Julia_in_Colab_RxInfer_for_Message_Passing_and_VB.ipynb

{
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  "metadata": {
    "colab": {
      "provenance": [],
      "include_colab_link": true
    },
    "kernelspec": {
      "display_name": "Julia",
      "language": "julia",
      "name": "julia"
    },
    "language_info": {
      "file_extension": ".jl",
      "mimetype": "application/julia",
      "name": "julia"
    },
    "gpuClass": "standard"
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/apashea/Julia-and-VB-in-Colab/blob/main/Julia_in_Colab_RxInfer_for_Message_Passing_and_VB.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tQ1r1bbb0yBv"
      },
      "source": [
        "# <img src=\"https://github.com/JuliaLang/julia-logo-graphics/raw/master/images/julia-logo-color.png\" height=\"100\" /> _Colab Notebook Template_\n",
        "\n",
        "## Instructions\n",
        "1. Work on a copy of this notebook: _File_ > _Save a copy in Drive_ (you will need a Google account). Alternatively, you can download the notebook using _File_ > _Download .ipynb_, then upload it to [Colab](https://colab.research.google.com/).\n",
        "2. If you need a GPU: _Runtime_ > _Change runtime type_ > _Harware accelerator_ = _GPU_.\n",
        "3. Execute the following cell (click on it and press Ctrl+Enter) to install Julia, IJulia and other packages (if needed, update `JULIA_VERSION` and the other parameters). This takes a couple of minutes.\n",
        "4. Reload this page (press Ctrl+R, or ⌘+R, or the F5 key) and continue to the next section.\n",
        "\n",
        "_Notes_:\n",
        "* If your Colab Runtime gets reset (e.g., due to inactivity), repeat steps 2, 3 and 4.\n",
        "* After installation, if you want to change the Julia version or activate/deactivate the GPU, you will need to reset the Runtime: _Runtime_ > _Factory reset runtime_ and repeat steps 3 and 4."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "GIeFXS0F0zww",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "57180d12-5479-4995-db6a-a5d7bcb9ee26"
      },
      "source": [
        "%%shell\n",
        "set -e\n",
        "\n",
        "#---------------------------------------------------#\n",
        "JULIA_VERSION=\"1.8.2\" # any version ≥ 0.7.0\n",
        "JULIA_PACKAGES=\"IJulia BenchmarkTools\"\n",
        "JULIA_PACKAGES_IF_GPU=\"CUDA\" # or CuArrays for older Julia versions\n",
        "JULIA_NUM_THREADS=2\n",
        "#---------------------------------------------------#\n",
        "\n",
        "if [ -z `which julia` ]; then\n",
        "  # Install Julia\n",
        "  JULIA_VER=`cut -d '.' -f -2 <<< \"$JULIA_VERSION\"`\n",
        "  echo \"Installing Julia $JULIA_VERSION on the current Colab Runtime...\"\n",
        "  BASE_URL=\"https://julialang-s3.julialang.org/bin/linux/x64\"\n",
        "  URL=\"$BASE_URL/$JULIA_VER/julia-$JULIA_VERSION-linux-x86_64.tar.gz\"\n",
        "  wget -nv $URL -O /tmp/julia.tar.gz # -nv means \"not verbose\"\n",
        "  tar -x -f /tmp/julia.tar.gz -C /usr/local --strip-components 1\n",
        "  rm /tmp/julia.tar.gz\n",
        "\n",
        "  # Install Packages\n",
        "  nvidia-smi -L &> /dev/null && export GPU=1 || export GPU=0\n",
        "  if [ $GPU -eq 1 ]; then\n",
        "    JULIA_PACKAGES=\"$JULIA_PACKAGES $JULIA_PACKAGES_IF_GPU\"\n",
        "  fi\n",
        "  for PKG in `echo $JULIA_PACKAGES`; do\n",
        "    echo \"Installing Julia package $PKG...\"\n",
        "    julia -e 'using Pkg; pkg\"add '$PKG'; precompile;\"' &> /dev/null\n",
        "  done\n",
        "\n",
        "  # Install kernel and rename it to \"julia\"\n",
        "  echo \"Installing IJulia kernel...\"\n",
        "  julia -e 'using IJulia; IJulia.installkernel(\"julia\", env=Dict(\n",
        "      \"JULIA_NUM_THREADS\"=>\"'\"$JULIA_NUM_THREADS\"'\"))'\n",
        "  KERNEL_DIR=`julia -e \"using IJulia; print(IJulia.kerneldir())\"`\n",
        "  KERNEL_NAME=`ls -d \"$KERNEL_DIR\"/julia*`\n",
        "  mv -f $KERNEL_NAME \"$KERNEL_DIR\"/julia\n",
        "\n",
        "  echo ''\n",
        "  echo \"Successfully installed `julia -v`!\"\n",
        "  echo \"Please reload this page (press Ctrl+R, ⌘+R, or the F5 key) then\"\n",
        "  echo \"jump to the 'Checking the Installation' section.\"\n",
        "fi"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Installing Julia 1.8.2 on the current Colab Runtime...\n",
            "2024-01-25 14:13:21 URL:https://storage.googleapis.com/julialang2/bin/linux/x64/1.8/julia-1.8.2-linux-x86_64.tar.gz [135859273/135859273] -> \"/tmp/julia.tar.gz\" [1]\n",
            "Installing Julia package IJulia...\n",
            "Installing Julia package BenchmarkTools...\n",
            "Installing IJulia kernel...\n",
            "\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mInstalling julia kernelspec in /root/.local/share/jupyter/kernels/julia-1.8\n",
            "\n",
            "Successfully installed julia version 1.8.2!\n",
            "Please reload this page (press Ctrl+R, ⌘+R, or the F5 key) then\n",
            "jump to the 'Checking the Installation' section.\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": []
          },
          "metadata": {},
          "execution_count": 1
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "-OS3Ac017T1i"
      },
      "source": [
        "# Checking the Installation\n",
        "The `versioninfo()` function should print your Julia version and some other info about the system:"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "EEzvvzCl1i0F",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "3535f675-0d96-4515-f6e5-7ccdda9f9275"
      },
      "source": [
        "versioninfo()"
      ],
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Julia Version 1.8.2\n",
            "Commit 36034abf260 (2022-09-29 15:21 UTC)\n",
            "Platform Info:\n",
            "  OS: Linux (x86_64-linux-gnu)\n",
            "  CPU: 2 × Intel(R) Xeon(R) CPU @ 2.20GHz\n",
            "  WORD_SIZE: 64\n",
            "  LIBM: libopenlibm\n",
            "  LLVM: libLLVM-13.0.1 (ORCJIT, broadwell)\n",
            "  Threads: 2 on 2 virtual cores\n",
            "Environment:\n",
            "  LD_LIBRARY_PATH = /usr/local/nvidia/lib:/usr/local/nvidia/lib64\n",
            "  JULIA_NUM_THREADS = 2\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "using BenchmarkTools\n",
        "\n",
        "M = rand(2^11, 2^11)\n",
        "\n",
        "@btime $M * $M;"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "YjM_qq54lCcs",
        "outputId": "f0cc2a11-cf1a-4ed8-afdd-f9070d189dce"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "  470.409 ms (2 allocations: 32.00 MiB)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "XciCcMAJOT3_",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "77436a76-7ecc-47c3-9adc-4effde178501"
      },
      "source": [
        "try\n",
        "    using CUDA\n",
        "catch\n",
        "    println(\"No GPU found.\")\n",
        "else\n",
        "    run(`nvidia-smi`)\n",
        "    # Create a new random matrix directly on the GPU:\n",
        "    M_on_gpu = CUDA.CURAND.rand(2^11, 2^11)\n",
        "    @btime $M_on_gpu * $M_on_gpu; nothing\n",
        "end"
      ],
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "No GPU found.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8RC1QNNqk6h1"
      },
      "source": [
        "# Need Help?\n",
        "\n",
        "* Learning: https://julialang.org/learning/\n",
        "* Documentation: https://docs.julialang.org/\n",
        "* Questions & Discussions:\n",
        "  * https://discourse.julialang.org/\n",
