{"cells": [{"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["# 分布的维度\n", "PyMC 提供了多种方式来指定其分布的维度。本文档提供了一个概述，并提供了一些用户提示。\n", "\n", "## 术语表\n", "在本文档中，我们将使用“维度”一词来指代维度的概念。以下每个术语在 PyMC 中都有特定的语义和计算定义。虽然我们在这里分享它们，但在下面的示例中查看时，它们会更有意义。\n", "\n", "+ *支持维度* → 分布的核心维度\n", "+ *批处理维度* → 超出分布支持维度的额外维度\n", "+ *隐含维度* → 源自分布参数的值或形状的维度\n", "+ *显式维度* → 由以下参数之一明确定义的维度：\n", "    + *形状* → 从分布中抽样的数量\n", "    + *维度* → 维度名称的数组\n", "+ *坐标* → 一个字典，将维度名称映射到坐标值\n"]}, {"cell_type": "code", "execution_count": 1, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [], "source": ["from functools import partial\n", "\n", "import pymc as pm\n", "import numpy as np\n", "import pytensor.tensor as pt\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["## 单变量分布示例\n", "我们可以从最简单的情况开始，单个正态分布。我们使用`.dist`来指定一个位于PyMC模型之外的分布。\n"]}, {"cell_type": "code", "execution_count": 2, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [], "source": ["normal_dist = pm.Normal.dist()\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["然后我们可以使用 {func}`~pymc.draw` 函数从该分布中进行随机抽样。\n"]}, {"cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": ["# 仅对绘图函数进行修补以确保可重复性\n", "rng = np.random.default_rng(seed=sum(map(ord, \"dimensionality\")))\n", "draw = partial(pm.draw, random_seed=rng)\n"]}, {"cell_type": "code", "execution_count": 4, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["(array(0.80189558), 0)"]}, "execution_count": 4, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_draw = draw(normal_dist)\n", "normal_draw, normal_draw.ndim\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["在这种情况下，我们最终得到一个单一的标量值。这意味着正态分布具有标量支撑维度，因为你可以进行的最小随机抽取是一个维度为零的标量。每个分布的支撑维度作为一个属性被硬编码。\n"]}, {"cell_type": "code", "execution_count": 5, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["0"]}, "execution_count": 5, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dist.owner.op.ndim_supp\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 显式批处理维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["如果需要进行多次抽取，通常的倾向是创建多个相同变量的副本并将它们堆叠在一起。\n"]}, {"cell_type": "code", "execution_count": 6, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([ 0.83636296, -0.33327414,  0.9434115 ])"]}, "execution_count": 6, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.math.stack([pm.Normal.dist() for _ in range(3)])\n", "draw(normal_dists)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["更简单地说，可以通过使用形状参数来创建来自同一分布族的*批量*独立抽样。\n"]}, {"cell_type": "code", "execution_count": 7, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([ 0.98810294, -0.07003785, -0.37962748])"]}, "execution_count": 7, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.Normal.dist(shape=(3,))\n", "draw(normal_dists)\n"]}, {"cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[ 7.99932116e-04, -1.94407945e+00,  3.90451962e-01],\n", "       [ 1.10657367e+00,  6.49042149e-01, -1.09450185e+00],\n", "       [-2.96226305e-01,  1.41884595e+00, -1.31589441e+00],\n", "       [ 1.53109449e+00, -7.73771737e-01,  2.37418367e+00]])"]}, "execution_count": 8, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.Normal.dist(shape=(4, 3))\n", "draw(normal_dists)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["这不仅更加简洁，而且生成的向量化代码效率更高。在PyMC中，我们很少使用堆栈方法，除非需要组合来自不同分布族的抽样。\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 隐式批次维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["还可以通过传递具有更高维度的参数来创建一批抽样，而无需指定形状。\n"]}, {"cell_type": "code", "execution_count": 9, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([ 0.81833093, -0.2891973 ,  1.2399946 ])"]}, "execution_count": 9, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.Normal.dist(mu=np.array([0, 0, 0]), sigma=np.array([1, 1, 1]))\n", "draw(normal_dists)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["这与之前的显式形状示例是等效的，我们本可以在这里显式地传递它。由于我们没有这样做，因此我们将这些批次维度称为 *隐式*。\n", "\n", "这在我们希望参数在批次维度上变化时变得非常有用。\n"]}, {"cell_type": "code", "execution_count": 10, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([  0.99989975,  10.00009874, 100.00004215])"]}, "execution_count": 10, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Normal.dist(mu=[1, 10, 100], sigma=0.0001))\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["当参数的形状不同时，它们会被广播，类似于NumPy的工作方式。在这种情况下，`sigma`被广播以匹配`mu`的形状。