diff --git a/Project.toml b/Project.toml
index bdec9be..868b4bc 100644
--- a/Project.toml
+++ b/Project.toml
@@ -8,12 +8,15 @@ JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
 
 [weakdeps]
+AbstractGPs = "99985d1d-32ba-4be9-9821-2ec096f28918"
 InfiniteOpt = "20393b10-9daf-11e9-18c9-8db751c92c57"
 
 [extensions]
+AbstractGPsDisjunctiveProgramming = "AbstractGPs"
 InfiniteDisjunctiveProgramming = "InfiniteOpt"
 
 [compat]
+AbstractGPs = "0.5"
 Aqua = "0.8"
 JuMP = "1.18"
 Reexport = "1"
@@ -30,4 +33,4 @@ Ipopt = "b6b21f68-93f8-5de0-b562-5493be1d77c9"
 Juniper = "2ddba703-00a4-53a7-87a5-e8b9971dde84"
 
 [targets]
-test = ["Aqua", "HiGHS", "Test", "Juniper", "Ipopt", "InfiniteOpt"]
+test = ["Aqua", "HiGHS", "Test", "Juniper", "Ipopt", "InfiniteOpt", "AbstractGPs"]
diff --git a/README.md b/README.md
index 543d1f7..97c29a2 100644
--- a/README.md
+++ b/README.md
@@ -176,6 +176,7 @@ The following reformulation methods are currently supported:
 
     - `optimizer`: Optimizer to use when solving subproblems to determine M values. This is a required value.
     - `default_M`: Default big-M value to use if no big-M is specified for a logical variable (1e9).
+    - `sampler`: M-value sampler for infinite models. Default: `nothing`, which solves an M subproblem at every support. Pass a `GPSampler` to instead solve a subset of the supports and fill the rest with a conservative Gaussian-process estimate. Ignored for finite models.
 
 5. [P-Split](https://arxiv.org/abs/2202.05198): This method reformulates each disjunct constraint into P constraints, each with a partitioned group defined by the user. This method requires that terms in the constraint be convex additively seperable with respect to each variable. The `PSplit` struct is created with the following required arguments:
 
@@ -223,6 +224,8 @@ optimize!(model, gdp_method = Hull())
 value(W)
 ```
 
+When the `MBM` reformulation is used on an infinite model, an M subproblem is solved at every support by default. Loading [AbstractGPs.jl](https://github.com/JuliaGaussianProcesses/AbstractGPs.jl) (`using AbstractGPs`) enables an additional extension that instead solves M at a subset of the supports and fills the rest with a conservative Gaussian-process estimate, which can substantially reduce the number of subproblem solves. To opt in, pass a `GPSampler` via the `sampler` keyword, e.g. `MBM(optimizer, sampler = GPSampler())` (squared exponential kernel with the lengthscale selected by marginal likelihood), `GPSampler(Matern52Kernel())` (a custom kernel, lengthscale selected the same way), or `GPSampler(Matern52Kernel(), lengthscales = [0.2])` (lengthscale pinned). The filled values are heuristic upper estimates rather than certificates; see the `GPSampler` docstring for the tuning keywords (`kappa`, `budget`, `detect_uniform_M`, `lengthscales`, `jitter`, `seeds`).
+
 ## Release Notes
 
 Prior to `v0.4.0`, the package did not leverage the JuMP extension capabilities and was not as robust. For these earlier releases, refer to [Perez, Joshi, and Grossmann, 2023](https://arxiv.org/abs/2304.10492v1) and the following [JuliaCon 2022 Talk](https://www.youtube.com/watch?v=AMIrgTTfUkI).
diff --git a/ext/AbstractGPsDisjunctiveProgramming.jl b/ext/AbstractGPsDisjunctiveProgramming.jl
new file mode 100644
index 0000000..00af68c
--- /dev/null
+++ b/ext/AbstractGPsDisjunctiveProgramming.jl
@@ -0,0 +1,119 @@
+module AbstractGPsDisjunctiveProgramming
+
+import AbstractGPs
+import AbstractGPs.KernelFunctions
+import DisjunctiveProgramming as DP
+
+################################################################################
+#                                 GP FITTING
+################################################################################
+# Normalized to [0, 1]^d so one lengthscale works across dimensions
+function _support_coords(
+    grids::Tuple{Vararg{Vector{Float64}}}
+    )::Vector{Vector{Float64}}
+    indices = CartesianIndices(length.(grids))
+    mins = [minimum(g) for g in grids]
+    ranges = [max(maximum(g) - minimum(g), eps()) for g in grids]
+    return [[(grids[d][I[d]] - mins[d]) / ranges[d]
+             for d in 1:length(grids)] for I in vec(indices)]
+end
+
+# Lengthscale selected by marginal likelihood over the candidates
+function _fit_posterior(
+    kernel::KernelFunctions.Kernel,
+    X::Vector{Vector{Float64}},
+    y::Vector{Float64},
+    sampler::DP.GPSampler
+    )
+    best_posterior, best_log_prob = nothing, -Inf
+    for lengthscale in sampler.lengthscales
+        scaled_kernel = KernelFunctions.with_lengthscale(
+            kernel, lengthscale)
+        finite_gp = AbstractGPs.GP(scaled_kernel)(X, sampler.jitter)
+        log_prob = AbstractGPs.logpdf(finite_gp, y)
+        if log_prob > best_log_prob
+            best_posterior = AbstractGPs.posterior(finite_gp, y)
+            best_log_prob = log_prob
+        end
+    end
+    return best_posterior
+end
+
+function _mean_sd(
+    sampler::DP.GPSampler,
+    X::Vector{Vector{Float64}},
+    solved::Dict{Int, Float64}
+    )
+    solved_indices = collect(keys(solved))
+    y = [solved[i] for i in solved_indices]
+    y_mean = sum(y) / length(y)
+    # floored so near-equal solved values still cushion the filled ones
+    y_scale = max(sqrt(sum(abs2, y .- y_mean) / max(length(y) - 1, 1)),
+        1e-2 * abs(y_mean), 1e-8)
+    kernel = something(sampler.kernel,
+        KernelFunctions.SqExponentialKernel())
+    posterior = _fit_posterior(
+        kernel, X[solved_indices], (y .- y_mean) ./ y_scale, sampler)
+    posterior_mean = AbstractGPs.mean(posterior, X)
+    posterior_var = max.(AbstractGPs.var(posterior, X), 0.0)
+    return posterior_mean .* y_scale .+ y_mean,
+        sqrt.(posterior_var) .* y_scale
+end
+
+################################################################################
+#                              M VALUE SAMPLING
+################################################################################
+# Solve M at max-UCB selected supports, fill the rest with the bound
+function DP.sample_M_values(
+    sampler::DP.GPSampler,
+    objectives::AbstractArray,
+    sub::DP.GDPSubmodel,
+    method::DP._MBM,
+    support_grids::Function
+    )
+    indices = collect(CartesianIndices(objectives))
+    n = length(indices)
+    solved = Dict{Int, Float64}()
+    solve_at(index::Int) = begin
+        M_val = DP.raw_M(sub, objectives[indices[index]], method)
+        M_val === nothing && return false
+        solved[index] = M_val
+        return true
+    end
+    # an evenly spaced seed count, or user-given seed fractions
+    fractions = sampler.seeds isa Int ?