        "  * http://julialang.slack.com/\n",
        "  * https://stackoverflow.com/questions/tagged/julia\n",
        "\n",
        "If you ever ask for help or file an issue about Julia, you should generally provide the output of `versioninfo()`."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "2UMidUQB03vJ"
      },
      "source": [
        "Add new code cells by clicking the `+ Code` button (or _Insert_ > _Code cell_).\n",
        "\n",
        "Have fun!\n",
        "\n",
        "<img src=\"https://raw.githubusercontent.com/JuliaLang/julia-logo-graphics/master/images/julia-logo-mask.png\" height=\"100\" />"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Message Passing\n",
        "\n",
        "## Tutorial reference: https://www.youtube.com/watch?v=_vVHWzK9NEI\n",
        "\n",
        "- See also: Dimtry Bagaev and Bert de Vries, \"Reactive Message Passing for Scalable Bayesian Inference\", *Hindawi; Scientific Programming* (2023).  https://doi.org/10.1155/2023/6601690"
      ],
      "metadata": {
        "id": "W8b0mkTy5-r1"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import Pkg; Pkg.add(\"RxInfer\")   # ~3min install"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vvsdzRO16F6q",
        "outputId": "b673539a-3bc9-41c9-e02b-a61fb4a8a590"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
            "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SIMDTypes ──────────────────────── v0.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Calculus ───────────────────────── v0.5.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m IrrationalConstants ────────────── v0.2.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m BitTwiddlingConvenienceFunctions ─ v0.1.5\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m MatrixFactorizations ───────────── v1.0.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Adapt ──────────────────────────── v3.7.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DiffRules ──────────────────────── v1.15.1\n",
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            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DualNumbers ────────────────────── v0.6.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Rmath ──────────────────────────── v0.7.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m HCubature ──────────────────────── v1.5.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FastCholesky ───────────────────── v1.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LayoutPointers ─────────────────── v0.1.15\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m HypergeometricFunctions ────────── v0.3.23\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m IntervalSets ───────────────────── v0.7.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m VectorizationBase ──────────────── v0.21.65\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CpuId ──────────────────────────── v0.3.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StatsFuns ──────────────────────── v1.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m PositiveFactorizations ─────────── v0.2.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m GraphPPL ───────────────────────── v3.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CompositeTypes ─────────────────── v0.1.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DiffResults ────────────────────── v1.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SpecialFunctions ───────────────── v2.3.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FastGaussQuadrature ────────────── v0.5.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m GaussQuadrature ────────────────── v0.5.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DomainSets ─────────────────────── v0.6.7\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m PDMats ─────────────────────────── v0.11.31\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m IfElse ─────────────────────────── v0.1.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CPUSummary ─────────────────────── v0.2.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DataAPI ────────────────────────── v1.16.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Unrolled ───────────────────────── v0.1.5\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ProgressMeter ──────────────────── v1.9.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StaticArraysCore ───────────────── v1.4.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Setfield ───────────────────────── v1.1.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LoopVectorization ──────────────── v0.12.166\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StaticArrays ───────────────────── v1.9.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m NaNMath ────────────────────────── v1.0.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Optim ──────────────────────────── v1.7.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ConstructionBase ───────────────── v1.5.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m HostCPUFeatures ────────────────── v0.1.16\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m OrderedCollections ─────────────── v1.6.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DomainIntegrals ────────────────── v0.4.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ThreadingUtilities ─────────────── v0.5.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ManualMemory ───────────────────── v0.1.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m NLSolversBase ──────────────────── v7.8.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Combinatorics ──────────────────── v1.0.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ChainRulesCore ─────────────────── v1.20.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m PolyesterWeave ─────────────────── v0.2.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ArrayInterface ─────────────────── v7.5.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m QuadGK ─────────────────────────── v2.9.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ForwardDiff ────────────────────── v0.10.36\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FillArrays ─────────────────────── v1.9.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LineSearches ───────────────────── v7.2.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Rocket ─────────────────────────── v1.7.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Reexport ───────────────────────── v1.2.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Rmath_jll ──────────────────────── v0.4.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LazyArrays ─────────────────────── v1.5.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DataStructures ─────────────────── v0.18.16\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CommonSubexpressions ───────────── v0.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Requires ───────────────────────── v1.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LogExpFunctions ────────────────── v0.3.26\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ReactiveMP ─────────────────────── v3.11.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CloseOpenIntervals ─────────────── v0.1.12\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TinyHugeNumbers ────────────────── v1.0.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m MacroTools ─────────────────────── v0.5.13\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Static ─────────────────────────── v0.8.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SLEEFPirates ───────────────────── v0.6.42\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StatsAPI ───────────────────────── v1.7.