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BUGS_demo.ipynb
{
"cells": [
{
"cell_type": "markdown",
"id": "857709b0-ea42-4611-bec6-dc57a0bbfbbb",
"metadata": {
"tags": []
},
"source": [
"## The BUGS Language\n",
"BUGS(Bayesian inference Using Gibbs Sampling) is one of the earliest probabilistic programming language. A legal program in the BUGS language describes a directed probabilistic graphical model. The static nature of these models enables developers to extract much information deterministically, resulting in efficient inference algorithm implementations. Syntactically, to guarantee the static property of the program, all the loop bounds must not depend on stochastic variables."
]
},
{
"cell_type": "markdown",
"id": "7892abb3-b5f7-4b4c-86e8-a5a59db5f7f8",
"metadata": {
"tags": []
},
"source": [
"## The Need for Partial Evaluation\n",
"As we discussed before, legal BUGS programs describe Directed Acyclic Graphical(DAG) models, and one can understand assignments in BUGS programs as describing edges in the DAGs. There are no chronological orders between all the assignments, which means, if we want to analyze if a certain variable is dependent on some other variables, the analysis has to consider all the assignments in the program. \n",
"\n",
"Our approaches rely on [Symbolics.jl](https://symbolics.juliasymbolics.org/dev/). The intuition behind our solution is that the RHS of a logical assignment can be constructed into a symbolic term, while the LHS of that logical assignments are potentially a variable referred in other symbolic terms. By utilizing [Symbolic.subsitute](https://symbolics.juliasymbolics.org/dev/manual/expression_manipulation/#SymbolicUtils.substitute) function, we can do partial evaluation of any variable. And if the model is fully specified, we can evaluate the variable to a concrete number. To test if a model is correctly specified, we only need to check if all the loop bounds and array indices can be resolved to a concrete number by only considering logical assignments. \n",
"\n",
"In the remaining of this chapter, we will detail some aspects of the implementation. "
]
},
{
"cell_type": "markdown",
"id": "048e0923-223f-45bc-844c-a4800452db36",
"metadata": {},
"source": [
"# Plan\n",
"\n",
"**We will only cover the model compilation from the Julia AST acquired from `@bugsast` or `bugsmodel` in this demo.**\n",
"\n",
"We will first demonstrate some key aspects of our implementation. Then we'll demonstrate the complete process of compiling and inference with the [Seeds](https://www.multibugs.org/examples/latest/Seeds.html) model from BUGS Examples Volume I."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "dffcfdee-7536-47ad-a02a-60dd2ad5be3e",
"metadata": {},
"outputs": [],
"source": [
"using SymbolicPPL\n",
"using SymbolicPPL: CompilerState, addlogicalrules!, addstochasticrules!, unroll!, tosymbolic, resolve, gen_output\n",
"using Symbolics\n",
"using Graphs, MetaGraphsNext"
]
},
{
"cell_type": "markdown",
"id": "d4c62c87-380e-4571-8469-a03ff1710d06",
"metadata": {
"tags": []
},
"source": [
"## Logical Assignments\n",
"\n",
"Let's consider a simple logical assignment."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "a3ebd4f4-830d-40d3-b56d-bfa0024c6303",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"quote\n",
" a = b + c * 3\n",
" b = 2\n",
" c = 0.5\n",
"end"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"logical_assignments = bugsmodel\"\"\"\n",
" a <- b + c * 3\n",
" b <- 2\n",
" c <- 0.5\n",
"\"\"\""
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "251ec458-e37a-47a1-b844-dc67a2a8192b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"quote\n",
" a = b + c * 3\n",
" b = 2\n",
" c = 0.5\n",
"end"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Equivalently\n",
"logical_assignments = @bugsast begin\n",
" a = b + c * 3\n",
" b = 2\n",
" c = 0.5\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "32993121-78f0-4cc4-bd25-1dfd894974cf",
"metadata": {},
"source": [
"In the code block above, `@bugsast` translates a BUGS program into Julia AST showed in the output. \n",
"The function `addlogicalrules!` is the workhorse that process logical assignments. \n",
"For every logical assignment, `addlogicalrules!` creates a mapping between the LHS symbolic variable and RHS symbolic term, and this mapping is stored as a dictionary. Let's see the function at work."
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "88ee086c-b531-4fc2-8b03-46259325d50e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 3 entries:\n",
" a => b + 3c\n",
" c => 0.5\n",
" b => 2"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compiler_state = CompilerState(logical_assignments) # keep information during te compilation process\n",
"addlogicalrules!(logical_assignments, compiler_state)\n",
"compiler_state.logicalrules"
]
},
{
"cell_type": "markdown",
"id": "bed21005-1939-4ef8-8c6c-ae5b802d29b6",
"metadata": {},
"source": [
"Then we can ask: what is the value of variable `a`, partially evaluated in the context of all logical assignments? The answer will be "
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "bc2b7a6e-938b-4e09-8c18-30405a5b6dad",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{equation}\n",
"b + 3 c\n",
"\\end{equation}\n"
],
"text/plain": [
"b + 3c"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"inter_result = substitute(tosymbolic(:a), compiler_state.logicalrules)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "6a960cb7-3830-4e3f-947a-41a4b9ffec85",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{equation}\n",
"3.5\n",
"\\end{equation}\n"
],
"text/plain": [
"3.5"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"substitute(inter_result, compiler_state.logicalrules)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "d0e4cdc7-4a6d-4968-9d03-2ee9b6a466ec",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3.5"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"resolve(tosymbolic(:a), compiler_state.logicalrules) # `resolve` repeatedly use Symbolics.substitut for evaluation, stop when the substitution result was seen previously"
]
},
{
"cell_type": "markdown",
"id": "e7d9bdc5-ec8a-463b-9c13-9a442f60ff2b",
"metadata": {},
"source": [
"and it's exactly what we expect."