+        range(0, 1, length = sampler.seeds) : sampler.seeds
+    for index in unique(1 .+ round.(Int, fractions .* (n - 1)))
+        solve_at(index) || return nothing
+    end
+    if sampler.detect_uniform_M
+        # a uniform M needs no fit, and so no support grid either
+        probes = collect(values(solved))
+        all(==(first(probes)), probes) && return first(probes)
+    end
+    budget = min(ceil(Int, sampler.budget * n), n)
+    X = _support_coords(support_grids())
+    while length(solved) < budget
+        means, sds = _mean_sd(sampler, X, solved)
+        acquisition = means .+ sampler.kappa .* sds
+        for index in keys(solved)
+            acquisition[index] = -Inf
+        end
+        solve_at(argmax(acquisition)) || return nothing
+    end
+    M_vals = Array{Float64}(undef, size(objectives))
+    if length(solved) == n # nothing left to estimate
+        for (index, I) in enumerate(indices)
+            M_vals[I] = solved[index]
+        end
+        return M_vals
+    end
+    means, sds = _mean_sd(sampler, X, solved)
+    for (index, I) in enumerate(indices) # exact M values are nonnegative
+        M_vals[I] = get(solved, index,
+            max(means[index] + sampler.kappa * sds[index], 0.0))
+    end
+    return M_vals
+end
+
+end
diff --git a/ext/InfiniteDisjunctiveProgramming.jl b/ext/InfiniteDisjunctiveProgramming.jl
index 5f77dce..83aed38 100644
--- a/ext/InfiniteDisjunctiveProgramming.jl
+++ b/ext/InfiniteDisjunctiveProgramming.jl
@@ -38,6 +38,67 @@ function DP.requires_disaggregation(vref::InfiniteOpt.GeneralVariableRef)
     return !_is_parameter(vref)
 end
 
+# Bound info for parameter refs in disjunct constraints: parameter
+# functions report their support extrema, finite parameters a point,
+# other parameters (-Inf, Inf). Returns nothing for variables.
+function _parameter_bound_info(
+    vref::InfiniteOpt.GeneralVariableRef
+    )::Union{Nothing, Tuple{Float64, Float64}}
+    _is_parameter(vref) || return nothing
+    dvref = InfiniteOpt.dispatch_variable_ref(vref)
+    if dvref isa InfiniteOpt.ParameterFunctionRef
+        prefs = InfiniteOpt.parameter_list(dvref)
+        length(prefs) == 1 || return (-Inf, Inf)
+        supports = InfiniteOpt.supports(first(prefs))
+        isempty(supports) && return (-Inf, Inf)
+        func = InfiniteOpt.raw_function(dvref)
+        func_values = [func(s) for s in supports]
+        return (minimum(func_values), maximum(func_values))
+    elseif dvref isa InfiniteOpt.FiniteParameterRef
+        value = InfiniteOpt.parameter_value(dvref)
+        return (value, value)
+    end
+    return (-Inf, Inf)
+end
+
+function DP.set_variable_bound_info(
+    vref::InfiniteOpt.GeneralVariableRef, ::DP.BigM)
+    info = _parameter_bound_info(vref)
+    info === nothing || return info
+    lb = JuMP.has_lower_bound(vref) ? JuMP.lower_bound(vref) : -Inf
+    ub = JuMP.has_upper_bound(vref) ? JuMP.upper_bound(vref) : Inf
+    return lb, ub
+end
+
+# Hull and PSplit require finite bounds that include 0
+function _zero_inclusive_bounds(
+    vref::InfiniteOpt.GeneralVariableRef,
+    info::Union{Nothing, Tuple{Float64, Float64}},
+    method_name::String
+    )::Tuple{Float64, Float64}
+    info === nothing || return (min(0, info[1]), max(0, info[2]))
+    if !JuMP.has_lower_bound(vref) || !JuMP.has_upper_bound(vref)
+        error("Variable $vref must have both lower and upper " *
+              "bounds defined when using the $method_name " *
+              "reformulation.")