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m OpenSpecFun_jll ────────────────── v0.5.5+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Compat ─────────────────────────── v4.12.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m UnPack ─────────────────────────── v1.0.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Distributions ──────────────────── v0.25.107\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Parameters ─────────────────────── v0.12.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m InverseFunctions ───────────────── v0.1.12\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StaticArrayInterface ───────────── v1.5.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TupleTools ─────────────────────── v1.4.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SortingAlgorithms ──────────────── v1.2.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Missings ───────────────────────── v1.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StatsBase ──────────────────────── v0.34.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m MatrixCorrectionTools ──────────── v1.2.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DensityInterface ───────────────── v0.4.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ArrayLayouts ───────────────────── v1.5.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FiniteDiff ─────────────────────── v2.22.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DocStringExtensions ────────────── v0.9.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m RxInfer ────────────────────────── v2.13.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ChangesOfVariables ─────────────── v0.1.8\n",
            "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.8/Project.toml`\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [86711068] \u001b[39m\u001b[92m+ RxInfer v2.13.2\u001b[39m\n",
            "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.8/Manifest.toml`\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [79e6a3ab] \u001b[39m\u001b[92m+ Adapt v3.7.2\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [4fba245c] \u001b[39m\u001b[92m+ ArrayInterface v7.5.1\u001b[39m\n",
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            "\u001b[32m⌃\u001b[39m\u001b[90m [cc6bae93] \u001b[39m\u001b[92m+ DomainIntegrals v0.4.4\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [5b8099bc] \u001b[39m\u001b[92m+ DomainSets v0.6.7\u001b[39m\n",
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            "\u001b[33m⌅\u001b[39m\u001b[90m [5078a376] \u001b[39m\u001b[92m+ LazyArrays v1.5.2\u001b[39m\n",
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            "\u001b[33m⌅\u001b[39m\u001b[90m [a3b82374] \u001b[39m\u001b[92m+ MatrixFactorizations v1.0.0\u001b[39m\n",
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            "\u001b[33m⌅\u001b[39m\u001b[90m [a194aa59] \u001b[39m\u001b[92m+ ReactiveMP v3.11.1\u001b[39m\n",
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            " \u001b[90m [ae029012] \u001b[39m\u001b[92m+ Requires v1.3.0\u001b[39m\n",
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            "\u001b[33m⌅\u001b[39m\u001b[90m [86711068] \u001b[39m\u001b[92m+ RxInfer v2.13.2\u001b[39m\n",
            " \u001b[90m [94e857df] \u001b[39m\u001b[92m+ SIMDTypes v0.1.0\u001b[39m\n",
            " \u001b[90m [476501e8] \u001b[39m\u001b[92m+ SLEEFPirates v0.6.42\u001b[39m\n",
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            " \u001b[90m [aedffcd0] \u001b[39m\u001b[92m+ Static v0.8.8\u001b[39m\n",
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            " \u001b[90m [90137ffa] \u001b[39m\u001b[92m+ StaticArrays v1.9.1\u001b[39m\n",
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            " \u001b[90m [783c9a47] \u001b[39m\u001b[92m+ TinyHugeNumbers v1.0.0\u001b[39m\n",
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            " \u001b[90m [8ba89e20] \u001b[39m\u001b[92m+ Distributed\u001b[39m\n",
            " \u001b[90m [9fa8497b] \u001b[39m\u001b[92m+ Future\u001b[39m\n",
            " \u001b[90m [4607b0f0] \u001b[39m\u001b[92m+ SuiteSparse\u001b[39m\n",
            " \u001b[90m [05823500] \u001b[39m\u001b[92m+ OpenLibm_jll v0.8.1+0\u001b[39m\n",
            "\u001b[36m\u001b[1m        Info\u001b[22m\u001b[39m Packages marked with \u001b[32m⌃\u001b[39m and \u001b[33m⌅\u001b[39m have new versions available, but those with \u001b[33m⌅\u001b[39m are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`\n",
            "\u001b[32m\u001b[1mPrecompiling\u001b[22m\u001b[39m project...\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mCompositeTypes\u001b[39m\n",
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            "\u001b[32m  ✓ \u001b[39m\u001b[90mPolyesterWeave\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mSpecialFunctions\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mFiniteDiff\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mHCubature\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mCloseOpenIntervals\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mLayoutPointers\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mDomainSets\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mGaussQuadrature\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mDualNumbers\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mDiffRules\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mFastGaussQuadrature\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mHypergeometricFunctions\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mArrayLayouts\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mDomainIntegrals\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mForwardDiff\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mStatsFuns\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mNLSolversBase\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mMatrixFactorizations\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mVectorizationBase\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mLineSearches\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mDistributions\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mSLEEFPirates\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mOptim\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mLazyArrays\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mLoopVectorization\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39m\u001b[90mReactiveMP\u001b[39m\n",
            "\u001b[32m  ✓ \u001b[39mRxInfer\n",
            "  87 dependencies successfully precompiled in 217 seconds. 20 already precompiled.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import Pkg; Pkg.add(\"StatsPlots\")   # ~5min install"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "9BamX82h8FzR",
        "outputId": "c3d54dde-f333-4a28-d0b3-f5883eeca479"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stderr",
          "text": [
