# hmcSampler

Hamiltonian Monte Carlo (HMC) sampler

## Syntax

``hmc = hmcSampler(logpdf,startpoint)``
``hmc = hmcSampler(___,Name,Value)``

## Description

example

````hmc = hmcSampler(logpdf,startpoint)` creates a Hamiltonian Monte Carlo (HMC) sampler, returned as a `HamiltonianSampler` object. `logpdf` is a function handle that evaluates the logarithm of the probability density of the equilibrium distribution and its gradient. The column vector `startpoint` is the initial point from which to start HMC sampling.After you create the sampler, you can compute MAP (maximum-a-posteriori) point estimates, tune the sampler, draw samples, and check convergence diagnostics using the methods of the `HamiltonianSampler` class. For an example of this workflow, see Bayesian Linear Regression Using Hamiltonian Monte Carlo.```
````hmc = hmcSampler(___,Name,Value)` specifies additional options using one or more name-value pair arguments. Specify name-value pair arguments after all other input arguments.```

## Examples

collapse all

Create a Hamiltonian Monte Carlo (HMC) sampler to sample from a normal distribution.

First, save a function `normalDistGrad` on the MATLAB® path that returns the multivariate normal log probability density and its gradient (`normalDistGrad` is defined at the end of this example). Then, call the function with arguments to define the `logpdf` input argument to the `hmcSampler` function.

```means = [1;-3]; standevs = [1;2]; logpdf = @(theta)normalDistGrad(theta,means,standevs);```

Choose a starting point for the HMC sampler.

`startpoint = randn(2,1);`

Create the HMC sampler and display its properties.

`smp = hmcSampler(logpdf,startpoint);`
`smp`
```smp = HamiltonianSampler with properties: StepSize: 0.1000 NumSteps: 50 MassVector: [2x1 double] JitterMethod: 'jitter-both' StepSizeTuningMethod: 'dual-averaging' MassVectorTuningMethod: 'iterative-sampling' LogPDF: @(theta)normalDistGrad(theta,means,standevs) VariableNames: {2x1 cell} StartPoint: [2x1 double] ```