+    end
+    return (min(0, JuMP.lower_bound(vref)),
+            max(0, JuMP.upper_bound(vref)))
+end
+
+function DP.set_variable_bound_info(
+    vref::InfiniteOpt.GeneralVariableRef, ::DP.Hull)
+    return _zero_inclusive_bounds(vref,
+        _parameter_bound_info(vref), "Hull")
+end
+
+function DP.set_variable_bound_info(
+    vref::InfiniteOpt.GeneralVariableRef,
+    ::Union{DP.PSplit, DP._PSplit})
+    return _zero_inclusive_bounds(vref,
+        _parameter_bound_info(vref), "PSplit")
+end
+
 function DP.VariableProperties(vref::InfiniteOpt.GeneralVariableRef)
     info = DP.get_variable_info(vref)
     name = JuMP.name(vref)
@@ -263,31 +324,62 @@ function _interpolate_at(
         )
 end
 
-# Transcribe mini_expr, solve per support on the transcribed JuMP
-# model, and aggregate to a scalar if uniform, else to a parameter
-# function on main.
+# The infinite parameters of `mini_expr` and their supports, in the
+# ascending order of `parameter_refs`. Only defined when M varies over
+# the supports, so it is deferred until sample_M_values needs it.
+function _support_grids(
+    sub::DP.GDPSubmodel, mini_expr::JuMP.AbstractJuMPScalar)
+    reverse_map = Dict(ws[1] => v for (v, ws) in sub.fwd_map)
+    prefs = Tuple(get(reverse_map, p) do
+            error("MBM cannot build a support grid over `$p`, which " *
+                  "is a group of dependent infinite parameters, so M " *
+                  "must not vary over its supports.")
+        end for p in InfiniteOpt.parameter_refs(mini_expr))
+    return prefs, Tuple(InfiniteOpt.supports(p) for p in prefs)
+end
+
+# Solve the M subproblem exactly at every support
+function DP.sample_M_values(
+    sampler::Nothing,
+    objectives::AbstractArray,
+    sub::DP.GDPSubmodel,
+    method::DP._MBM,
+    support_grids::Function
+    )
+    M_vals = Array{Float64}(undef, size(objectives))
+    for I in eachindex(objectives)
+        m = DP.raw_M(sub, objectives[I], method)
+        m === nothing && return nothing
+        M_vals[I] = m
+    end
+    return M_vals
+end
+
+# Transcribe mini_expr, compute the per-support M values with the
+# method's sampler, and aggregate to a scalar if uniform, else to a
+# parameter function on main.
 function DP.raw_M(
     sub::DP.GDPSubmodel{<:InfiniteOpt.InfiniteModel},
     mini_expr::JuMP.AbstractJuMPScalar,
     method::DP._MBM
     )
     objectives = InfiniteOpt.transformation_expression(mini_expr)
-    transcribed = InfiniteOpt.transformation_model(sub.model)
-    inner_sub = DP.GDPSubmodel(transcribed,JuMP.VariableRef[],
-        Dict{JuMP.VariableRef, Vector{JuMP.VariableRef}}()
-        )
-    M_vals = Array{typeof(method.default_M)}(undef, size(objectives))
-    for I in eachindex(objectives)
-        m = DP.raw_M(inner_sub, objectives[I], method)
-        m === nothing && return nothing
-        M_vals[I] = m
+    # transcription orders the dimensions by parameter group, which is
+    # not the ascending order `parameter_refs` gives the grids below
+    group_idxs = InfiniteOpt.parameter_group_int_indices(mini_expr)
+    if length(group_idxs) > 1 && ndims(objectives) == length(group_idxs)
+        objectives = permutedims(objectives, sortperm(group_idxs))
     end
+    transcribed = InfiniteOpt.transformation_model(sub.model)
+    inner_sub = DP.GDPSubmodel(transcribed, JuMP.VariableRef[],
+        Dict{JuMP.VariableRef, Vector{JuMP.VariableRef}}())
+    M_vals = DP.sample_M_values(method.sampler, objectives,
+        inner_sub, method, () -> _support_grids(sub, mini_expr)[2])
+    M_vals === nothing && return nothing
+    M_vals isa Number && return M_vals
     all(==(first(M_vals)), M_vals) && return first(M_vals)
-    mini_prefs = InfiniteOpt.parameter_refs(mini_expr)
-    reverse_map = Dict(ws[1] => v for (v, ws) in sub.fwd_map)
-    prefs = Tuple(reverse_map[p] for p in mini_prefs)
+    prefs, grids = _support_grids(sub, mini_expr)
     main = JuMP.owner_model(first(prefs))
-    grids = Tuple(InfiniteOpt.supports(p) for p in prefs)
     param_func = InfiniteOpt.build_parameter_function(
         error, _interpolate(grids, M_vals), prefs)
     return InfiniteOpt.add_parameter_function(main, param_func)
diff --git a/src/datatypes.jl b/src/datatypes.jl
index bdcd4c2..ef82bce 100644
--- a/src/datatypes.jl
+++ b/src/datatypes.jl
@@ -368,21 +368,120 @@ struct BigM{T} <: AbstractReformulationMethod
 end
 
 """
-    MBM{O, T, L <: LogicalVariableRef} <: AbstractReformulationMethod
+    MBM{O, T} <: AbstractReformulationMethod
 
 A type for using the multiple big-M reformulation approach for disjunctive constraints.
 
 **Fields**
 - `optimizer::O`: Optimizer to use when solving mini-models (required).
 - `default_M::T`: Default big-M value to use if no big-M is specified for a logical variable (1e9).
+- `sampler::Any`: M-value sampler for infinite models (`nothing`).