]
},
{
"cell_type": "markdown",
"id": "4bb7569a-67ef-4741-a2c5-9a6de2a42d55",
"metadata": {},
"source": [
"## Stochastic Assignments\n",
"\n",
"Stochastic assignments are handled in a similar way. BUGS syntax requires the RHS of a stochastic assignment to be a probability distribution function. \n",
"At this stage, the processing of stochastic assignments is very straightforward and mechanical, just like in the case of logical assignments. \n",
"The function that handle stochastic assignments is called `addstochasticrules!`."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "14da7fb7-3b08-41e3-9dbb-3bee4c431038",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"quote\n",
" a = b + c * 3\n",
" b = 2\n",
" c = 0.5\n",
" $(Expr(:~, :y, :(dnorm(a, 1))))\n",
"end"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mixed_assignments = @bugsast begin\n",
" a = b + c * 3\n",
" b = 2\n",
" c = 0.5\n",
" y ~ dnorm(a, 1)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "85e4dd48-33d6-4ec1-91c4-dc1d54fe6bc8",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 1 entry:\n",
" y => dnorm(a, 1)"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compiler_state = CompilerState(mixed_assignments);\n",
"addlogicalrules!(mixed_assignments, compiler_state)\n",
"addstochasticrules!(mixed_assignments, compiler_state)\n",
"compiler_state.stochasticrules"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "26ceaf25-4fdb-43c0-ba6e-15aa79f20d0a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
":(function ()\n",
" Distributions.Normal{Float64}(μ=3.5, σ=1.0)\n",
" end)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"gen_output(compiler_state)[:y][2]"
]
},
{
"cell_type": "markdown",
"id": "c1bb34d3-5e73-4b72-bb54-ff65b7f148d8",
"metadata": {},
"source": [
"## Array \n",
"The challenge of support BUGS' array interface is that every element of an array can be either logical or stochastic, so we need to treat every array element as a separate variable. We won't go deep into the inner mechanism, rather let's see how array indexing work with some demos."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "3dc5e682-0303-48a2-a49f-39650bffa726",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 2 entries:\n",
" g[1:3] => [1, 2, 3]\n",
" h[2, 3] => 2g[1]"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"array_demo = @bugsast begin\n",
" h[2, 3] = 2 * g[1]\n",
"end\n",
"data = (; g = [1, 2 ,3])\n",
"\n",
"compiler_state = CompilerState(array_demo)\n",
"addlogicalrules!(data, compiler_state)\n",
"addlogicalrules!(array_demo, compiler_state)\n",
"compiler_state.logicalrules"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "4f096db0-b671-4328-b079-7e1f2a51da7c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Symbol, Symbolics.Arr{Num}} with 1 entry:\n",
" :h => h[1:2,1:3]"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compiler_state.arrays"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "6e6a46a6-0b43-4510-820a-230b13dae80b",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Symbol, Symbolics.Arr{Num}} with 1 entry:\n",
" :g => g[1:3]"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"compiler_state.data_arrays"
]
},
{
"cell_type": "markdown",
"id": "082ca899-e1c0-4cec-87e5-f1297f98fc01",
"metadata": {},
"source": [
"There are several things worth explaining here:\n",
"- The array indexing is translated into a variable with an easy-to-identify name (`h[2,3]` -> `var\"h[2, 3]\"`).\n",
"- Array shape deduction is carried out as side-effects of the previous translation: the largest index seen is treated as the size of the array\n",
"- No shape deduction is done for data arrays, indexing beyond the given range or modification will cause an error"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "319fbcc1-7ee3-4860-b15a-97cd9ccf028e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 3 entries:\n",
" x => (2//3)*h[2, 1] + (2//3)*h[2, 2] + (2//3)*h[2, 3]\n",
" g[1:3] => [1, 2, 3]\n",
" h[2, 3] => 2g[1]"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x_assignment = @bugsast begin\n",
" h[2, 3] = 2 * g[1]\n",
" x = 2 * mean(h[2, :]) # slicing h\n",
"end\n",
"\n",
"data = (; g = [1, 2 ,3])\n",
"\n",
"compiler_state = CompilerState(x_assignment)\n",
"addlogicalrules!(data, compiler_state)\n",
"addlogicalrules!(x_assignment, compiler_state, false) \n",
"compiler_state.logicalrules"
]
},
{
"cell_type": "markdown",
"id": "bd006d3f-c4ed-4d0d-8ce4-dda4b477b8e5",
"metadata": {},
"source": [
"The compiler will infer the size of `h` to be (2, 3), because that's the largest indices seen. We can also see `mean()` function is interpreted and simplified. \n",
"Indexing an array with other array indexing is also supported."
