Home > Single parameter > Brute force Bayes for one parameter

Brute force Bayes for one parameter

Although we talk a lot about conjugate analyses, one doesn’t need to restrict oneself to the use of conjugate priors.  Here we illustrate learning about a Poisson mean using a normal prior on the mean \lambda .

Suppose I observe the following number of fire calls for a particular community each week:  0, 0, 1, 3, 2, 2.  If we assume y_i, the number of fire calls in week i, is Poisson(\lambda), then the likelihood function is given by

L(\lambda) = \exp(- 6 \lambda) \lambda^8, \, \, \lambda>0

Suppose I represent my prior beliefs about \lambda by a normal curve with mean 3 and standard deviation 1.  Then the posterior is given by (up to an unknown proportionality constant) by

g(\lambda | data) = L(\lambda) \times \exp(-(\lambda - 3)^2)

Here is a simple brute-force method of summarizing this posterior.

1.  Choose a grid of \lambda values that covers the region where the posterior is concentrated (this might take some trial and error).

2.  On this grid, compute the prior, likelihood, and posterior.

3.  Using the R sample function, take a large sample from the grid where the probabilities of the points are proportional to the like x prior values.

4.  Summarize this posterior simulated sample to learn about the location of the posterior.

Here’s some R code for this example.  I use the plot function to make sure the grid does cover the posterior.  The vector L contains 1000 draws from the posterior.

lambda = seq(0, 5, by=0.1)
like = exp(-6*lambda)*lambda^8
prior = dnorm(lambda, 3, 1)
post = like * prior
plot(lambda, post)
L = sample(lambda, size=1000, prob=post, replace=TRUE)
plot(density(L))  # density graph of simulated draws
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Categories: Single parameter
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