+  `nothing` solves an M subproblem at every support; a
+  [`GPSampler`](@ref) solves a subset of the supports and fills the
+  rest with a conservative Gaussian-process estimate (see
+  [`sample_M_values`](@ref)). Ignored for finite models.
 """
 mutable struct MBM{O, T} <: AbstractReformulationMethod
     optimizer::O
     default_M::T
-    
+    sampler::Any
+
     # Constructor with optimizer (required) and optional default_M
-    function MBM(optimizer::O, default_M::T = 1e9) where {O, T}
-        new{O, T}(optimizer, default_M)
+    function MBM(
+        optimizer::O, default_M::T = 1e9; sampler = nothing
+        ) where {O, T}
+        new{O, T}(optimizer, default_M, sampler)
+    end
+end
+
+"""
+    GPSampler(
+        kernel = nothing;
+        kappa::Real = 2.5,
+        budget::Real = 0.25,
+        detect_uniform_M::Bool = true,
+        lengthscales = [0.05, 0.1, 0.2, 0.4, 0.8],
+        jitter::Real = 1e-8,
+        seeds = 4
+        )
+
+A Gaussian-process M sampler for the `sampler` field of [`MBM`](@ref)
+on infinite models. Instead of solving an M subproblem at every
+support, it solves the seed supports, then the supports selected by
+an upper-confidence-bound acquisition until the budget is spent, and
+fills the remaining supports with the posterior upper confidence
+bound `mean + kappa * sd`. The filled values are heuristic upper
+estimates of the exact M values, not certificates. Using it requires
+that AbstractGPs be loaded.
+
+**Arguments**
+- `kernel`: Covariance kernel for the GP fit; `nothing` (the
+  default) uses a squared exponential kernel. The lengthscale is
+  selected from `lengthscales` by marginal likelihood.
+- `kappa::Real`: Upper-confidence-bound multiplier used to select the
+  next support to solve and to fill unsolved supports (2.5).
+- `budget::Real`: Fraction of the supports to solve exactly, in
+  `(0, 1]` (0.25). The seed supports are always solved;
+  `budget = 1.0` solves every support.
+- `detect_uniform_M::Bool`: If `true` (the default), M values that
+  agree at the seed supports are taken to be uniform and used for
+  every support. Set it to `false` to always fit the GP, which
+  leaves the usual `kappa * sd` cushion on the unsolved supports at
+  the cost of the extra solves, and which requires that every
+  infinite parameter have a support grid.
+- `lengthscales`: Candidate lengthscales for the marginal-likelihood
+  kernel fit, relative to support coordinates normalized to
+  `[0, 1]` ([0.05, 0.1, 0.2, 0.4, 0.8]). Give a single candidate to
+  pin the lengthscale.
+- `jitter::Real`: Observation-noise nugget added to the GP prior
+  when fitting (1e-8).
+- `seeds`: Number of evenly spaced supports solved before the first
+  GP fit (4), or a vector of fractions in `[0, 1]` giving their
+  positions along the support grid. With `detect_uniform_M`, evenly
+  spaced seeds can read a periodic M as uniform; pass unevenly
+  spaced fractions to guard against that.
+
+**Example**
+```julia
+julia> using DisjunctiveProgramming, InfiniteOpt, AbstractGPs, HiGHS
+
+julia> method = MBM(HiGHS.Optimizer, sampler = GPSampler(kappa = 4.0))
+```
+"""
+struct GPSampler
+    # Typed Any so base needs no AbstractGPs dependency
+    kernel::Any
+    kappa::Float64
+    budget::Float64
+    detect_uniform_M::Bool
+    lengthscales::Vector{Float64}
+    jitter::Float64
+    seeds::Union{Int, Vector{Float64}}
+
+    function GPSampler(
+        kernel = nothing;
+        kappa::Real = 2.5,
+        budget::Real = 0.25,
+        detect_uniform_M::Bool = true,
+        lengthscales = [0.05, 0.1, 0.2, 0.4, 0.8],
+        jitter::Real = 1e-8,
+        seeds = 4
+        )
+        kappa >= 0 || error("`kappa` must be nonnegative.")
+        0 < budget <= 1 || error("`budget` must be in `(0, 1]`.")
+        lengthscales = collect(Float64, lengthscales)
+        (!isempty(lengthscales) && all(>(0), lengthscales)) ||
+            error("`lengthscales` must be positive and nonempty.")
+        jitter >= 0 || error("`jitter` must be nonnegative.")
+        if seeds isa Int
+            seeds >= 2 || error("`seeds` must be at least 2.")
+        else
+            seeds = collect(Float64, seeds)
+            (!isempty(seeds) && all(f -> 0 <= f <= 1, seeds)) ||
+                error("`seeds` must be fractions in `[0, 1]`.")
+        end
+        new(kernel, Float64(kappa), Float64(budget),
+            detect_uniform_M, lengthscales, Float64(jitter), seeds)
     end
 end
 
@@ -390,6 +489,7 @@ mutable struct _MBM{O, T, M <: JuMP.AbstractModel} <: AbstractReformulationMetho