            "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m JpegTurbo_jll ──────────────── v3.0.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LERC_jll ───────────────────── v3.0.0+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m x265_jll ───────────────────── v3.5.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libfdk_aac_jll ─────────────── v2.0.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m GR_jll ─────────────────────── v0.72.10+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LoggingExtras ──────────────── v1.0.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FFTW ───────────────────────── v1.8.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m RelocatableFolders ─────────── v1.0.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Opus_jll ───────────────────── v1.3.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xkbcomp_jll ───────────── v1.4.6+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Unitful ────────────────────── v1.19.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Measures ───────────────────── v0.3.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Grisu ──────────────────────── v1.0.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ConcurrentUtilities ────────── v2.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Contour ────────────────────── v0.6.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m NearestNeighbors ───────────── v0.4.16\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Formatting ─────────────────── v0.4.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_wm_jll ───────── v0.4.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_image_jll ────── v0.4.0+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m PlotUtils ──────────────────── v1.4.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m RecipesPipeline ────────────── v0.6.12\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libpthread_stubs_jll ──── v0.1.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libSM_jll ─────────────── v1.2.3+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m OpenSSL ────────────────────── v1.4.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Cairo_jll ──────────────────── v1.16.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m HTTP ───────────────────────── v1.10.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Fontconfig_jll ─────────────── v2.13.93+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_jll ──────────── v0.4.0+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m IntelOpenMP_jll ────────────── v2024.0.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libgpg_error_jll ───────────── v1.42.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libxkbfile_jll ────────── v1.1.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m EpollShim_jll ──────────────── v0.0.20230411+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXau_jll ────────────── v1.0.11+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXinerama_jll ───────── v1.1.4+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FFMPEG ─────────────────────── v0.4.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Distances ──────────────────── v0.10.11\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_keysyms_jll ──── v0.4.0+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Showoff ────────────────────── v1.0.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Bzip2_jll ──────────────────── v1.0.8+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m xkbcommon_jll ──────────────── v1.4.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Clustering ─────────────────── v0.15.7\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Pipe ───────────────────────── v1.3.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m XZ_jll ─────────────────────── v5.4.5+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SimpleBufferStream ─────────── v1.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m SentinelArrays ─────────────── v1.4.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m HarfBuzz_jll ───────────────── v2.8.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LZO_jll ────────────────────── v2.10.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m PlotThemes ─────────────────── v3.1.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m fzf_jll ────────────────────── v0.43.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TranscodingStreams ─────────── v0.10.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FriBidi_jll ────────────────── v1.0.10+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m GLFW_jll ───────────────────── v3.3.9+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m UnicodeFun ─────────────────── v0.4.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Ratios ─────────────────────── v0.4.5\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m x264_jll ───────────────────── v2021.5.5+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FreeType2_jll ──────────────── v2.13.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ColorSchemes ───────────────── v3.24.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m CodecZlib ──────────────────── v0.7.3\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m JLFzf ──────────────────────── v0.1.7\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Colors ─────────────────────── v0.12.10\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libpng_jll ─────────────────── v1.6.40+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libxcb_jll ────────────── v1.15.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libaom_jll ─────────────────── v3.4.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m mtdev_jll ──────────────────── v1.1.6+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ExceptionUnwrapping ────────── v0.1.10\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m gperf_jll ──────────────────── v3.1.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m eudev_jll ──────────────────── v3.2.9+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m StatsPlots ─────────────────── v0.15.6\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ColorTypes ─────────────────── v0.11.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Scratch ────────────────────── v1.2.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Plots ──────────────────────── v1.39.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m GR ─────────────────────────── v0.72.10\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXext_jll ───────────── v1.3.4+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Zstd_jll ───────────────────── v1.5.5+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_cursor_jll ───── v0.1.4+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TensorCore ─────────────────── v0.1.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TableTraits ────────────────── v1.0.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Expat_jll ──────────────────── v2.5.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m MKL_jll ────────────────────── v2024.0.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libtiff_jll ────────────────── v4.5.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m DataValueInterfaces ────────── v1.0.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libffi_jll ─────────────────── v3.2.2+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m ColorVectorSpace ───────────── v0.10.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m AbstractFFTs ───────────────── v1.5.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libmount_jll ───────────────── v2.35.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXrender_jll ────────── v0.9.10+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libevdev_jll ───────────────── v1.11.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libinput_jll ───────────────── v1.18.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Vulkan_Loader_jll ──────────── v1.3.243+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXi_jll ─────────────── v1.7.10+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Ogg_jll ────────────────────── v1.3.5+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m KernelDensity ──────────────── v0.6.