Generating Samples from your Hakaru Program

The Hakaru language is designed so that it is easy to express probabilistic models programmatically. In particular, Hakaru makes makes it easy to use Monte Carlo methods, which aim to generate individual samples and estimate expectation functions, or expectation, from a given distribution. The first task, drawing samples from a distribution, is often difficult because it might not be possible to sample from the target distribution directly. This can be exasperated as the dimensionality of the sample space increases. In these scenarios, a comparable distribution can be selected to draw samples from. Importance sampling is a Monte Carlo method that is used to generate samples by estimating an expectation for the target distribution instead. The estimated expectation is then used to generate samples. To account for the knowledge that the samples were not generated from the target distribution, a weight is assigned so that each sample’s contribution to the estimator is adjusted according to its relevance. However, this method only works well if the distribution proposed by the expectation is similar to the target distribution. For more complex distributions, a different approach, such as the Metropolis Hastings method should be used¹.

The hakaru command is used to indefinitely generate samples from a Hakaru program using importance sampling. Each sample is assigned a weight, and a sample’s weight is initialized to 1.0. Weights are changed by Hakaru primitives and processes such as weight.

Usage

The hakaru command can take up to two Hakaru programs as arguments. If only one program is provided, the hakaru command generates samples based on the model described in the Hakaru program. In this case, the hakaru command can be invoked in the command-line by calling:

hakaru hakaru_program.hk

If a second program is given to the hakaru command, it will treat the two programs as the start of a Markov Chain. This is used when you have created a transition kernel using the Metropolis Hastings transformation. To invoke the hakaru command with a transition kernel, you would call:

hakaru --transition-kernel transition.hk init.hk

The first program, transition.hk, is treated as the transition kernel and the second program, init.hk, is treated as the initial state of the Markov Chain. When the hakaru command is run, a sample is drawn from init.hk. This sample is then passed to transition.hk to generate the second sample. After this point, samples generated from transition.hk are passed back into itself to generate further samples.

The Dash (-) Operator

You might encounter some scenarios where you wish to run a Hakaru command or transformation on a program and then send the resulting output to another command or transform. In these cases, you can take advantage of the dash (-) command-line notation.

The dash notation is a shortcut used to pass standard inputs and outputs to another command in the same line of script. For example, if you wanted to run the disintegrate Hakaru command followed by the hk-maple -c Simplify command, you would enter:

disintegrate program.hk | hk-maple -c Simplify -

This command is equivalent to entering:

disintegrate program.hk > temp.hk
hk-maple -c Simplify temp.hk

Note: The > operator redirects the output from disintegrate program.hk to a new file called temp.hk.

Example

To demonstrate weights in Hakaru, a sample problem of a burglary alarm is adapted from Pearl’s textbook on probabilistic reasoning (page 35)²:

Imagine being awakened one night by the shrill sound of your burglar alarm. What is your degree of belief that a burglary attempt has taken place? For illustrative purposes we make the following judgements: (a) There is a 95% chance that an attempted burglary will trigger the alarm system – P(Alarm|Burglary) = 0.95; (b) based on previous false alarms, there is a slight (1 percent) chance that the alarm will be triggered by a mechanism other than an attempted burglary – P(Alarm|No Burglary) = 0.01; (c) previous crime patterns indicate that there is a one in ten thousand chance that a given house will be burglarized on a given night – P(Burglary) = 10^-4.

This can be modelled in Hakaru by the program:

burglary <~ categorical([0.0001, 0.9999])
weight([0.95, 0.01][burglary],
return [true,false][burglary])

If you save this program as weight_burglary.hk, you can generate samples from it by calling:

$ hakaru weight_burglary.hk
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
1.0000000000000004e-2   false
...

The hakaru command will print a continuous stream of samples drawn from this program. In this example, true and false samples have different weights. This will not be immediately apparent if you manually sift through the samples. If you wanted to see the ratio of weights for a series of samples, you can use an awk script that tallies the weights for a limited set of samples:

$ hakaru weight_burglary.hk | head -n 100000 | awk '{a[$2]+=$1}END{for (i in a) print i, a[i]}'
false 999.87
true 12.35

If you were only interested in counting how many times the alarm was triggered correctly and erroneously, modify the awk script to be a counter instead:

$ hakaru weight_burglary.hk | head -n 100000 | awk '{a[$2]+=1}END{for (i in a) print i, a[i]}'
false 99987
true 13

In this case, the printing of sample weights might not be important. To suppress the printing of weights during sample generation, you can use the --no-weights or -w option:

$ hakaru --no-weights weight_burglary.hk
false
false
false
false
false
false
false
false
...

Hakaru can sample from more complex distributions using the Metropolis Hastings transform. The hakaru command can then be invoked using a transition kernel. For an example of the hakaru command usage in this context, refer to the Metropolis Hastings transform page.