     optimizer::O
     M::Dict{LogicalVariableRef{M}, Any}
     default_M::T
+    sampler::Any
     subproblem_indicators::Vector{LogicalVariableRef{M}}
     # Cached submodels: indicator => GDPSubmodel.
     # Typed Any so extensions can store different types.
@@ -400,6 +500,7 @@ mutable struct _MBM{O, T, M <: JuMP.AbstractModel} <: AbstractReformulationMetho
             method.optimizer,
             Dict{LogicalVariableRef{M}, Any}(),
             method.default_M,
+            method.sampler,
             Vector{LogicalVariableRef{M}}(),
             Dict{LogicalVariableRef{M}, Any}()
         )
diff --git a/src/extension_api.jl b/src/extension_api.jl
index ad3566f..aa1c517 100644
--- a/src/extension_api.jl
+++ b/src/extension_api.jl
@@ -38,3 +38,29 @@ Y(t, x)
 ```
 """
 function InfiniteLogical end
+
+"""
+    sample_M_values(sampler, objectives, sub, method, support_grids)
+
+Compute the MBM M values at the transcription supports of an infinite
+model. `sampler` is the `sampler` field of [`MBM`](@ref),
+`objectives` is the array of per-support objective expressions, `sub`
+is the transcribed submodel wrapped as a `GDPSubmodel`, `method` is
+the `_MBM` data, and `support_grids` is a function returning the
+support vectors of the infinite parameters. It is a function because
+the supports are only well defined once M is known to vary over them,
+so methods that return early (or never need coordinates) must not
+call it. Returns an array of M values shaped like `objectives`, a
+scalar when M is uniform across the supports, or `nothing` if an M
+subproblem is infeasible. Extensions implement methods that dispatch
+on `sampler`: `nothing` solves an M subproblem at every support,
+while a [`GPSampler`](@ref) solves a subset of the supports and
+fills the rest with a Gaussian-process upper confidence bound.
+"""
+function sample_M_values(sampler, objectives, sub, method, support_grids)
+    error("Unrecognized `sampler` value `$(repr(sampler))` for MBM " *
+          "on an infinite model. Use `nothing` to solve an M " *
+          "subproblem at every support, or a `GPSampler` (with " *
+          "AbstractGPs loaded) to estimate M values with a " *
+          "Gaussian process.")
+end
diff --git a/src/mbm.jl b/src/mbm.jl
index 23c5096..8f74269 100644
--- a/src/mbm.jl
+++ b/src/mbm.jl
@@ -87,7 +87,9 @@ function reformulate_disjunct_constraint(
         },
     method::_MBM
     )
-    ref_cons = reformulate_disjunction(model, con, MBM(method.optimizer))
+    ref_cons = reformulate_disjunction(model, con, MBM(
+        method.optimizer, method.default_M,
+        sampler = method.sampler))
     new_ref_cons = Vector{JuMP.AbstractConstraint}()
     for ref_con in ref_cons
         append!(new_ref_cons,
diff --git a/test/extensions/AbstractGPsDisjunctiveProgramming.jl b/test/extensions/AbstractGPsDisjunctiveProgramming.jl
new file mode 100644
index 0000000..2f0e818
--- /dev/null
+++ b/test/extensions/AbstractGPsDisjunctiveProgramming.jl
@@ -0,0 +1,248 @@
+using InfiniteOpt, HiGHS, AbstractGPs
+import DisjunctiveProgramming as DP
+
+function test_gp_sampler_kwargs()
+    @test MBM(HiGHS.Optimizer).sampler === nothing
+    sampler = GPSampler()
+    @test sampler.kernel === nothing
+    @test sampler.kappa == 2.5
+    @test sampler.budget == 0.25
+    @test sampler.detect_uniform_M
+    @test sampler.lengthscales == [0.05, 0.1, 0.2, 0.4, 0.8]
+    @test sampler.jitter == 1e-8
+    @test sampler.seeds == 4
+    kern = SqExponentialKernel()
+    sampler = GPSampler(kern, kappa = 4.0, budget = 0.1,
+        detect_uniform_M = false, lengthscales = (0.1, 0.3),
+        jitter = 1e-6, seeds = [0.0, 0.3, 1.0])
+    @test sampler.kernel === kern
+    @test sampler.kappa == 4.0
+    @test sampler.budget == 0.1
+    @test !sampler.detect_uniform_M
+    @test sampler.lengthscales == [0.1, 0.3]
+    @test sampler.jitter == 1e-6
+    @test sampler.seeds == [0.0, 0.3, 1.0]
+    @test GPSampler(seeds = 6).seeds == 6
+    @test MBM(HiGHS.Optimizer, sampler = sampler).sampler === sampler
+    @test_throws ErrorException GPSampler(kappa = -1)
+    @test_throws ErrorException GPSampler(budget = 0)
+    @test_throws ErrorException GPSampler(budget = 1.5)
+    @test_throws ErrorException GPSampler(lengthscales = Float64[])
+    @test_throws ErrorException GPSampler(lengthscales = [-0.1])
+    @test_throws ErrorException GPSampler(jitter = -1)
+    @test_throws ErrorException GPSampler(seeds = 1)
+    @test_throws ErrorException GPSampler(seeds = [1.5])
+    @test_throws ErrorException GPSampler(seeds = Float64[])
+end
+
+# Mirror of test_raw_M_infinite_scalar: uniform seed M values collapse
+# to the exactly-solved scalar under the GP sampler
+function test_gp_raw_M_scalar()
+    model = InfiniteGDPModel()