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXcursor_jll ────────── v1.2.0+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libuuid_jll ────────────────── v2.36.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m XSLT_jll ───────────────────── v1.1.34+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Wayland_protocols_jll ──────── v1.31.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FFTW_jll ───────────────────── v3.3.10+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Widgets ────────────────────── v0.6.6\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m MultivariateStats ──────────── v0.10.2\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xcb_util_renderutil_jll ─ v0.3.9+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libICE_jll ────────────── v1.0.10+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Pixman_jll ─────────────────── v0.42.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Graphite2_jll ──────────────── v1.3.14+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libass_jll ─────────────────── v0.15.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m XML2_jll ───────────────────── v2.12.2+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Latexify ───────────────────── v0.16.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Gettext_jll ────────────────── v0.21.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Wayland_jll ────────────────── v1.21.0+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xtrans_jll ────────────── v1.5.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m OpenSSL_jll ────────────────── v3.0.12+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m BitFlags ───────────────────── v0.1.8\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Arpack_jll ─────────────────── v3.5.1+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FFMPEG_jll ─────────────────── v4.4.4+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libgcrypt_jll ──────────────── v1.8.7+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_xkeyboard_config_jll ──── v2.39.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXrandr_jll ─────────── v1.5.2+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LLVMOpenMP_jll ─────────────── v15.0.7+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m FixedPointNumbers ──────────── v0.8.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m RecipesBase ────────────────── v1.3.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Tables ─────────────────────── v1.11.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m TableOperations ────────────── v1.2.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXfixes_jll ─────────── v5.0.3+4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LAME_jll ───────────────────── v3.100.1+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Interpolations ─────────────── v0.14.7\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Qt6Base_jll ────────────────── v6.5.3+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m IteratorInterfaceExtensions ── v1.0.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libiconv_jll ───────────────── v1.17.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m URIs ───────────────────────── v1.5.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Glib_jll ───────────────────── v2.76.5+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m LaTeXStrings ───────────────── v1.3.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m libvorbis_jll ──────────────── v1.3.7+1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Arpack ─────────────────────── v0.5.4\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Libglvnd_jll ───────────────── v1.6.0+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libX11_jll ────────────── v1.8.6+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Xorg_libXdmcp_jll ──────────── v1.1.4+0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m AxisAlgorithms ─────────────── v1.0.1\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Unzip ──────────────────────── v0.2.0\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m WoodburyMatrices ───────────── v0.5.6\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m Observables ────────────────── v0.5.5\n",
            "\u001b[32m\u001b[1m   Installed\u001b[22m\u001b[39m UnitfulLatexify ────────────── v1.6.3\n",
            "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.8/Project.toml`\n",
            " \u001b[90m [f3b207a7] \u001b[39m\u001b[92m+ StatsPlots v0.15.6\u001b[39m\n",
            "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.8/Manifest.toml`\n",
            " \u001b[90m [621f4979] \u001b[39m\u001b[92m+ AbstractFFTs v1.5.0\u001b[39m\n",
            " \u001b[90m [7d9fca2a] \u001b[39m\u001b[92m+ Arpack v0.5.4\u001b[39m\n",
            "\u001b[32m⌃\u001b[39m\u001b[90m [13072b0f] \u001b[39m\u001b[92m+ AxisAlgorithms v1.0.1\u001b[39m\n",
            " \u001b[90m [d1d4a3ce] \u001b[39m\u001b[92m+ BitFlags v0.1.8\u001b[39m\n",
            " \u001b[90m [aaaa29a8] \u001b[39m\u001b[92m+ Clustering v0.15.7\u001b[39m\n",
            " \u001b[90m [944b1d66] \u001b[39m\u001b[92m+ CodecZlib v0.7.3\u001b[39m\n",
            " \u001b[90m [35d6a980] \u001b[39m\u001b[92m+ ColorSchemes v3.24.0\u001b[39m\n",
            " \u001b[90m [3da002f7] \u001b[39m\u001b[92m+ ColorTypes v0.11.4\u001b[39m\n",
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            " \u001b[90m [5ae59095] \u001b[39m\u001b[92m+ Colors v0.12.10\u001b[39m\n",
            " \u001b[90m [f0e56b4a] \u001b[39m\u001b[92m+ ConcurrentUtilities v2.3.0\u001b[39m\n",
            " \u001b[90m [d38c429a] \u001b[39m\u001b[92m+ Contour v0.6.2\u001b[39m\n",
            " \u001b[90m [e2d170a0] \u001b[39m\u001b[92m+ DataValueInterfaces v1.0.0\u001b[39m\n",
            " \u001b[90m [b4f34e82] \u001b[39m\u001b[92m+ Distances v0.10.11\u001b[39m\n",
            " \u001b[90m [460bff9d] \u001b[39m\u001b[92m+ ExceptionUnwrapping v0.1.10\u001b[39m\n",
            " \u001b[90m [c87230d0] \u001b[39m\u001b[92m+ FFMPEG v0.4.1\u001b[39m\n",
            " \u001b[90m [7a1cc6ca] \u001b[39m\u001b[92m+ FFTW v1.8.0\u001b[39m\n",
            " \u001b[90m [53c48c17] \u001b[39m\u001b[92m+ FixedPointNumbers v0.8.4\u001b[39m\n",
            " \u001b[90m [59287772] \u001b[39m\u001b[92m+ Formatting v0.4.2\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [28b8d3ca] \u001b[39m\u001b[92m+ GR v0.72.10\u001b[39m\n",
            " \u001b[90m [42e2da0e] \u001b[39m\u001b[92m+ Grisu v1.0.2\u001b[39m\n",
            " \u001b[90m [cd3eb016] \u001b[39m\u001b[92m+ HTTP v1.10.1\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [a98d9a8b] \u001b[39m\u001b[92m+ Interpolations v0.14.7\u001b[39m\n",
            " \u001b[90m [82899510] \u001b[39m\u001b[92m+ IteratorInterfaceExtensions v1.0.0\u001b[39m\n",
            " \u001b[90m [1019f520] \u001b[39m\u001b[92m+ JLFzf v0.1.7\u001b[39m\n",
            " \u001b[90m [5ab0869b] \u001b[39m\u001b[92m+ KernelDensity v0.6.8\u001b[39m\n",
            " \u001b[90m [b964fa9f] \u001b[39m\u001b[92m+ LaTeXStrings v1.3.1\u001b[39m\n",
            " \u001b[90m [23fbe1c1] \u001b[39m\u001b[92m+ Latexify v0.16.1\u001b[39m\n",
            " \u001b[90m [e6f89c97] \u001b[39m\u001b[92m+ LoggingExtras v1.0.3\u001b[39m\n",
            " \u001b[90m [442fdcdd] \u001b[39m\u001b[92m+ Measures v0.3.2\u001b[39m\n",
            " \u001b[90m [6f286f6a] \u001b[39m\u001b[92m+ MultivariateStats v0.10.2\u001b[39m\n",
            " \u001b[90m [b8a86587] \u001b[39m\u001b[92m+ NearestNeighbors v0.4.16\u001b[39m\n",
            " \u001b[90m [510215fc] \u001b[39m\u001b[92m+ Observables v0.5.5\u001b[39m\n",
            " \u001b[90m [4d8831e6] \u001b[39m\u001b[92m+ OpenSSL v1.4.1\u001b[39m\n",
