# Importance Sampling

Suppose we have an random variable with pdf and we wish to find its second moment numerically. However, the random-fu package does not support sampling from such as distribution. We notice that

So we can sample from and evaluate

```
> {-# OPTIONS_GHC -Wall #-}
> {-# OPTIONS_GHC -fno-warn-name-shadowing #-}
> {-# OPTIONS_GHC -fno-warn-type-defaults #-}
> {-# OPTIONS_GHC -fno-warn-unused-do-bind #-}
> {-# OPTIONS_GHC -fno-warn-missing-methods #-}
> {-# OPTIONS_GHC -fno-warn-orphans #-}
```

```
> module Importance where
```

```
> import Control.Monad
> import Data.Random.Source.PureMT
> import Data.Random
> import Data.Random.Distribution.Binomial
> import Data.Random.Distribution.Beta
> import Control.Monad.State
> import qualified Control.Monad.Writer as W
```

```
> sampleImportance :: RVarT (W.Writer [Double]) ()
> sampleImportance = do
> x <- rvarT $ Normal 0.0 2.0
> let x2 = x^2
> u = x2 * 0.5 * exp (-(abs x))
> v = (exp ((-x2)/8)) * (recip (sqrt (8*pi)))
> w = u / v
> lift $ W.tell [w]
> return ()
```

```
> runImportance :: Int -> [Double]
> runImportance n =
> snd $
> W.runWriter $
> evalStateT (sample (replicateM n sampleImportance))
> (pureMT 2)
```

We can run this 10,000 times to get an estimate.

```
ghci> import Formatting
ghci> format (fixed 2) (sum (runImportance 10000) / 10000)
"2.03"
```

Since we know that the -th moment of the exponential distribution is where is the rate (1 in this example), the exact answer is 2 which is not too far from our estimate using importance sampling.

The value of

is called the weight, is the pdf from which we wish to sample and is the pdf of the importance distribution.

# Importance Sampling Approximation of the Posterior

Suppose that the posterior distribution of a model in which we are interested has a complicated functional form and that we therefore wish to approximate it in some way. First assume that we wish to calculate the expectation of some arbitrary function of the parameters.

Using Bayes

where is some normalizing constant.

As before we can re-write this using a proposal distribution

We can now sample repeatedly to obtain

where the weights are defined as before by

We follow Alex Cook and use the example from (Rerks-Ngarm et al. 2009). We take the prior as and use as the proposal distribution. In this case the proposal and the prior are identical just expressed differently and therefore cancel.

Note that we use the log of the pdf in our calculations otherwise we suffer from (silent) underflow, e.g.,

```
ghci> pdf (Binomial nv (0.4 :: Double)) xv
0.0
```

On the other hand if we use the log pdf form

```
ghci> logPdf (Binomial nv (0.4 :: Double)) xv
-3900.8941170876574
```

```
> xv, nv :: Int
> xv = 51
> nv = 8197
```

```
> sampleUniform :: RVarT (W.Writer [Double]) ()
> sampleUniform = do
> x <- rvarT StdUniform
> lift $ W.tell [x]
> return ()
```

```
> runSampler :: RVarT (W.Writer [Double]) () ->
> Int -> Int -> [Double]
> runSampler sampler seed n =
> snd $
> W.runWriter $
> evalStateT (sample (replicateM n sampler))
> (pureMT (fromIntegral seed))
```

```
> sampleSize :: Int
> sampleSize = 1000
```

```
> pv :: [Double]
> pv = runSampler sampleUniform 2 sampleSize
```

```
> logWeightsRaw :: [Double]
> logWeightsRaw = map (\p -> logPdf (Beta 1.0 1.0) p +
> logPdf (Binomial nv p) xv -
> logPdf StdUniform p) pv
```

```
> logWeightsMax :: Double
> logWeightsMax = maximum logWeightsRaw
>
> weightsRaw :: [Double]
> weightsRaw = map (\w -> exp (w - logWeightsMax)) logWeightsRaw
```

```
> weightsSum :: Double
> weightsSum = sum weightsRaw
```

```
> weights :: [Double]
> weights = map (/ weightsSum) weightsRaw
```

```
> meanPv :: Double
> meanPv = sum $ zipWith (*) pv weights
>
> meanPv2 :: Double
> meanPv2 = sum $ zipWith (\p w -> p * p * w) pv weights
>
> varPv :: Double
> varPv = meanPv2 - meanPv * meanPv
```

We get the answer

```
ghci> meanPv
6.400869727227364e-3
```

But if we look at the size of the weights and the effective sample size

```
ghci> length $ filter (>= 1e-6) weights
9
ghci> (sum weights)^2 / (sum $ map (^2) weights)
4.581078458313967
```

so we may not be getting a very good estimate. Let’s try

```
> sampleNormal :: RVarT (W.Writer [Double]) ()
> sampleNormal = do
> x <- rvarT $ Normal meanPv (sqrt varPv)
> lift $ W.tell [x]
> return ()
```

```
> pvC :: [Double]
> pvC = runSampler sampleNormal 3 sampleSize
```

```
> logWeightsRawC :: [Double]
> logWeightsRawC = map (\p -> logPdf (Beta 1.0 1.0) p +
> logPdf (Binomial nv p) xv -
> logPdf (Normal meanPv (sqrt varPv)) p) pvC
```

```
> logWeightsMaxC :: Double
> logWeightsMaxC = maximum logWeightsRawC
>
> weightsRawC :: [Double]
> weightsRawC = map (\w -> exp (w - logWeightsMaxC)) logWeightsRawC
```

```
> weightsSumC :: Double
> weightsSumC = sum weightsRawC
```

```
> weightsC :: [Double]
> weightsC = map (/ weightsSumC) weightsRawC
```

```
> meanPvC :: Double
> meanPvC = sum $ zipWith (*) pvC weightsC
```

```
> meanPvC2 :: Double
> meanPvC2 = sum $ zipWith (\p w -> p * p * w) pvC weightsC
>
> varPvC :: Double
> varPvC = meanPvC2 - meanPvC * meanPvC
```

Now the weights and the effective size are more re-assuring

```
ghci> length $ filter (>= 1e-6) weightsC
1000
ghci> (sum weightsC)^2 / (sum $ map (^2) weightsC)
967.113872888872
```

And we can take more confidence in the estimate

```
ghci> meanPvC
6.371225269833208e-3
```

# Bibliography

Rerks-Ngarm, Supachai, Punnee Pitisuttithum, Sorachai Nitayaphan, Jaranit Kaewkungwal, Joseph Chiu, Robert Paris, Nakorn Premsri, et al. 2009. “Vaccination with ALVAC and AIDSVAX to Prevent HIV-1 Infection in Thailand.” *New England Journal of Medicine* 361 (23) (December 3): 2209–2220. doi:10.1056/nejmoa0908492. http://dx.doi.org/10.1056/nejmoa0908492.