+    @infinite_parameter(model, t ∈ [0, 1], num_supports = 5)
+    @variable(model, 0 <= x <= 10, Infinite(t))
+    @variable(model, Y[1:2], InfiniteLogical(t))
+    @constraint(model, con, x >= 5, Disjunct(Y[1]))
+    @constraint(model, con2, x <= 3, Disjunct(Y[2]))
+    @disjunction(model, Y)
+    mbm = DP._MBM(
+        MBM(HiGHS.Optimizer, sampler = GPSampler()), model)
+    sub = DP.copy_model_with_constraints(
+        model, DP.DisjunctConstraintRef[con2], mbm)
+    obj = DP.prepare_max_M_objective(
+        model, JuMP.constraint_object(con), sub)
+    @test DP.raw_M(sub, obj, mbm) == 5.0
+end
+
+# With budget = 1.0 every support is solved exactly, so the GP
+# sampler must reproduce the exact grid parameter function
+function test_gp_raw_M_matches_exact()
+    function pfunc_values(sampler, supports)
+        model = InfiniteGDPModel()
+        @infinite_parameter(model, t ∈ [0, 1], supports = supports)
+        @variable(model, 0 <= x <= 10, Infinite(t))
+        @variable(model, Y[1:2], InfiniteLogical(t))
+        @parameter_function(model, f == t -> 2*t)
+        @constraint(model, con, x <= f, Disjunct(Y[1]))
+        @constraint(model, con2, x >= 0.5, Disjunct(Y[2]))
+        @disjunction(model, Y)
+        mbm = DP._MBM(
+            MBM(HiGHS.Optimizer, sampler = sampler), model)
+        sub = DP.copy_model_with_constraints(
+            model, DP.DisjunctConstraintRef[con2], mbm)
+        obj = DP.prepare_max_M_objective(
+            model, JuMP.constraint_object(con), sub)
+        M = DP.raw_M(sub, obj, mbm)
+        @test M isa InfiniteOpt.GeneralVariableRef
+        return [InfiniteOpt.raw_function(M)(t_val) for t_val in supports]
+    end
+    supports = [0.0, 0.25, 0.5, 0.75, 1.0]
+    exact_vals = pfunc_values(nothing, supports)
+    # default kernel, user kernel, and pinned lengthscale all solve
+    # the same supports here, so the M values match exactly
+    @test pfunc_values(GPSampler(budget = 1.0), supports) ==
+        exact_vals
+    @test pfunc_values(
+        GPSampler(SqExponentialKernel(), budget = 1.0), supports) ==
+        exact_vals
+    @test pfunc_values(GPSampler(SqExponentialKernel(),
+        lengthscales = [0.2], budget = 1.0), supports) == exact_vals
+end
+
+# an empty disjunct region makes the M subproblems infeasible; both
+# samplers propagate that up to the reformulation error
+function test_gp_infeasible_disjunct()
+    function build()
+        model = InfiniteGDPModel(HiGHS.Optimizer)
+        set_silent(model)
+        @infinite_parameter(model, t ∈ [0, 1], num_supports = 5)
+        @variable(model, 0 <= x <= 10, Infinite(t))
+        @variable(model, Y[1:2], InfiniteLogical(t))
+        @parameter_function(model, f == t -> 2*t)
+        @constraint(model, x <= f, Disjunct(Y[1]))
+        @constraint(model, x >= 8, Disjunct(Y[2]))
+        @constraint(model, x <= 3, Disjunct(Y[2]))
+        @disjunction(model, Y)
+        @objective(model, Max, 𝔼(x, t))
+        return model
+    end
+    for sampler in (nothing, GPSampler())
+        model = build()
+        @test_throws ErrorException optimize!(model,
+            gdp_method = MBM(HiGHS.Optimizer, sampler = sampler))
+    end
+end
+
+# optimum (10) needs M(t) >= 10 - 2t pointwise; the GP fill is heuristic
+function test_gp_mbm_solve_equivalence()
+    function solve_with(sampler)
+        model = InfiniteGDPModel(HiGHS.Optimizer)
+        set_silent(model)
+        @infinite_parameter(model, t ∈ [0, 1], num_supports = 20)
+        @variable(model, 0 <= x <= 10, Infinite(t))
+        @variable(model, Y[1:2], InfiniteLogical(t))
+        @parameter_function(model, f == t -> 2*t)
+        @constraint(model, x <= f, Disjunct(Y[1]))
+        @constraint(model, x >= 0.5, Disjunct(Y[2]))
+        @disjunction(model, Y)
+        @objective(model, Max, 𝔼(x, t))
+        optimize!(model,
+            gdp_method = MBM(HiGHS.Optimizer, sampler = sampler))
+        @test termination_status(model) == MOI.OPTIMAL
+        return objective_value(model)
+    end
+    obj_exact = solve_with(nothing)
+    obj_gp = solve_with(GPSampler())
+    obj_tuned = solve_with(GPSampler(kappa = 4.0, budget = 0.2))
+    @test obj_exact ≈ 10.0 atol = 1e-4
+    # over-M can't raise the optimum, under-M can only shave it a bit
+    @test obj_gp <= obj_exact + 1e-6
+    @test obj_gp ≈ obj_exact atol = 1e-2
+    @test obj_tuned <= obj_exact + 1e-6
+    @test obj_tuned ≈ obj_exact atol = 1e-2
+end
+
+# Seed placement vs a periodic M. With f(t) = 2|cos(2*pi*t)| on these
+# supports, M = 10 - f is 8 at supports 1, 3, 5 and 10 at supports
+# 2, 4. Seeds that only hit the M = 8 supports alias the periodic M
+# to uniform 8, which caps x at 8 and cuts the optimum from 10 down
+# to 9; the default and denser seed grids see both values and stay
+# exact.