            " \u001b[90m [b98c9c47] \u001b[39m\u001b[92m+ Pipe v1.3.0\u001b[39m\n",
            " \u001b[90m [ccf2f8ad] \u001b[39m\u001b[92m+ PlotThemes v3.1.0\u001b[39m\n",
            " \u001b[90m [995b91a9] \u001b[39m\u001b[92m+ PlotUtils v1.4.0\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [91a5bcdd] \u001b[39m\u001b[92m+ Plots v1.39.0\u001b[39m\n",
            " \u001b[90m [c84ed2f1] \u001b[39m\u001b[92m+ Ratios v0.4.5\u001b[39m\n",
            " \u001b[90m [3cdcf5f2] \u001b[39m\u001b[92m+ RecipesBase v1.3.4\u001b[39m\n",
            " \u001b[90m [01d81517] \u001b[39m\u001b[92m+ RecipesPipeline v0.6.12\u001b[39m\n",
            " \u001b[90m [05181044] \u001b[39m\u001b[92m+ RelocatableFolders v1.0.1\u001b[39m\n",
            " \u001b[90m [6c6a2e73] \u001b[39m\u001b[92m+ Scratch v1.2.1\u001b[39m\n",
            " \u001b[90m [91c51154] \u001b[39m\u001b[92m+ SentinelArrays v1.4.1\u001b[39m\n",
            " \u001b[90m [992d4aef] \u001b[39m\u001b[92m+ Showoff v1.0.3\u001b[39m\n",
            " \u001b[90m [777ac1f9] \u001b[39m\u001b[92m+ SimpleBufferStream v1.1.0\u001b[39m\n",
            " \u001b[90m [f3b207a7] \u001b[39m\u001b[92m+ StatsPlots v0.15.6\u001b[39m\n",
            " \u001b[90m [ab02a1b2] \u001b[39m\u001b[92m+ TableOperations v1.2.0\u001b[39m\n",
            " \u001b[90m [3783bdb8] \u001b[39m\u001b[92m+ TableTraits v1.0.1\u001b[39m\n",
            " \u001b[90m [bd369af6] \u001b[39m\u001b[92m+ Tables v1.11.1\u001b[39m\n",
            " \u001b[90m [62fd8b95] \u001b[39m\u001b[92m+ TensorCore v0.1.1\u001b[39m\n",
            " \u001b[90m [3bb67fe8] \u001b[39m\u001b[92m+ TranscodingStreams v0.10.2\u001b[39m\n",
            " \u001b[90m [5c2747f8] \u001b[39m\u001b[92m+ URIs v1.5.1\u001b[39m\n",
            " \u001b[90m [1cfade01] \u001b[39m\u001b[92m+ UnicodeFun v0.4.1\u001b[39m\n",
            " \u001b[90m [1986cc42] \u001b[39m\u001b[92m+ Unitful v1.19.0\u001b[39m\n",
            " \u001b[90m [45397f5d] \u001b[39m\u001b[92m+ UnitfulLatexify v1.6.3\u001b[39m\n",
            " \u001b[90m [41fe7b60] \u001b[39m\u001b[92m+ Unzip v0.2.0\u001b[39m\n",
            " \u001b[90m [cc8bc4a8] \u001b[39m\u001b[92m+ Widgets v0.6.6\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [efce3f68] \u001b[39m\u001b[92m+ WoodburyMatrices v0.5.6\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [68821587] \u001b[39m\u001b[92m+ Arpack_jll v3.5.1+1\u001b[39m\n",
            " \u001b[90m [6e34b625] \u001b[39m\u001b[92m+ Bzip2_jll v1.0.8+1\u001b[39m\n",
            " \u001b[90m [83423d85] \u001b[39m\u001b[92m+ Cairo_jll v1.16.1+1\u001b[39m\n",
            " \u001b[90m [2702e6a9] \u001b[39m\u001b[92m+ EpollShim_jll v0.0.20230411+0\u001b[39m\n",
            " \u001b[90m [2e619515] \u001b[39m\u001b[92m+ Expat_jll v2.5.0+0\u001b[39m\n",
            " \u001b[90m [b22a6f82] \u001b[39m\u001b[92m+ FFMPEG_jll v4.4.4+1\u001b[39m\n",
            " \u001b[90m [f5851436] \u001b[39m\u001b[92m+ FFTW_jll v3.3.10+0\u001b[39m\n",
            " \u001b[90m [a3f928ae] \u001b[39m\u001b[92m+ Fontconfig_jll v2.13.93+0\u001b[39m\n",
            " \u001b[90m [d7e528f0] \u001b[39m\u001b[92m+ FreeType2_jll v2.13.1+0\u001b[39m\n",
            " \u001b[90m [559328eb] \u001b[39m\u001b[92m+ FriBidi_jll v1.0.10+0\u001b[39m\n",
            " \u001b[90m [0656b61e] \u001b[39m\u001b[92m+ GLFW_jll v3.3.9+0\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [d2c73de3] \u001b[39m\u001b[92m+ GR_jll v0.72.10+0\u001b[39m\n",
            " \u001b[90m [78b55507] \u001b[39m\u001b[92m+ Gettext_jll v0.21.0+0\u001b[39m\n",
            " \u001b[90m [7746bdde] \u001b[39m\u001b[92m+ Glib_jll v2.76.5+0\u001b[39m\n",
            " \u001b[90m [3b182d85] \u001b[39m\u001b[92m+ Graphite2_jll v1.3.14+0\u001b[39m\n",
            " \u001b[90m [2e76f6c2] \u001b[39m\u001b[92m+ HarfBuzz_jll v2.8.1+1\u001b[39m\n",
            " \u001b[90m [1d5cc7b8] \u001b[39m\u001b[92m+ IntelOpenMP_jll v2024.0.2+0\u001b[39m\n",
            " \u001b[90m [aacddb02] \u001b[39m\u001b[92m+ JpegTurbo_jll v3.0.1+0\u001b[39m\n",
            " \u001b[90m [c1c5ebd0] \u001b[39m\u001b[92m+ LAME_jll v3.100.1+0\u001b[39m\n",
            " \u001b[90m [88015f11] \u001b[39m\u001b[92m+ LERC_jll v3.0.0+1\u001b[39m\n",
            " \u001b[90m [1d63c593] \u001b[39m\u001b[92m+ LLVMOpenMP_jll v15.0.7+0\u001b[39m\n",
            " \u001b[90m [dd4b983a] \u001b[39m\u001b[92m+ LZO_jll v2.10.1+0\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [e9f186c6] \u001b[39m\u001b[92m+ Libffi_jll v3.2.2+1\u001b[39m\n",
            " \u001b[90m [d4300ac3] \u001b[39m\u001b[92m+ Libgcrypt_jll v1.8.7+0\u001b[39m\n",
            " \u001b[90m [7e76a0d4] \u001b[39m\u001b[92m+ Libglvnd_jll v1.6.0+0\u001b[39m\n",
            " \u001b[90m [7add5ba3] \u001b[39m\u001b[92m+ Libgpg_error_jll v1.42.0+0\u001b[39m\n",
            " \u001b[90m [94ce4f54] \u001b[39m\u001b[92m+ Libiconv_jll v1.17.0+0\u001b[39m\n",
            " \u001b[90m [4b2f31a3] \u001b[39m\u001b[92m+ Libmount_jll v2.35.0+0\u001b[39m\n",
            "\u001b[33m⌅\u001b[39m\u001b[90m [89763e89] \u001b[39m\u001b[92m+ Libtiff_jll v4.5.1+1\u001b[39m\n",
            " \u001b[90m [38a345b3] \u001b[39m\u001b[92m+ Libuuid_jll v2.36.0+0\u001b[39m\n",
            " \u001b[90m [856f044c] \u001b[39m\u001b[92m+ MKL_jll v2024.0.0+0\u001b[39m\n",
            " \u001b[90m [e7412a2a] \u001b[39m\u001b[92m+ Ogg_jll v1.3.5+1\u001b[39m\n",
            " \u001b[90m [458c3c95] \u001b[39m\u001b[92m+ OpenSSL_jll v3.0.12+0\u001b[39m\n",
            " \u001b[90m [91d4177d] \u001b[39m\u001b[92m+ Opus_jll v1.3.2+0\u001b[39m\n",
            " \u001b[90m [30392449] \u001b[39m\u001b[92m+ Pixman_jll v0.42.2+0\u001b[39m\n",
            " \u001b[90m [c0090381] \u001b[39m\u001b[92m+ Qt6Base_jll v6.5.3+1\u001b[39m\n",
            " \u001b[90m [a44049a8] \u001b[39m\u001b[92m+ Vulkan_Loader_jll v1.3.243+0\u001b[39m\n",
            " \u001b[90m [a2964d1f] \u001b[39m\u001b[92m+ Wayland_jll v1.21.0+1\u001b[39m\n",
            " \u001b[90m [2381bf8a] \u001b[39m\u001b[92m+ Wayland_protocols_jll v1.31.0+0\u001b[39m\n",
            " \u001b[90m [02c8fc9c] \u001b[39m\u001b[92m+ XML2_jll v2.12.2+0\u001b[39m\n",
            " \u001b[90m [aed1982a] \u001b[39m\u001b[92m+ XSLT_jll v1.1.34+0\u001b[39m\n",
            " \u001b[90m [ffd25f8a] \u001b[39m\u001b[92m+ XZ_jll v5.4.5+0\u001b[39m\n",
            " \u001b[90m [f67eecfb] \u001b[39m\u001b[92m+ Xorg_libICE_jll v1.0.10+1\u001b[39m\n",
            " \u001b[90m [c834827a] \u001b[39m\u001b[92m+ Xorg_libSM_jll v1.2.3+0\u001b[39m\n",
            " \u001b[90m [4f6342f7] \u001b[39m\u001b[92m+ Xorg_libX11_jll v1.8.6+0\u001b[39m\n",
            " \u001b[90m [0c0b7dd1] \u001b[39m\u001b[92m+ Xorg_libXau_jll v1.0.11+0\u001b[39m\n",
            " \u001b[90m [935fb764] \u001b[39m\u001b[92m+ Xorg_libXcursor_jll v1.2.0+4\u001b[39m\n",
            " \u001b[90m [a3789734] \u001b[39m\u001b[92m+ Xorg_libXdmcp_jll v1.1.4+0\u001b[39m\n",
            " \u001b[90m [1082639a] \u001b[39m\u001b[92m+ Xorg_libXext_jll v1.3.4+4\u001b[39m\n",
            " \u001b[90m [d091e8ba] \u001b[39m\u001b[92m+ Xorg_libXfixes_jll v5.0.3+4\u001b[39m\n",
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            " \u001b[90m [d1454406] \u001b[39m\u001b[92m+ Xorg_libXinerama_jll v1.1.4+4\u001b[39m\n",
            " \u001b[90m [ec84b674] \u001b[39m\u001b[92m+ Xorg_libXrandr_jll v1.5.2+4\u001b[39m\n",
            " \u001b[90m [ea2f1a96] \u001b[39m\u001b[92m+ Xorg_libXrender_jll v0.9.10+4\u001b[39m\n",
            " \u001b[90m [14d82f49] \u001b[39m\u001b[92m+ Xorg_libpthread_stubs_jll v0.1.1+0\u001b[39m\n",
            " \u001b[90m [c7cfdc94] \u001b[39m\u001b[92m+ Xorg_libxcb_jll v1.15.0+0\u001b[39m\n",
