 # DynamicHMC for Bayesian inference where product distribution is observed - Possible?

Here is a generative model I am working on. Think of a game show where your winnings are determined by the product of two RV’s: winningsMultiplier(X) and maxWinnings(Y). We observe winnings w_i = x_i * y. Our prior on y/10^5 ~ Beta(2,8) corresponding to winnings up to \$100k, but more likely around \$20-\$30k. X_i ~ Beta(2,2). Y does not change from observation to observation, but x_i and w_i do.

Here is a pic of the generative DAG:

Anyway, my attempt to code it is below. I get an error after the last line (`ERROR: type UnionAll has no field w`):

I need two pieces of help for which I would be very appreciative: 1) verify the logic of the implemented math is correct and 2) help me eliminate the error so I can get a posterior for y. Thanks! ``````## simulate some data to see if we can recover params
Random.seed!(1234)
## make y_winnings equal to 35000 for simulation
nObs = 20
x = rand(Beta(2,2),nObs))

## find posterior for y, the max winnings (we know it is 35k)
struct spinProblem{T <: AbstractVector}
w::T # winnings of previous players
end
# https://en.wikipedia.org/wiki/Product_distribution

function (problem::spinProblem)(θ)
@unpack x, y_raw = θ   # extract the parameters
@unpack w, = spinProblem   # extract the data
# log likelihood accumulated in llSpinProb
# prior
llSpinProb = logpdf(LocationScale(0, 100000, Beta(2,8)),y)
yDraw = 10^5 * y_raw
for i in 1:length(w)
x = w / y
llSpinProb += logpdf(Beta(2,2),x) #data
end
return(llSpinProb)
end

p2((x = 0.5, y_raw = 0.25,)) # gives ERROR
``````

I think this line should be

``````@unpack w, = problem
``````

Looks like it’s referencing the type, not the instance of the type.

This was indeed the error I was stuck on. After fixing it, I found many more errors. Those are all fixed. The fixed code is below for anyone viewing this thread. The posterior looks close to correct, but something is still off as the true value from simulated data is not recovered during a posterior predictive check. If anyone sees the math error that leads to this, please comment. I will mark this solved though as the initial Julia issue I was having was graciously solved by @cpfiffer . Thank you for that!!!

``````using Revise, TransformVariables, LogDensityProblems, DynamicHMC,  DynamicHMC.Diagnostics, Parameters, Statistics, Random, DataFrames, Distributions, Gadfly

## simulate some data to see if we can recover params
Random.seed!(234)
## make y_winnings equal to 35000 for simulation
nObs = 2000

## find posterior for y, the max winnings (we know it is 35k)
struct spinProblem{T <: AbstractVector}
w::T # winnings of previous players
end
# https://en.wikipedia.org/wiki/Product_distribution

function (problem::spinProblem)(θ)
@unpack y_raw = θ   # extract the parameters
@unpack w = problem   # extract the data

# log likelihood accumulated in llSpinProb
llSpinProb = logpdf(Beta(2,8),y_raw)
yDraw = y_raw * 10^5

# likelihood
for i in 1:length(w)
x_i = min(w[i] / yDraw,0.999)
# use min below to restrict data to domain
# otherwise get domain error
llSpinProb += logpdf(Beta(2,2),min(x_i,0.999)) #data
end

return(llSpinProb)
end

p2((y_raw = 0.25,)) # infeasible
p2((y_raw = 0.40,)) # feasible

trans2 = as((y_raw = as𝕀,))  ## xtra comma to make it a named tuple
P2 = TransformedLogDensity(trans2, p2)

results = mcmc_with_warmup(Random.GLOBAL_RNG, ∇P2, 8000; reporter = NoProgressReport())

## transform not defined
posterior = TransformVariables.transform.(trans2, results.chain)
posterior_y_raw = first.(posterior)
posterior_y = posterior_y_raw * 10^5 ##estimates seem high - math error?