+function test_gp_periodic_M_seeds()
+    supports = [0.0, 0.25, 0.5, 0.75, 1.0]
+    function solve_with(sampler)
+        model = InfiniteGDPModel(HiGHS.Optimizer)
+        set_silent(model)
+        @infinite_parameter(model, t ∈ [0, 1], supports = supports)
+        @variable(model, 0 <= x <= 10, Infinite(t))
+        @variable(model, Y[1:2], InfiniteLogical(t))
+        @parameter_function(model, f == t -> 2 * abs(cos(2 * pi * t)))
+        @constraint(model, x <= f, Disjunct(Y[1]))
+        @constraint(model, x >= 0.5, Disjunct(Y[2]))
+        @disjunction(model, Y)
+        @objective(model, Max, 𝔼(x, t))
+        optimize!(model,
+            gdp_method = MBM(HiGHS.Optimizer, sampler = sampler))
+        return objective_value(model)
+    end
+    @test solve_with(nothing) ≈ 10.0 atol = 1e-6
+    @test solve_with(GPSampler()) ≈ 10.0 atol = 1e-6
+    @test solve_with(GPSampler(seeds = 5)) ≈ 10.0 atol = 1e-6
+    @test solve_with(GPSampler(seeds = [0.0, 0.5, 1.0])) ≈
+        9.0 atol = 1e-6
+end
+
+# With detection off the uniform M is not collapsed to a scalar: the
+# GP is fit and the unsolved supports keep their kappa * sd cushion,
+# which must sit above the M that detection would have returned.
+function test_gp_detect_uniform_M_off()
+    function raw_M_with(detect)
+        model = InfiniteGDPModel()
+        @infinite_parameter(model, t ∈ [0, 1], num_supports = 20)
+        @variable(model, 0 <= x <= 10, Infinite(t))
+        @variable(model, Y[1:2], InfiniteLogical(t))
+        @constraint(model, con, x >= 5, Disjunct(Y[1]))
+        @constraint(model, con2, x <= 3, Disjunct(Y[2]))
+        @disjunction(model, Y)
+        mbm = DP._MBM(MBM(HiGHS.Optimizer,
+            sampler = GPSampler(detect_uniform_M = detect)), model)
+        sub = DP.copy_model_with_constraints(
+            model, DP.DisjunctConstraintRef[con2], mbm)
+        obj = DP.prepare_max_M_objective(
+            model, JuMP.constraint_object(con), sub)
+        return DP.raw_M(sub, obj, mbm)
+    end
+    @test raw_M_with(true) == 5.0
+    M = raw_M_with(false)
+    @test M isa InfiniteOpt.GeneralVariableRef
+    raw_fn = InfiniteOpt.raw_function(M)
+    vals = [raw_fn(t) for t in range(0, 1, length = 20)]
+    @test all(vals .>= 5.0 - 1e-6)
+    @test maximum(vals) > 5.0
+end
+
+# Dependent parameters have no support grid, so turning detection off
+# leaves the GP with nothing to fit over
+function test_gp_detect_uniform_M_off_dependent()
+    model = InfiniteGDPModel()
+    @infinite_parameter(model, ξ[1:2] ∈ [0, 1], num_supports = 4)
+    @variable(model, 0 <= x <= 10, Infinite(ξ))
+    @variable(model, Y[1:2], InfiniteLogical(ξ))
+    @constraint(model, con, x >= 5, Disjunct(Y[1]))
+    @constraint(model, con2, x <= 3, Disjunct(Y[2]))
+    @disjunction(model, Y)
+    mbm = DP._MBM(MBM(HiGHS.Optimizer,
+        sampler = GPSampler(detect_uniform_M = false)), model)
+    sub = DP.copy_model_with_constraints(
+        model, DP.DisjunctConstraintRef[con2], mbm)
+    obj = DP.prepare_max_M_objective(
+        model, JuMP.constraint_object(con), sub)
+    @test_throws ErrorException DP.raw_M(sub, obj, mbm)
+end
+
+function test_gp_unknown_sampler_error()
+    model = InfiniteGDPModel(HiGHS.Optimizer)
+    set_silent(model)
+    @infinite_parameter(model, t ∈ [0, 1], num_supports = 5)
+    @variable(model, 0 <= x <= 10, Infinite(t))
+    @variable(model, Y[1:2], InfiniteLogical(t))
+    @parameter_function(model, f == t -> 2*t)
+    @constraint(model, x <= f, Disjunct(Y[1]))
+    @constraint(model, x >= 0.5, Disjunct(Y[2]))
+    @disjunction(model, Y)
+    @objective(model, Max, 𝔼(x, t))
+    @test_throws ErrorException optimize!(model,
+        gdp_method = MBM(HiGHS.Optimizer, sampler = :grid))
+end
+
+@testset "AbstractGPsDisjunctiveProgramming" begin
+    test_gp_sampler_kwargs()
+    test_gp_raw_M_scalar()
+    test_gp_raw_M_matches_exact()
+    test_gp_mbm_solve_equivalence()
+    test_gp_periodic_M_seeds()
+    test_gp_detect_uniform_M_off()
+    test_gp_detect_uniform_M_off_dependent()
+    test_gp_unknown_sampler_error()
+    test_gp_infeasible_disjunct()
+end
diff --git a/test/extensions/InfiniteDisjunctiveProgramming.jl b/test/extensions/InfiniteDisjunctiveProgramming.jl
index 47bb884..6e96d8c 100644
--- a/test/extensions/InfiniteDisjunctiveProgramming.jl
+++ b/test/extensions/InfiniteDisjunctiveProgramming.jl
@@ -121,6 +121,36 @@ function test_requires_disaggregation()
     @test DP.requires_disaggregation(y) == true
 end
 
+# Bound info for parameter refs in disjunct constraints: parameter
+# functions report their support extrema, finite parameters a point,
+# other parameters (-Inf, Inf); Hull/PSplit clamp the bounds to
+# include 0 and error when a variable is missing bounds.