            " \u001b[90m [cc61e674] \u001b[39m\u001b[92m+ Xorg_libxkbfile_jll v1.1.2+0\u001b[39m\n",
            " \u001b[90m [e920d4aa] \u001b[39m\u001b[92m+ Xorg_xcb_util_cursor_jll v0.1.4+0\u001b[39m\n",
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            " \u001b[90m [975044d2] \u001b[39m\u001b[92m+ Xorg_xcb_util_keysyms_jll v0.4.0+1\u001b[39m\n",
            " \u001b[90m [0d47668e] \u001b[39m\u001b[92m+ Xorg_xcb_util_renderutil_jll v0.3.9+1\u001b[39m\n",
            " \u001b[90m [c22f9ab0] \u001b[39m\u001b[92m+ Xorg_xcb_util_wm_jll v0.4.1+1\u001b[39m\n",
            " \u001b[90m [35661453] \u001b[39m\u001b[92m+ Xorg_xkbcomp_jll v1.4.6+0\u001b[39m\n",
            " \u001b[90m [33bec58e] \u001b[39m\u001b[92m+ Xorg_xkeyboard_config_jll v2.39.0+0\u001b[39m\n",
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            "\u001b[36m\u001b[1m        Info\u001b[22m\u001b[39m Packages marked with \u001b[32m⌃\u001b[39m and \u001b[33m⌅\u001b[39m have new versions available, but those with \u001b[33m⌅\u001b[39m are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`\n",
            "\u001b[32m\u001b[1mPrecompiling\u001b[22m\u001b[39m project...\n",
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            "\u001b[32m  ✓ \u001b[39mStatsPlots\n",
            "  141 dependencies successfully precompiled in 237 seconds. 107 already precompiled.\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# load packages you will be using, set random seed\n",
        "using RxInfer, StatsPlots\n",
        "using Random; Random.seed!(1)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8o3Os0Ww6qdn",
        "outputId": "c862f044-32b4-43e4-87af-5b2be4931976"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "TaskLocalRNG()"
            ]
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# generate random data (ex. coin tosses with  outcomes {heads, tails})\n",
        "\n",
        "true_success_rate = 0.75\n",
        "distribution = Bernoulli(true_success_rate)\n",
        "tosses = 10\n",
        "myobservations = rand(distribution, tosses)\n",
        "heads = sum(myobservations)\n",
        "println(heads)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "AwKAKyNGAhod",
        "outputId": "30e426e1-345f-4cdd-bc7c-7f90f6f5fc61"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "7\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# model\n",
        "\n",
        "# defining a model i.e. Forney-style factor graph (FFG) which will perform message passing\n",
        "\n",
        "# RxInfer takes in an observation to update the prior to a posterior, then uses latter as the\n",
        "# new prior when computing the posterior given the next observation\n",
        "\n",
        "\n",
        "@model function mymodel(tosses)\n",
        "  # prior\n",
        "  success_rate ~ Beta(1, 1)   # assign distribution to our prior, here Beta(alpha, beta) = Beta(n_successes, n_failures)\n",
        "  # likelihood\n",
        "  observations = datavar(Bool, tosses)   # 'data variables'; allows passing data thru the model, here Boolean values\n",
        "  for i in 1:tosses    # fill a vector with a likelihood distribution\n",
        "    observations[i] ~ Bernoulli(success_rate)  # defining process (distribution) from which observations are derived\n",
        "  end\n",
        "end"
      ],
      "metadata": {
        "id": "Q6oKRcBnBGQh"
      },
      "execution_count": 8,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# find posterior distribution\n",
        "\n",
        "model = mymodel(tosses)\n",
        "data = (observations = myobservations, )   # set data to pass, in form of tuple with trailing comma\n",
        "result = inference(model = model, data = data)  # use RxInfer's inference() function\n",
        "\n",
        "post = result.posteriors[:success_rate]   # store posteriors in a variable\n",
        "println(post)   # returns an inferred Beta distribution B(8.0, 4.0) as the prior after sequentially showing the model 7 heads and 3 tails\n",
        "# Notice our distributions for the prior, likelihood, and posterior are Beta, Bernoulli, and Beta, respectively.\n",
        "# When the prior and posterior distributions are of the same family, these are of\n",
        "# a 'conjugate distribution' (here, both are Beta distributions), where the prior is a 'conjugate prior'.\n",
        "\n",
        "post_mean = mean(post)   # store the mean\n",
        "println(post_mean)    # our updated belief after 10 observations is a success_rate of 2/3\n",
        "post_std = std(post)  # store the standard deviation\n",
        "println(post_std)     # our updated belief after 10 observations is a standard deviation of approx. 0.13"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "E-kHg5zaCkAi",
        "outputId": "4b7b5cbd-d905-4be3-9565-9d298007018d"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Beta{Float64}(α=8.0, β=4.0)\n",
            "0.6666666666666666\n",
            "0.1307440900921227\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# visualization\n",
        "\n",
        "p1 = plot(Beta(1, 1);    # prior (our original Beta(1, 1) i.e. a unform, weakly informative prior)\n",
        "  legend = false,\n",
        "  xlims = (0, 1),\n",
        "  title = \"Prior\",\n",
        "  fill = true,\n",
        "  alpha = 0.3\n",
        ")\n",
        "\n",
        "# plotting posteriors: we can directly plot the output previous result.posteriors[:success_rate]\n",
        "p2 = plot(post;\n",
        "  label = \"\",\n",
        "  xlims = (0, 1),\n",
        "  title = \"Posterior\",\n",
        "  fill = true,\n",
        "  alpha = 0.3,\n",
        "  color = 3\n",
        ")\n",
        "\n",
        "# add vertical line denoting posterior mean found\n",
        "vline!(p2,\n",
        "  [post_mean],\n",
        "  label = \"Posterior Mean\",\n",
        "  color = 1,\n",
        "  linestyle = :dash\n",
        ")\n",
        "\n",
        "# compute std above and below posterior mean, add each as vertical lines\n",
        "std_left = post_mean - post_std\n",
        "std_right = post_mean + post_std\n",
        "vline!(p2,\n",
        "  [std_left, std_right],\n",
        "  label = \"Posterior Std\",\n",
        "  color = 2,\n",
        "  linestyle = :dash\n",
        ")\n",
        "# add true_success_rate as verticle line\n",
        "vline!(p2,\n",
        "  [true_success_rate],\n",
        "  label = \"True Success Rate\",\n",
        "  color = :black,\n",
        "  linestyle = :dash\n",
        ")\n",
        "# plot prior plot and posterior plot with vlines together, stacked\n",
        "plot(p1, p2;\n",
        "  layout = (2, 1),\n",
        "  size = (600, 600)\n",
        ")\n",
        "\n",
        "# See that after just 10 observations, the true mean is within 1 standard deviation\n",
        "# around the mean of the posterior distribution"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 621
        },
        "id": "blRWXBoBCkJQ",
        "outputId": "96fc8473-aa56-471a-936b-ff3f1105c629"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
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          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Final thoughts:\n",
        "\n",
        "# Notice how, unlike MCMC, we were able to compute a relatively accurate and conjugate posterior\n",
        "# distribution without needing a random sampler.\n",
        "\n",
        "# RxInfer also supports non-conjugate inference. It also is capable of running various\n",
        "# Bayesian inference algorithms in different parts of the factor graph of a single\n",
        "# probabilistic model."