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "d79fe952-ccb3-4883-bfdc-9a4ac7f429bd",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 2 entries:\n",
" k[1] => 3\n",
" g[3] => 2"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"nested_indexing = @bugsast begin\n",
" g[k[1]] = 2\n",
" k[1] = 3\n",
"end\n",
"compiler_state = CompilerState(nested_indexing)\n",
"addlogicalrules!(nested_indexing, compiler_state) # The first one add `k[1]` \n",
"addlogicalrules!(nested_indexing, compiler_state) # so the index can be evaluated now\n",
"compiler_state.logicalrules"
]
},
{
"cell_type": "markdown",
"id": "6c42a35a-9710-40f5-99b2-59a4da6c9021",
"metadata": {},
"source": [
"## Unrolling\n",
"Internally, we unroll all the loops. There are several reasons for this decision:\n",
"1. If we can unroll all the loops, then all the expression are either logical or stochastic assignments, and we have established that we can handle them well.\n",
"2. For a BUGS program to be legal, the loop bounds would need to be evaluated to a concrete number. Otherwise, either the model is under-specified, or the loop bound is dependent on stochastic variables.\n",
"\n",
"`unrollforloops!` handles the unrolling."
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "2fcc1268-68c7-4196-87bc-5de34e8a2a5e",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"quote\n",
" array_variable_0[1] = 1 + 1\n",
" array_variable_2[1, 1, 1] = 2\n",
" array_variable_2[1, 2, 1] = 2\n",
" array_variable_2[1, 2, 2] = 2\n",
" array_variable_0[2] = 2 + 1\n",
" array_variable_2[2, 1, 1] = 2\n",
" array_variable_2[2, 2, 1] = 2\n",
" array_variable_2[2, 2, 2] = 2\n",
" array_variable_0[3] = 3 + 1\n",
" array_variable_2[3, 1, 1] = 2\n",
" array_variable_2[3, 2, 1] = 2\n",
" array_variable_2[3, 2, 2] = 2\n",
" $(Expr(:processed))\n",
"end"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"unroll_example = @bugsast begin \n",
" # nested loop\n",
" for i in 1:N\n",
" # assignment using loop variable\n",
" array_variable_0[i] = i + 1\n",
" \n",
" # nested loops in another for loop\n",
" for j in 1:2\n",
" # loop bound depend on loop variable\n",
" for k in 1:j\n",
" array_variable_2[i, j, k] = 2\n",
" end\n",
" end\n",
" end\n",
"\n",
" N = 3\n",
"end\n",
"compiler_state = CompilerState(unroll_example)\n",
"addlogicalrules!(unroll_example, compiler_state) # Add `N => 3` to the rules\n",
"unroll!(unroll_example, compiler_state)\n",
"unroll_example"
]
},
{
"cell_type": "markdown",
"id": "54e59ac1-ced9-4408-a9da-63a9df3c0b20",
"metadata": {},
"source": [
"# Seeds"
]
},
{
"cell_type": "markdown",
"id": "a7257bb1-4fa8-4c2b-88ca-40ae606f2604",
"metadata": {},
"source": [
"`Seeds` concerns the proportion of seeds that germinated on each of 21 plates. "
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "8f31463a-3062-4d08-ad61-0059a3e0996d",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(r = [10, 23, 23, 26, 17], n = [39, 62, 81, 51, 39], x1 = [0, 0, 0, 0, 0], x2 = [0, 0, 0, 0, 0], N = 5)"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"data = (\n",
" r = [10, 23, 23, 26, 17, 5, 53, 55, 32, 46, 10, 8, 10, 8, 23, 0, 3, 22, 15, 32, 3],\n",
" n = [39, 62, 81, 51, 39, 6, 74, 72, 51, 79, 13, 16, 30, 28, 45, 4, 12, 41, 30, 51, 7],\n",
" x1 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],\n",
" x2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1],\n",
" N = 21,\n",
")\n",
"\n",
"# we will take the first five plates to make model plot easier to look at\n",
"partial_data = (\n",
" r = [10, 23, 23, 26, 17],\n",
" n = [39, 62, 81, 51, 39],\n",
" x1 = [0, 0, 0, 0, 0],\n",
" x2 = [0, 0, 0, 0, 0],\n",
" N = 5,\n",
")"
]
},
{
"cell_type": "markdown",
"id": "b68b6648-76a2-45d3-bfcb-821a8e15792d",
"metadata": {},
"source": [
"where `r[i]` is the number of germinated seeds and `n[i]` is the total number of the seeds on the $i$-th plate. \n",
"The model is constructed such that, let $p_i$ be the probability of germination on the $i$-th plate, \n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"r_i &\\sim \\operatorname{Binomial}(p_i, n_i) \\\\\n",