+function test_parameter_bound_info()
+    model = InfiniteGDPModel()
+    @infinite_parameter(model, t in [0, 1], supports = [0.0, 0.5, 1.0])
+    @infinite_parameter(model, s in [0, 1])
+    @finite_parameter(model, p == 2.0)
+    @variable(model, 0 <= x <= 10, Infinite(t))
+    @variable(model, y, Infinite(t))
+    @parameter_function(model, pf == t -> 2t - 1)
+    @parameter_function(model, pf2 == (t, s) -> t + s)
+    @parameter_function(model, pf3 == s -> s)
+    @test DP.set_variable_bound_info(pf, BigM()) == (-1.0, 1.0)
+    @test DP.set_variable_bound_info(p, BigM()) == (2.0, 2.0)
+    @test DP.set_variable_bound_info(t, BigM()) == (-Inf, Inf)
+    @test DP.set_variable_bound_info(x, BigM()) == (0.0, 10.0)
+    @test DP.set_variable_bound_info(y, BigM()) == (-Inf, Inf)
+    # multi-parameter and support-less parameter functions fall back
+    @test DP.set_variable_bound_info(pf2, BigM()) == (-Inf, Inf)
+    @test DP.set_variable_bound_info(pf3, BigM()) == (-Inf, Inf)
+    # Hull and PSplit clamp the bounds to include 0
+    @test DP.set_variable_bound_info(pf, Hull()) == (-1.0, 1.0)
+    @test DP.set_variable_bound_info(p, Hull()) == (0.0, 2.0)
+    @test DP.set_variable_bound_info(x, Hull()) == (0.0, 10.0)
+    @test DP.set_variable_bound_info(p, PSplit([[x]])) == (0.0, 2.0)
+    @test_throws ErrorException DP.set_variable_bound_info(y, Hull())
+end
+
 function test_all_variables_infiniteopt()
     model = InfiniteGDPModel()
     @infinite_parameter(model, t ∈ [0, 1])
@@ -413,6 +443,53 @@ function test_raw_M_infinite_param_function()
     end
 end
 
+# raw_M over two infinite parameters with different support counts.
+# Transcription orders the objective dimensions by parameter group,
+# which need not be the ascending order of the grids, so the M values
+# must be permuted to line up. Setup: x(t, s) in [0, 10],
+# disj1: x <= t + s, disj2: x >= 0.5. Slack r(x) = x - t - s
+# maximized over x in [0.5, 10]: 10 - t - s.
+function test_raw_M_infinite_two_params()
+    model = InfiniteGDPModel()
+    @infinite_parameter(model, t ∈ [0, 1], supports = [0.0, 0.5, 1.0])
+    @infinite_parameter(model, s ∈ [0, 1], supports = [0.0, 1.0])
+    @variable(model, 0 <= x <= 10, Infinite(t, s))
+    @variable(model, Y[1:2], InfiniteLogical(t, s))
+    @constraint(model, con, x <= t + s, Disjunct(Y[1]))
+    @constraint(model, con2, x >= 0.5, Disjunct(Y[2]))
+    @disjunction(model, Y)
+    mbm = DP._MBM(MBM(HiGHS.Optimizer), model)
+    sub = DP.copy_model_with_constraints(
+        model, DP.DisjunctConstraintRef[con2], mbm)
+    obj = DP.prepare_max_M_objective(
+        model, JuMP.constraint_object(con), sub)
+    M = DP.raw_M(sub, obj, mbm)
+    @test M isa InfiniteOpt.GeneralVariableRef
+    raw_fn = InfiniteOpt.raw_function(M)
+    for t_val in [0.0, 0.5, 1.0], s_val in [0.0, 1.0]
+        @test raw_fn(t_val, s_val) >= 10.0 - t_val - s_val - 1e-6
+    end
+end
+
+# Dependent parameters have no per-parameter support grid, so raw_M
+# must not need one when M does not vary. Setup as in
+# test_raw_M_infinite_scalar, over a dependent parameter array.
+function test_raw_M_infinite_dependent_params()
+    model = InfiniteGDPModel()
+    @infinite_parameter(model, ξ[1:2] ∈ [0, 1], num_supports = 4)
+    @variable(model, 0 <= x <= 10, Infinite(ξ))
+    @variable(model, Y[1:2], InfiniteLogical(ξ))
+    @constraint(model, con, x >= 5, Disjunct(Y[1]))
+    @constraint(model, con2, x <= 3, Disjunct(Y[2]))
+    @disjunction(model, Y)
+    mbm = DP._MBM(MBM(HiGHS.Optimizer), model)
+    sub = DP.copy_model_with_constraints(
+        model, DP.DisjunctConstraintRef[con2], mbm)
+    obj = DP.prepare_max_M_objective(
+        model, JuMP.constraint_object(con), sub)
+    @test DP.raw_M(sub, obj, mbm) == 5.0
+end
+
 # Piecewise-constant max-of-corners: returns the maximum value over
 # the 2^n corners of the cell containing the query.
 function test_interpolate()
@@ -806,6 +883,7 @@ end
         test_is_parameter()
         test_is_parameter_concrete_dispatches()
         test_requires_disaggregation()
+        test_parameter_bound_info()
         test_variable_properties_infiniteopt()
         test_variable_properties_from_expr()
         test_variable_properties_from_quad_expr()
@@ -834,6 +912,8 @@ end
         test_interpolate()
         test_raw_M_infinite_scalar()
         test_raw_M_infinite_param_function()
+        test_raw_M_infinite_two_params()
+        test_raw_M_infinite_dependent_params()
         test_mbm_finite_and_integer_var()
         test_mbm_infinite_simple()
         test_mbm_infinite_param_dependent()
diff --git a/test/runtests.jl b/test/runtests.jl
index 06e8813..8034f6e 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -24,4 +24,5 @@ include("constraints/disjunction.jl")
 include("print.jl")
 include("solve.jl")
 include("extensions/InfiniteDisjunctiveProgramming.jl")
+include("extensions/AbstractGPsDisjunctiveProgramming.jl")