) )
##Likelihood
dynamicsᵉ = [Normal(μ[iter], σ[iter]) for iter in eachindex(μ)]
dynamicsˢ = Categorical(p)
ll = 0.0
#FOR PMCMC ~ target p(θ ∣ latent_1:t, data_1:t)
for iter in eachindex(data)
ll += logpdf(dynamicsᵉ[latent[iter]], data[iter])
ll += logpdf(dynamicsˢ, latent[iter] )
end
#=
# FOR MCMC ~ target p(θ ∣ data_1:t) by integrating out latent_1:t
for time in eachindex(data)
ll += logsumexp(logpdf(dynamicsˢ, iter) + logpdf(dynamicsᵉ[iter], grab(data, time)) for iter in eachindex(dynamicsᵉ))
end
=#
return ll + lprior
end
myobjective_mcmc(myobjective_mcmc.model.val)
# Note - It is good to benchmark this function, as it will allocate >98% of the mcmc kernel time
using BenchmarkTools
$myobjective_mcmc($myobjective_mcmc.model.val) #13.600 μs (2 allocations: 176 bytes)
```

As we can analytically compute the marginal likelihood of a univariate mixture, I could also write down (and comment out) the corresponding objective function in the MCMC case. This should help understanding my comments above. Once our objective is defined, we can intialize a `ParticleGibbs`

struct and sample with it:

```
# Assign an objective for both a particle filter and an mcmc kernel:
myobjective_pf = Objective(mymodel, data, :latent)
myobjective_mcmc = Objective(mymodel, data, (:μ, :σ, :p))
# Assign Particle Gibbs sampler
pfdefault = ParticleFilterDefault(referencing = Conditional())
pgibbs = ParticleGibbs(
#Conditional Particle filter
ParticleFilter(_rng, myobjective_pf, pfdefault
),
#MCMC kernel -> can use more advanced kernels
MCMC(_rng, NUTS, myobjective_mcmc)
)
# Proposal steps work exactly as in BaytesFilters.jl and BaytesMCMC.jl
_val, _diagnostics = propose!(_rng, pgibbs, mymodel, data)
```

This package is still highly experimental - suggestions and comments are always welcome!