The `normalDistGrad` function returns the logarithm of the multivariate normal probability density with means in `Mu` and standard deviations in `Sigma`, specified as scalars or columns vectors the same length as `startpoint`. The second output argument is the corresponding gradient.

```function [lpdf,glpdf] = normalDistGrad(X,Mu,Sigma) Z = (X - Mu)./Sigma; lpdf = sum(-log(Sigma) - .5*log(2*pi) - .5*(Z.^2)); glpdf = -Z./Sigma; end```

## Input Arguments

collapse all

Logarithm of target density and its gradient, specified as a function handle.

`logpdf` must return two output arguments: ```[lpdf,glpdf] = logpdf(X)```. Here, `lpdf` is the base-e log probability density (up to an additive constant), `glpdf` is the gradient of the log density, and the point `X` is a column vector with the same number of elements as `startpoint`.

The input argument `X` to `logpdf` must be unconstrained, meaning that every element of `X` can be any real number. Transform any constrained sampling parameters into unconstrained variables before using the HMC sampler.

If the `'UseNumericalGradient'` value is set to `true`, then `logpdf` does not need to return the gradient as the second output. Using a numerical gradient can be easier since `logpdf` does not need to compute the gradient, but it can make sampling slower.

Data Types: `function_handle`

Initial point to start sampling from, specified as a numeric column vector.

Data Types: `single` | `double`

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `'VariableNames',{'Intercept','Beta'},'MassVectorTuningMethod','hessian'` specifies sampling variable names and the mass vector tuning method to be `'hessian'`.

Step size of Hamiltonian dynamics, specified as the comma-separated pair consisting of `'StepSize'` and a positive scalar.

To propose a new state for the Markov chain, the HMC sampler integrates the Hamiltonian dynamics using leapfrog integration. This argument controls the step size of that leapfrog integration.

You can automatically tune the step size using `tuneSampler`.

Example: `'StepSize',0.2`

Data Types: `single` | `double`

Number of steps of Hamiltonian dynamics, specified as the comma-separated pair consisting of `'NumSteps'` and a positive integer.

To propose a new state for the Markov chain, the HMC sampler integrates the Hamiltonian dynamics using leapfrog integration. This argument controls the number of steps of that leapfrog integration.

You can automatically tune the number of steps using `tuneSampler`.

Example: `'NumSteps',20`

Data Types: `single` | `double`

Mass vector of momentum variables, specified as the comma-separated pair consisting of `'MassVector'` and a numeric column vector with positive values and the same length as `startpoint`.

The “masses” of the momentum variables associated with the variables of interest control the Hamiltonian dynamics in each Markov chain proposal.

You can automatically tune the mass vector using `tuneSampler`.

Example: `'MassVector',rand(3,1)`

Data Types: `single` | `double`

Method for jittering the step size and the number of steps, specified as the comma-separated pair consisting of `'JitterMethod'` and one of the following values.

ValueDescription
`'jitter-both'`

Randomly jitter the step size and number of steps for each leapfrog trajectory.

`'jitter-numsteps'`

Jitter only the number of steps of each leapfrog trajectory.

`'none'`

Perform no jittering.

With jittering, the sampler randomly selects the step size or the number of steps of each leapfrog trajectory as values smaller than the `'StepSize'` and `'NumSteps'` values. Use jittering to improve the stability of the leapfrog integration of the Hamiltonian dynamics.

Example: `'JitterMethod','jitter-both'`

Method for tuning the sampler step size, specified as the comma-separated pair consisting of `'StepSizeTuningMethod'` and `'dual-averaging'` or `'none'`.

If the `'StepSizeTuningMethod'` value is set to `'dual-averaging'`, then `tuneSampler` tunes the leapfrog step size of the HMC sampler to achieve a certain acceptance ratio for a fixed value of the simulation length. The simulation length equals the step size multiplied by the number of steps. To set the target acceptance ratio, use the `'TargetAcceptanceRatio'` name-value pair argument of the `tuneSampler` method.

Example: `'StepSizeTuningMethod','none'`

Method for tuning the sampler mass vector, specified as the comma-separated pair consisting of `'MassVectorTuningMethod'` and one of the following values.

ValueDescription
`'iterative-sampling'`

Tune the `MassVector` via successive approximations by drawing samples using a sequence of mass vector estimates.

`'hessian'`

Set the `MassVector` equal to the negative diagonal Hessian of the `logpdf` at the `startpoint`.

`'none'`

Perform no tuning of the `MassVector`.

To perform the tuning, use the `tuneSampler` method.

Example: `'MassVectorTuningMethod','hessian'`

Flag for checking the analytical gradient, specified as the comma-separated pair consisting of `'CheckGradient'` and either `true` (or `1`) or `false` (or `0`).

If `'CheckGradient'` is `true`, then the sampler calculates the numerical gradient at the `startpoint` and compares it to the analytical gradient returned by `logpdf`.

Example: `'CheckGradient',true`

Sampling variable names, specified as the comma-separated pair consisting of `'VariableNames'` and a string array or a cell array of character vectors. Elements of the array must be unique. The length of the array must be the same as the length of `startpoint`.

Supply a `'VariableNames'` value to label the components of the vector you want to sample using the HMC sampler.

Example: `'VariableNames',{'Intercept','Beta'}`

Data Types: `string` | `cell`

Flag for using numerical gradient, specified as the comma-separated pair consisting of `'UseNumericalGradient'` and either `true` (or `1`) or `false` (or `0`).

If you set the `'UseNumericalGradient'` value to `true`, then the HMC sampler numerically estimates the gradient from the log density returned by `logpdf`. In this case, the `logpdf` function does not need to return the gradient of the log density as the second output. Using a numerical gradient makes HMC sampling slower.

Example: `'UseNumericalGradient',true`

## Output Arguments

collapse all

Hamiltonian Monte Carlo sampler, returned as a `HamiltonianSampler` object.

## Version History

Introduced in R2017a