"\\operatorname{logit}(p_i) &\\sim \\alpha_0 + \\alpha_1 x_{1 i} + \\alpha_2 x_{2i} + \\alpha_{12} x_{1i} x_{2i} + b_{i} \\\\\n",
"b_i &\\sim \\operatorname{Normal}(0, \\tau)\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"where $x_{1i}$ and $x_{2i}$ are the seed type and root extract of the $i$-th plate. \n",
"We describe the model as"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "e86231f5-7596-4214-a94d-c0e227af5288",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"quote\n",
" for i = 1:N\n",
" $(Expr(:~, :(r[i]), :(dbin(p[i], n[i]))))\n",
" $(Expr(:~, :(b[i]), :(dnorm(0.0, tau))))\n",
" p[i] = logistic(alpha0 + alpha1 * x1[i] + alpha2 * x2[i] + alpha12 * x1[i] * x2[i] + b[i])\n",
" end\n",
" $(Expr(:~, :alpha0, :(dnorm(0.0, 1.0e-6))))\n",
" $(Expr(:~, :alpha1, :(dnorm(0.0, 1.0e-6))))\n",
" $(Expr(:~, :alpha2, :(dnorm(0.0, 1.0e-6))))\n",
" $(Expr(:~, :alpha12, :(dnorm(0.0, 1.0e-6))))\n",
" $(Expr(:~, :tau, :(dgamma(0.001, 0.001))))\n",
" sigma = 1 / sqrt(tau)\n",
"end"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"seeds = @bugsast begin\n",
" for i in 1:N\n",
" r[i] ~ dbin(p[i],n[i])\n",
" b[i] ~ dnorm(0.0,tau)\n",
" p[i] = logistic(alpha0 + alpha1 * x1[i] + alpha2 * x2[i] + alpha12 * x1[i] * x2[i] + b[i])\n",
" end\n",
" alpha0 ~ dnorm(0.0,1.0E-6)\n",
" alpha1 ~ dnorm(0.0,1.0E-6)\n",
" alpha2 ~ dnorm(0.0,1.0E-6)\n",
" alpha12 ~ dnorm(0.0,1.0E-6)\n",
" tau ~ dgamma(0.001,0.001)\n",
" sigma = 1 / sqrt(tau)\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "d1fce937-8a99-497c-9abb-f72dfe8ccf22",
"metadata": {},
"source": [
"The main function for compilation is \n",
"\n",
"```julia\n",
"compile(model_def::Expr, data::NamedTuple, target::Symbol),\n",
"```\n",
"\n",
"which takes three arguments: \n",
"- the first argument is the output of `@bugsast` or `bugsmodel`, \n",
"- the second argument is the data \n",
"- the third argument is a `Symbol` indicating the compilation target."
]
},
{
"cell_type": "markdown",
"id": "10edbde3-c263-4ba2-8cb2-cb9a5fcc6b87",
"metadata": {},
"source": [
"To compile into a `CompilerState` object"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "ac82824d-1e94-4593-b584-1b66e646d888",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Dict{Any, Any} with 27 entries:\n",
" x1[1:21] => [0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]\n",
" p[14] => 1 / (1 + exp(-alpha0 - b[14] - alpha1*x1[14] - alpha2*x2[14] - al…\n",
" p[12] => 1 / (1 + exp(-alpha0 - b[12] - alpha1*x1[12] - alpha2*x2[12] - al…\n",
" x2[1:21] => [0, 0, 0, 0, 0, 1, 1, 1, 1, 1 … 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]\n",
" p[8] => 1 / (1 + exp(-alpha0 - b[8] - alpha1*x1[8] - alpha2*x2[8] - alpha…\n",
" r[1:21] => [10, 23, 23, 26, 17, 5, 53, 55, 32, 46 … 8, 10, 8, 23, 0, 3, 22…\n",
" p[15] => 1 / (1 + exp(-alpha0 - b[15] - alpha1*x1[15] - alpha2*x2[15] - al…\n",
" p[9] => 1 / (1 + exp(-alpha0 - b[9] - alpha1*x1[9] - alpha2*x2[9] - alpha…\n",
" p[5] => 1 / (1 + exp(-alpha0 - b[5] - alpha1*x1[5] - alpha2*x2[5] - alpha…\n",
" n[1:21] => [39, 62, 81, 51, 39, 6, 74, 72, 51, 79 … 16, 30, 28, 45, 4, 12,…\n",
" p[11] => 1 / (1 + exp(-alpha0 - b[11] - alpha1*x1[11] - alpha2*x2[11] - al…\n",
" N => 21\n",
" p[17] => 1 / (1 + exp(-alpha0 - b[17] - alpha1*x1[17] - alpha2*x2[17] - al…\n",
" p[3] => 1 / (1 + exp(-alpha0 - b[3] - alpha1*x1[3] - alpha2*x2[3] - alpha…\n",
" p[7] => 1 / (1 + exp(-alpha0 - b[7] - alpha1*x1[7] - alpha2*x2[7] - alpha…\n",
" p[10] => 1 / (1 + exp(-alpha0 - b[10] - alpha1*x1[10] - alpha2*x2[10] - al…\n",
" p[4] => 1 / (1 + exp(-alpha0 - b[4] - alpha1*x1[4] - alpha2*x2[4] - alpha…\n",
" p[18] => 1 / (1 + exp(-alpha0 - b[18] - alpha1*x1[18] - alpha2*x2[18] - al…\n",
" p[2] => 1 / (1 + exp(-alpha0 - b[2] - alpha1*x1[2] - alpha2*x2[2] - alpha…\n",
" p[16] => 1 / (1 + exp(-alpha0 - b[16] - alpha1*x1[16] - alpha2*x2[16] - al…\n",
" p[19] => 1 / (1 + exp(-alpha0 - b[19] - alpha1*x1[19] - alpha2*x2[19] - al…\n",
" sigma => 1 / sqrt(tau)\n",
" p[1] => 1 / (1 + exp(-alpha0 - b[1] - alpha1*x1[1] - alpha2*x2[1] - alpha…\n",
" p[13] => 1 / (1 + exp(-alpha0 - b[13] - alpha1*x1[13] - alpha2*x2[13] - al…\n",
" p[20] => 1 / (1 + exp(-alpha0 - b[20] - alpha1*x1[20] - alpha2*x2[20] - al…\n",