      ],
      "metadata": {
        "id": "oG4NwygVH3LR"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# VB\n",
        "\n",
        "## Tutorial referenced: https://www.youtube.com/watch?v=qnj0PKc734g\n",
        "____\n",
        "____"
      ],
      "metadata": {
        "id": "DDGalPZRAlFg"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# generate observations\n",
        "true_mean = 3.0\n",
        "true_precision = 5.0    # 1/sigma^2, typically denoted as tau\n",
        "true_sigma = 1 / sqrt(true_precision)   # std\n",
        "true_distribution = Normal(true_mean, true_sigma)\n",
        "n = 100_000    # produce 100000 observations\n",
        "observations = rand(true_distribution, n)   # random samples from Normal distribution"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "eQiQOhZ19X1a",
        "outputId": "b69d1d8e-1c08-4ff0-e3a2-197fc63147aa"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "100000-element Vector{Float64}:\n",
              " 3.027697163475025\n",
              " 3.1245068651603756\n",
              " 2.7335392209183818\n",
              " 3.020866713374945\n",
              " 3.4855818483467274\n",
              " 2.294938732431209\n",
              " 3.078682756101985\n",
              " 3.3870100614637537\n",
              " 1.7521483990530564\n",
              " 2.1538649194639294\n",
              " 3.0974383553296065\n",
              " 2.706649991077448\n",
              " 3.1196359187656606\n",
              " ⋮\n",
              " 3.256219671340333\n",
              " 2.865511272641008\n",
              " 3.195018853884503\n",
              " 2.815648996383269\n",
              " 3.4348245931002093\n",
              " 2.874194232645634\n",
              " 2.9340230948674932\n",
              " 2.406729626842361\n",
              " 2.5373781580477295\n",
              " 3.3383339253743394\n",
              " 2.7023855908881953\n",
              " 2.9673882423239664"
            ]
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# visualize observations\n",
        "p_observations = histogram(observations,\n",
        "  bins = 1.0:0.1:5.0,\n",
        "  title = \"Observations\",\n",
        "  legend = false,\n",
        "  color = 2,\n",
        "  alpha = 0.5\n",
        ")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 421
        },
        "id": "M-Lu7jtI9JyB",
        "outputId": "618f738d-fbac-445c-b6de-2471b85d2080"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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          },
          "metadata": {},
          "execution_count": 11
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Model\n",
        "@model function mymodel(n)\n",
        "  # prior\n",
        "  # Following two lines determine the factor graph (model)\n",
        "  mu ~ NormalMeanPrecision(0.0, 1.0)   # mean parameter (via RxInfer.jl; args=mean,precision)\n",
        "  tau ~ GammaShapeRate(1.0, 1.0)       # precision parameter (values randomly drawn from gamma dist; args=shape, rate; where rate=1/scale)\n",
        "  # likelihood\n",
        "  y = datavar(Float64, n)   # establish data variable (observations) y\n",
        "  for i in 1:n    # loop to fill a vector of distributions\n",
        "    y[i] ~ NormalMeanPrecision(mu, tau)   # used because we want to discover mu and tau\n",
        "  end\n",
        "end\n"
      ],
      "metadata": {
        "id": "q4on8dT7_ET_"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Constraints on model\n",
        "# Mean-field approximation assumption: parameters work independently\n",
        "# i.e. assume that [ q(mu, tau) = q(mu)q(tau) ]\n",
        "\n",
        "# define constraints\n",
        "constraints = @constraints begin\n",
        "  q(mu, tau) = q(mu)q(tau)        # where Q is posterior approximation of which q is an element\n",
        "end"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "4EEoKjHY_kgp",
        "outputId": "3e50d1c9-0c64-444b-b2b2-af1e35b0627b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Constraints:\n",
              "  marginals form:\n",
              "  messages form:\n",
              "  factorisation:\n",
              "    q(mu, tau) = q(mu)q(tau)\n",
              "Options:\n",
              "  warn = true\n"
            ]
          },
          "metadata": {},
          "execution_count": 14
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Variational Inference\n",
        "\n",
        "# approximate posteriors\n",
        "result = inference(\n",
        "  model = mymodel(n),   # model we defined using @model\n",
        "  data = (y = observations, ),   # our data (observations); must be a julia tuple incl. ending comma\n",
        "  constraints = constraints,   # constraints we defined\n",
        "  initmarginals = (     # initial marginal distributions to initialize inference process\n",
        "    mu = vague(NormalMeanPrecision),    # vague() function allows initialization at some un-biased starting point\n",
        "    tau = vague(GammaShapeRate)\n",
        "  ),\n",
        "  returnvars = (mu = KeepLast(), tau = KeepLast()),   # only return variables of final iteration\n",
        "  iterations = 5,   # set n of iterations for inference process\n",
        "  free_energy = true    # use Bethe Free Energy algorithm as minimization objective\n",
        ")\n",
        "\n",
        "# Summary:\n",
        "#   posteriors are available for mu and tau (per returnvars, only of final iteration)\n",
        "#   Free energy: BFE values after each iteration; notice convergence after only 3 iterations"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Lqz7DZIVBZFP",
        "outputId": "d441b446-1d17-4118-cf77-a8a238bb0db3"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Inference results:\n",
              "  Posteriors       | available for (mu, tau)\n",
              "  Free Energy:     | Real[1.47348e6, 2.0908e5, 61568.8, 61568.8, 61568.8]\n"
            ]
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# view results for mu\n",
        "mu_posterior = result.posteriors[:mu]\n",
        "mu_mean = mean(mu_posterior)     # notice true_mean is 3.0\n",
        "println(mu_mean)   # print variable followed by new line\n",
        "mu_std = std(mu_posterior)\n",
        "print(mu_std)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "3mA605-RBzfQ",
        "outputId": "30d090af-0883-4842-eeea-565469d7ca18"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "3.0023025736112445\n",
            "0.0014160852609955302"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# view results for tau\n",
        "tau_posterior = result.posteriors[:tau]\n",
        "tau_mean = mean(tau_posterior)\n",
        "println(tau_mean)    # notice true_precision is 5.0\n",
        "tau_std = std(tau_posterior)\n",
        "print(tau_std)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "d3rVY6ABDfm3",
        "outputId": "a57fc584-9383-46ec-cfcc-984cc06c9e4e"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "4.986781320253787\n",
            "0.02230134102973199"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# VI approximation using our model's derived param values\n",
        "\n",
        "vi_sigma = 1 / sqrt(tau_mean)    # compute std from model's approximated precision\n",
        "println(vi_sigma)\n",
        "vi_approximation = Normal(mu_mean, vi_sigma)   # generate Normal distribution using approximated params"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "_e66La37DjCJ",
        "outputId": "9070ebaa-9676-4d2c-b916-ea51d2f96039"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.4478059275659404\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Normal{Float64}(μ=3.0023025736112445, σ=0.4478059275659404)"
            ]
          },
          "metadata": {},
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# true distribution\n",
        "p_distributions = plot(true_distribution;\n",
        "  label = \"True Distribution\",\n",
        "  title = \"Variational Inference (VI)\",\n",
        "  xlims = (1,5),\n",
        "  color = 1,\n",
        "  fill = true,\n",
        "  alpha = 0.3\n",
        ")\n",
        "\n",
        "# VI approximation\n",
        "\n",
        "plot!(p_distributions, vi_approximation;\n",
        "  label = \"VI Approximation\",\n",
        "  color = 3,\n",
        "  linewidth = 2,\n",
        "  linestyle = :dash\n",
        ")\n",
        "\n",
        "# observations + distributions\n",
        "\n",
        "plot(p_observations, p_distributions;\n",
        "  layout = (2,1),\n",
        "  size = (600, 600)\n",
        ")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 621
        },
        "id": "LJp5MQ2dE3OX",
        "outputId": "ea610c5a-494b-48e6-f61e-93d2da3c4d5a"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
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          },
          "metadata": {},
          "execution_count": 31
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Final notes:\n",
        "\n",
        "# While MCMC will be accurate despite using a much lower (ex. n=100) observations,\n",
        "# MCMC is inherently much more computationally expensive and inefficient compared to VI.\n",
        "# In real-world applications, VI may be sufficient despite less accuracy."
      ],
      "metadata": {
        "id": "YRPoINZPFFZN"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}