plot(x = posterior_y, Geom.density)  ## histogram
plot(x = 1:length(posterior_y),
y = posterior_y, Geom.line)

# posterior predictive check
type = "Observed",
group = "0")

postSimDF = DataFrame()
for i in 1:20
tempDF = DataFrame(
draw = rand(posterior_y) .* rand(Beta(2,2),nObs),
type = "Simulated",
group = string(i))
postSimDF = vcat(postSimDF,tempDF)
end

obsLayer = layer(postPredDF, x = :draw, color = [colorant"cadetblue"], style(line_width=6pt), Geom.density, order = 10)

plot(postSimDF, x = :draw, color = :group, style(line_width=1pt), Geom.density,
obsLayer)
## estimates look biased, not sure why?  I would think the prior should be made negligible in relation to the likelihood.  See any errors?
``````

Here is the posterior distribution (unfortunately, 35,000 is the true answer used to simulate the data):

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The posterior predictive check looks off as well. Something seems amiss

Just a quick observation… The mean of Beta(2,2) is .5 Not sure if that is what you intended when multiplying by 35k.

First, the prior won’t be negligible with a sample size of 20; a bigger sample might help. Second, can we see some diagnostic plots? (Traceplot, divergences, energy)

Multiplying by 35,000 is to create some sample data where the true “max winnings” (Y) is 35,000. I then want to use the model to recover this \$35,000 number. Unfortunately, the model returns that it is confident the true value for Y is closer to \$40,000. Being unable to recover this \$35,000 number makes me very skeptical of my model, but I do not see the error leading to this over-estimate.

The second block of code, for which the posterior predictive check plot is shown above, uses an observed sample of 2,000 observations. So the prior should be negligible at this point. Here is a traceplot showing the space is well-explored (code available when running the second code block I posted), just not at the “true” value of \$35k. My guess is I am not handling the loglikelihood properly for draws of y_raw that are completely infeasible given the data. For example, a draw of y_raw of 0.25 corresponds to max winnings(Y) of \$25,000. Yet, if we observed someone has won \$30k, then we know \$25k is completely infeasible - hence, the log density is negative infinity,. Investigations to continue That traceplot looks possibly ok, but not great either; what about other diagnostic plots? Here’s a list of the diagnostic plots I usually use to check:

``````    # Diagnostics
plot_trace(trace; var_names=params, compact=True)
plot_mcse(trace; var_names=params)
plot_ess(trace; var_names=params, kind="local")
plot_ess(trace, var_names=params, kind="quantile")
plot_ess(trace, var_names=params, kind="evolution")
plot_rank(trace, var_names=params)
plot_energy(trace)
``````

Here’s an example to show that sometimes, your trace can look great even if your inferences are pretty awful – the energy transition plot is clearly much more sharply peaked than the marginal energy plot: Meanwhile, an ESS plot shows exactly where the problem is – the centered parametrization for my data is making sampling of standard deviations extremely difficult. At some points, my effective sample size is even trending downwards as the value gets stuck at a low value for the standard deviation.

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Found the error. Missing the Jacobian for the likelihood. Found this post helpful in regards to knowing when to use a Jacobian and when not to: Demystifying Jacobian adjustment for transformed parameters in Stan

Here is the block of code that adds the Jacobian:

``````function (problem::spinProblem)(θ)
@unpack y_raw = θ   # extract the parameters
@unpack w = problem   # extract the data

# log likelihood accumulated in llSpinProb
# prior on y_raw
llSpinProb = logpdf(Beta(2,8),y_raw)
yDraw = y_raw * 10^5

# likelihood
for i in 1:length(w)
x_i = min(w[i] / yDraw,0.999)
# use min below to restrict data to domain
# otherwise get domain error
llSpinProb += logpdf(Beta(2,2),min(x_i,0.999)) #data
llSpinProb += log(1/yDraw) # Jacobian
end

return(llSpinProb)
end
``````

Now the posterior predictive check looks much better:

and the trace plot is centered in the right spot (i.e. \$35K):

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