" ⋮ => ⋮"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"copmiler_state = compile(seeds, data, :IR);\n",
"copmiler_state.logicalrules"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "8a8a3835-dd5b-48ae-beff-0a7bba32bf96",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"\\begin{equation}\n",
"\\frac{1}{1 + e^{ - alpha0 - b_{1 4} - alpha1 x1_{1 4} - alpha2 x2_{1 4} - alpha12 x1_{1 4} x2_{1 4}}}\n",
"\\end{equation}\n"
],
"text/plain": [
"1 / (1 + exp(-alpha0 - b[14] - alpha1*x1[14] - alpha2*x2[14] - alpha12*x1[14]*x2[14]))"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"copmiler_state.logicalrules[tosymbolic(Symbol(\"p[14]\"))]"
]
},
{
"cell_type": "markdown",
"id": "4b73f487-f149-4ecc-96de-eec97bb33ba1",
"metadata": {},
"source": [
"To compile into a graph representation"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "733c7912-be19-4d97-aed7-43e678a1192a",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Variable Name: b[17]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[6]\n",
"Variable Type: Observation\n",
"Data: 5\n",
"Parent Nodes: alpha2, alpha0, b[6]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[6]\"))))), 6))\n",
"\n",
"Variable Name: r[1]\n",
"Variable Type: Observation\n",
"Data: 10\n",
"Parent Nodes: alpha0, b[1]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[1]\"))))), 39))\n",
"\n",
"Variable Name: alpha0\n",
"Variable Type: Assumption\n",
"Parent Nodes: \n",
"Node Function: Distributions.Normal{Float64}(μ=0.0, σ=1000.0)\n",
"\n",
"Variable Name: b[4]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[7]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[13]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[15]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[3]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[21]\n",
"Variable Type: Observation\n",
"Data: 3\n",
"Parent Nodes: alpha12, alpha2, alpha0, b[21], alpha1\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)((+)((+)(0, (*)(-1, alpha12)), (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[21]\")), (*)(-1, alpha1))))), 7))\n",
"\n",
"Variable Name: r[17]\n",
"Variable Type: Observation\n",
"Data: 3\n",
"Parent Nodes: alpha12, alpha2, alpha0, b[17], alpha1\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)((+)((+)(0, (*)(-1, alpha12)), (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[17]\")), (*)(-1, alpha1))))), 12))\n",
"\n",
"Variable Name: tau\n",
"Variable Type: Assumption\n",
"Parent Nodes: \n",
"Node Function: Distributions.Gamma{Float64}(α=0.001, θ=1000.0)\n",
"\n",
"Variable Name: b[12]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[10]\n",
"Variable Type: Observation\n",
"Data: 46\n",
"Parent Nodes: alpha2, alpha0, b[10]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[10]\"))))), 79))\n",
"\n",
"Variable Name: r[12]\n",
"Variable Type: Observation\n",
"Data: 8\n",
"Parent Nodes: alpha0, alpha1, b[12]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, alpha1)), (*)(-1, var\"b[12]\"))))), 16))\n",
"\n",
"Variable Name: r[20]\n",
"Variable Type: Observation\n",
"Data: 32\n",
"Parent Nodes: alpha12, alpha2, alpha0, alpha1, b[20]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)((+)((+)(0, (*)(-1, alpha12)), (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, alpha1)), (*)(-1, var\"b[20]\"))))), 51))\n",
"\n",
"Variable Name: r[18]\n",
"Variable Type: Observation\n",
"Data: 22\n",
"Parent Nodes: alpha12, alpha2, alpha0, b[18], alpha1\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)((+)((+)(0, (*)(-1, alpha12)), (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[18]\")), (*)(-1, alpha1))))), 41))\n",
"\n",
"Variable Name: b[2]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[13]\n",
"Variable Type: Observation\n",
"Data: 10\n",
"Parent Nodes: alpha0, alpha1, b[13]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, alpha1)), (*)(-1, var\"b[13]\"))))), 30))\n",
"\n",
"Variable Name: r[14]\n",
"Variable Type: Observation\n",
"Data: 8\n",
"Parent Nodes: alpha0, b[14], alpha1\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[14]\")), (*)(-1, alpha1))))), 28))\n",
"\n",
"Variable Name: b[11]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[16]\n",
"Variable Type: Observation\n",
"Data: 0\n",
"Parent Nodes: alpha0, b[16], alpha1\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[16]\")), (*)(-1, alpha1))))), 4))\n",
"\n",
"Variable Name: b[1]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[15]\n",
"Variable Type: Observation\n",
"Data: 23\n",
"Parent Nodes: alpha0, alpha1, b[15]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, alpha1)), (*)(-1, var\"b[15]\"))))), 45))\n",
"\n",
"Variable Name: r[7]\n",
"Variable Type: Observation\n",
"Data: 53\n",
"Parent Nodes: b[7], alpha2, alpha0\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, var\"b[7]\")), (*)(-1, alpha2)), (*)(-1, alpha0))))), 74))\n",
"\n",
"Variable Name: b[21]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[4]\n",
"Variable Type: Observation\n",
"Data: 26\n",
"Parent Nodes: alpha0, b[4]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[4]\"))))), 51))\n",
"\n",
"Variable Name: b[10]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[6]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[9]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[8]\n",
"Variable Type: Observation\n",
"Data: 55\n",
"Parent Nodes: alpha2, b[8], alpha0\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha2)), (*)(-1, var\"b[8]\")), (*)(-1, alpha0))))), 72))\n",
"\n",
"Variable Name: b[20]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[11]\n",
"Variable Type: Observation\n",
"Data: 10\n",
"Parent Nodes: alpha2, b[11], alpha0\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha2)), (*)(-1, var\"b[11]\")), (*)(-1, alpha0))))), 13))\n",
"\n",
"Variable Name: b[18]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[3]\n",
"Variable Type: Observation\n",
"Data: 23\n",
"Parent Nodes: alpha0, b[3]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[3]\"))))), 81))\n",
"\n",
"Variable Name: r[5]\n",
"Variable Type: Observation\n",
"Data: 17\n",
"Parent Nodes: alpha0, b[5]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[5]\"))))), 39))\n",
"\n",
"Variable Name: b[8]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: r[9]\n",
"Variable Type: Observation\n",
"Data: 32\n",
"Parent Nodes: alpha2, alpha0, b[9]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)(0, (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, var\"b[9]\"))))), 51))\n",
"\n",
"Variable Name: b[16]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[14]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: alpha2\n",
"Variable Type: Assumption\n",
"Parent Nodes: \n",
"Node Function: Distributions.Normal{Float64}(μ=0.0, σ=1000.0)\n",
"\n",
"Variable Name: b[19]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: b[5]\n",
"Variable Type: Assumption\n",
"Parent Nodes: tau\n",
"Node Function: :((SymbolicPPL.dnorm)(0.0, tau))\n",
"\n",
"Variable Name: alpha12\n",
"Variable Type: Assumption\n",
"Parent Nodes: \n",
"Node Function: Distributions.Normal{Float64}(μ=0.0, σ=1000.0)\n",
"\n",
"Variable Name: r[19]\n",
"Variable Type: Observation\n",
"Data: 15\n",
"Parent Nodes: alpha12, alpha2, alpha0, alpha1, b[19]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)((+)((+)((+)(0, (*)(-1, alpha12)), (*)(-1, alpha2)), (*)(-1, alpha0)), (*)(-1, alpha1)), (*)(-1, var\"b[19]\"))))), 30))\n",
"\n",
"Variable Name: alpha1\n",
"Variable Type: Assumption\n",
"Parent Nodes: \n",
"Node Function: Distributions.Normal{Float64}(μ=0.0, σ=1000.0)\n",
"\n",
"Variable Name: r[2]\n",
"Variable Type: Observation\n",
"Data: 23\n",
"Parent Nodes: alpha0, b[2]\n",
"Node Function: :((SymbolicPPL.dbin)((/)(1, (+)(1, (exp)((+)((+)(0, (*)(-1, alpha0)), (*)(-1, var\"b[2]\"))))), 62))\n",
"\n"
]
}
],
"source": [
"g = compile(seeds, data, :Graph);\n",
"\n",
"for v in vertices(g)\n",
" println(g[label_for(g, v)]); println()\n",
"end"
]
},
{
"cell_type": "markdown",
"id": "9b94d6b4-0b20-4f2a-b144-1cff10924d9d",
"metadata": {},
"source": [
"Julia graph ecosystem provides myriad of plotting utilities, here we use the TikzGraphs.jl package to generate a Tikz picture."
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "2838b484-bb8c-4d1b-a217-5fef28c84712",
"metadata": {},
"outputs": [],
"source": [
"using TikzGraphs, TikzPictures\n",
"function plot(g)\n",
" color_dict = Dict{Int, String}()\n",
" for (i, node) in enumerate(vertices(g))\n",
" if g[label_for(g, node)].is_data\n",
" color_dict[i] = \"fill=green!10\"\n",
" else\n",
" color_dict[i] = \"fill=yellow!10\"\n",
" end\n",
" end\n",
"\n",
" TikzGraphs.plot(\n",
" g.graph, \n",
" map(x->string(label_for(g, x)), vertices(g)), \n",
" node_style=\"draw, rounded corners, fill=blue!10\", \n",
" node_styles=color_dict,\n",
" edge_style=\"black\"\n",
" )\n",
"end;"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "968341f4-6ecc-42a8-8dbc-d3a45ca5214d",
"metadata": {},
"outputs": [
{
"data": {
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"<defs>\n",
"<g>\n",
"<symbol overflow=\"visible\" id=\"glyph-1667476809358020-0-0\">\n",
"<path style=\"stroke:none;\" d=\"\"/>\n",
"</symbol>\n",
"<symbol overflow=\"visible\" id=\"glyph-1667476809358020-0-1\">\n",
"<path style=\"stroke:none;\" d=\"M 5.171875 -2.140625 C 5.171875 -3.40625 4.203125 -4.390625 3.078125 -4.390625 C 2.296875 -4.390625 1.875 -3.921875 1.703125 -3.75 L 1.703125 -6.890625 L 0.28125 -6.78125 L 0.28125 -6.484375 C 0.96875 -6.484375 1.046875 -6.40625 1.046875 -5.921875 L 1.046875 0 L 1.296875 0 L 1.65625 -0.609375 C 1.8125 -0.390625 2.21875 0.109375 2.96875 0.109375 C 4.140625 0.109375 5.171875 -0.859375 5.171875 -2.140625 Z M 4.359375 -2.15625 C 4.359375 -1.78125 4.328125 -1.1875 4.046875 -0.75 C 3.828125 -0.4375 3.453125 -0.109375 2.921875 -0.109375 C 2.46875 -0.109375 2.109375 -0.34375 1.875 -0.71875 C 1.734375 -0.921875 1.734375 -0.953125 1.734375 -1.140625 L 1.734375 -3.1875 C 1.734375 -3.375 1.734375 -3.375 1.84375 -3.53125 C 2.234375 -4.09375 2.78125 -4.171875 3.015625 -4.171875 C 3.46875 -4.171875 3.828125 -3.921875 4.0625 -3.53125 C 4.328125 -3.125 4.359375 -2.5625 4.359375 -2.15625 Z M 4.359375 -2.15625 \"/>\n",
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]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(g)"
]
},
{
"cell_type": "markdown",
"id": "7101058a-1ac3-42dc-bc74-46a7b54590c1",
"metadata": {},
"source": [
"The details are difficult to make out, let's plot with partial data. (`x1` and `x2` selects random effects, so some random variables are singular in this partial graph.)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "88392d11-b0e3-449d-9c4a-a5bfdfb39acf",
"metadata": {},
"outputs": [
{
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]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"partial_g = compile(seeds, partial_data, :Graph);\n",
"plot(partial_g)"
]
},
{
"cell_type": "markdown",
"id": "3b4c358a-d135-4fe4-a897-7cdc9335b2fe",
"metadata": {},
"source": [
"Compiling to a Turing model is supported too,"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "63c601b5-76af-49c7-b6ab-873c7da28cac",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"function model(; var\"r[6]\" = 5, var\"r[1]\" = 10, var\"r[21]\" = 3, var\"r[17]\" = 3, var\"r[10]\" = 46, var\"r[12]\" = 8, var\"r[20]\" = 32, var\"r[18]\" = 22, var\"r[13]\" = 10, var\"r[14]\" = 8, var\"r[16]\" = 0, var\"r[15]\" = 23, var\"r[7]\" = 53, var\"r[4]\" = 26, var\"r[8]\" = 55, var\"r[11]\" = 10, var\"r[3]\" = 23, var\"r[5]\" = 17, var\"r[9]\" = 32, var\"r[19]\" = 15, var\"r[2]\" = 23)\n",
" alpha1 ~ Normal(0.0, 1000.0)\n",
" alpha12 ~ Normal(0.0, 1000.0)\n",
" alpha2 ~ Normal(0.0, 1000.0)\n",
" tau ~ Gamma(0.001, 1000.0)\n",
" var\"b[5]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[19]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[14]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[16]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[8]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[18]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[20]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[9]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[6]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[10]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[21]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[1]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[11]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[2]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[12]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[3]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[15]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[13]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[7]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"b[4]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" alpha0 ~ Normal(0.0, 1000.0)\n",
" var\"r[2]\" ~ SymbolicPPL.dbin(1 / (1 + exp((0 + -1alpha0) + -1var\"b[2]\")), 62)\n",
" var\"r[19]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((((0 + -1alpha12) + -1alpha2) + -1alpha0) + -1alpha1) + -1var\"b[19]\")), 30)\n",
" var\"r[9]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha2) + -1alpha0) + -1var\"b[9]\")), 51)\n",
" var\"r[5]\" ~ SymbolicPPL.dbin(1 / (1 + exp((0 + -1alpha0) + -1var\"b[5]\")), 39)\n",
" var\"r[3]\" ~ SymbolicPPL.dbin(1 / (1 + exp((0 + -1alpha0) + -1var\"b[3]\")), 81)\n",
" var\"r[11]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha2) + -1var\"b[11]\") + -1alpha0)), 13)\n",
" var\"r[8]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha2) + -1var\"b[8]\") + -1alpha0)), 72)\n",
" var\"r[4]\" ~ SymbolicPPL.dbin(1 / (1 + exp((0 + -1alpha0) + -1var\"b[4]\")), 51)\n",
" var\"r[7]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1var\"b[7]\") + -1alpha2) + -1alpha0)), 74)\n",
" var\"r[15]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha0) + -1alpha1) + -1var\"b[15]\")), 45)\n",
" var\"r[16]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha0) + -1var\"b[16]\") + -1alpha1)), 4)\n",
" var\"r[14]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha0) + -1var\"b[14]\") + -1alpha1)), 28)\n",
" var\"r[13]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha0) + -1alpha1) + -1var\"b[13]\")), 30)\n",
" var\"r[18]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((((0 + -1alpha12) + -1alpha2) + -1alpha0) + -1var\"b[18]\") + -1alpha1)), 41)\n",
" var\"r[20]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((((0 + -1alpha12) + -1alpha2) + -1alpha0) + -1alpha1) + -1var\"b[20]\")), 51)\n",
" var\"r[12]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha0) + -1alpha1) + -1var\"b[12]\")), 16)\n",
" var\"r[10]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha2) + -1alpha0) + -1var\"b[10]\")), 79)\n",
" var\"r[21]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((((0 + -1alpha12) + -1alpha2) + -1alpha0) + -1var\"b[21]\") + -1alpha1)), 7)\n",
" var\"r[1]\" ~ SymbolicPPL.dbin(1 / (1 + exp((0 + -1alpha0) + -1var\"b[1]\")), 39)\n",
" var\"r[6]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((0 + -1alpha2) + -1alpha0) + -1var\"b[6]\")), 6)\n",
" var\"b[17]\" ~ SymbolicPPL.dnorm(0.0, tau)\n",
" var\"r[17]\" ~ SymbolicPPL.dbin(1 / (1 + exp(((((0 + -1alpha12) + -1alpha2) + -1alpha0) + -1var\"b[17]\") + -1alpha1)), 12)\n",
"end\n"
]
},
{
"data": {
"text/plain": [
"model (generic function with 2 methods)"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"turing_model = compile(seeds, data, :DynamicPPL) # the expression is only printed for demonstration"
]
},
{
"cell_type": "markdown",
"id": "b0c85b89-6d19-4e38-a3d4-1808b154fbfe",
"metadata": {},
"source": [
"Inference with `Turing` is simple,"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "09f682b2-94cf-4fe5-9654-d38cd99f74de",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"┌ Info: Found initial step size\n",
"│ ϵ = 0.2\n",
"└ @ Turing.Inference /home/sunxd/.julia/packages/Turing/szPqN/src/inference/hmc.jl:191\n",
"┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
"│ isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
"└ @ AdvancedHMC /home/sunxd/.julia/packages/AdvancedHMC/iWHPQ/src/hamiltonian.jl:47\n",
"\u001b[32mSampling: 100%|█████████████████████████████████████████| Time: 0:00:17\u001b[39m\n"
]
},
{
"data": {
"text/plain": [
"Chains MCMC chain (11000×5×1 Array{Float64, 3}):\n",
"\n",
"Iterations = 1001:1:12000\n",
"Number of chains = 1\n",
"Samples per chain = 11000\n",
"Wall duration = 141.05 seconds\n",
"Compute duration = 141.05 seconds\n",
"parameters = alpha1, alpha12, alpha2, tau, alpha0\n",
"internals = \n",
"\n",
"Summary Statistics\n",
" \u001b[1m parameters \u001b[0m \u001b[1m mean \u001b[0m \u001b[1m std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m mcse \u001b[0m \u001b[1m ess \u001b[0m \u001b[1m rhat \u001b[0m \u001b[1m \u001b[0m ⋯\n",
" \u001b[90m Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m \u001b[0m ⋯\n",
"\n",
" alpha1 0.0827 0.3130 0.0030 0.0046 4686.8932 1.0001 ⋯\n",
" alpha12 -0.8267 0.4340 0.0041 0.0068 4787.3431 1.0004 ⋯\n",
" alpha2 1.3521 0.2709 0.0026 0.0042 5115.3071 1.0005 ⋯\n",
" tau 36.3036 75.5741 0.7206 5.4885 115.2294 1.0185 ⋯\n",
" alpha0 -0.5532 0.1914 0.0018 0.0027 5306.6887 1.0001 ⋯\n",
"\u001b[36m 1 column omitted\u001b[0m\n",
"\n",
"Quantiles\n",
" \u001b[1m parameters \u001b[0m \u001b[1m 2.5% \u001b[0m \u001b[1m 25.0% \u001b[0m \u001b[1m 50.0% \u001b[0m \u001b[1m 75.0% \u001b[0m \u001b[1m 97.5% \u001b[0m\n",
" \u001b[90m Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m\n",
"\n",
" alpha1 -0.5618 -0.1112 0.0874 0.2844 0.6821\n",
" alpha12 -1.7074 -1.1059 -0.8275 -0.5507 0.0154\n",
" alpha2 0.8233 1.1807 1.3432 1.5224 1.8953\n",
" tau 2.7737 7.0629 12.8249 27.4362 296.6706\n",
" alpha0 -0.9274 -0.6750 -0.5549 -0.4323 -0.1643\n"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Turing; chn = sample(turing_model(), NUTS(), 11000, discard_initial = 1000);\n",
"chn[[:alpha0, :alpha1, :alpha12, :alpha2, :tau]]"
]
},
{
"cell_type": "markdown",
"id": "6742e777-bc48-4d45-9ae1-b92169614006",
"metadata": {},
"source": [
"Compare with the result from `MultiBUGS`, we can see the result is very close."
]
},
{
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"
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"
}
},
"cell_type": "markdown",
"id": "a64bb4bb-c1ad-41b1-8c16-7f122f038fd1",
"metadata": {},
"source": [
"![image.png](attachment:af53351e-c7bc-4718-b0ec-2a41569d25c9.png)![image.png](attachment:fa925936-7280-4f86-ab64-8cb2d2463e58.png)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.7.3",
"language": "julia",
"name": "julia-1.7"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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