\n"]}, {"cell_type": "code", "execution_count": 11, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["[array([  1,  10, 100]), array([0.0001, 0.0001, 0.0001])]"]}, "execution_count": 11, "metadata": {}, "output_type": "execute_result"}], "source": ["np.broadcast_arrays([1, 10, 100], 0.0001)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["了解 NumPy {ref}`广播 <numpy:basics.broadcasting>` 的工作原理是很重要的。当你做一些无效的操作时，容易遇到这种错误：\n"]}, {"cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["shape mismatch: objects cannot be broadcast to a single shape.  Mismatch is between arg 0 with shape (3,) and arg 1 with shape (2,).\n", "Apply node that caused the error: normal_rv{0, (0, 0), floatX, True}(RandomGeneratorSharedVariable(<Generator(PCG64) at 0x7F6427F8CAC0>), [], 11, [  1  10 100], [0.1 0.1])\n", "Toposort index: 0\n", "Inputs types: [RandomGeneratorType, TensorType(int64, shape=(0,)), TensorType(int64, shape=()), TensorType(int64, shape=(3,)), TensorType(float64, shape=(2,))]\n", "Inputs shapes: ['No shapes', (0,), (), (3,), (2,)]\n", "Inputs strides: ['No strides', (0,), (), (8,), (8,)]\n", "Inputs values: [Generator(PCG64) at 0x7F6427F8CAC0, array([], dtype=int64), array(11), array([  1,  10, 100]), array([0.1, 0.1])]\n", "Outputs clients: [['output'], ['output']]\n", "\n", "HINT: Re-running with most PyTensor optimizations disabled could provide a back-trace showing when this node was created. This can be done by setting the PyTensor flag 'optimizer=fast_compile'. If that does not work, PyTensor optimizations can be disabled with 'optimizer=None'.\n", "HINT: Use the PyTensor flag `exception_verbosity=high` for a debug print-out and storage map footprint of this Apply node.\n"]}], "source": ["try:\n", "    # 形状为 (3,) 和 (2,) 的数组无法一起广播\n", "    draw(pm.Normal.dist(mu=[1, 10, 100], sigma=[0.1, 0.1]))\n", "except ValueError as error:\n", "    print(error)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["### 结合隐式和显式批次维度\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["您可以将显式形状维度与隐式批次维度结合使用。如上所提到的，它们可以提供相同的信息。\n"]}, {"cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([-0.49526775, -0.94608062,  1.66397913])"]}, "execution_count": 13, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.Normal.dist(mu=np.array([0, 1, 2]), sigma=1, shape=(3,))\n", "draw(normal_dists)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["但是形状也可以用来扩展超出任何隐式批次维度。\n"]}, {"cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[ 2.22626513,  2.12938134,  0.49074886],\n", "       [ 0.08312601,  1.05049093,  1.91718083],\n", "       [-0.68191815,  1.43771096,  1.76780399],\n", "       [-0.59883241,  0.26954893,  2.74319335]])"]}, "execution_count": 14, "metadata": {}, "output_type": "execute_result"}], "source": ["normal_dists = pm.Normal.dist(mu=np.array([0, 1, 2]), sigma=1, shape=(4, 3))\n", "draw(normal_dists)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["请注意，由于广播规则，显式的批次维度必须始终“放在左侧”，在任何隐式维度之前。因此，在前面的例子中，`shape=(4, 3)` 是有效的，但 `shape=(3, 4)` 是无效的，因为 `mu` 参数可以广播到第一个形状，而不能广播到第二个形状。\n"]}, {"cell_type": "code", "execution_count": 15, "metadata": {"scrolled": true}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["shape mismatch: objects cannot be broadcast to a single shape.  Mismatch is between arg 0 with shape (3, 4) and arg 1 with shape (3,).\n", "Apply node that caused the error: normal_rv{0, (0, 0), floatX, True}(RandomGeneratorSharedVariable(<Generator(PCG64) at 0x7F64280725E0>), [3 4], 11, [0 1 2], 1.0)\n", "Toposort index: 0\n", "Inputs types: [RandomGeneratorType, TensorType(int64, shape=(2,)), TensorType(int64, shape=()), TensorType(int64, shape=(3,)), TensorType(float64, shape=())]\n", "Inputs shapes: ['No shapes', (2,), (), (3,), ()]\n", "Inputs strides: ['No strides', (8,), (), (8,), ()]\n", "Inputs values: [Generator(PCG64) at 0x7F64280725E0, array([3, 4]), array(11), array([0, 1, 2]), array(1.)]\n", "Outputs clients: [['output'], ['output']]\n", "\n", "HINT: Re-running with most PyTensor optimizations disabled could provide a back-trace showing when this node was created. This can be done by setting the PyTensor flag 'optimizer=fast_compile'. If that does not work, PyTensor optimizations can be disabled with 'optimizer=None'.\n", "HINT: Use the PyTensor flag `exception_verbosity=high` for a debug print-out and storage map footprint of this Apply node.\n"]}], "source": ["try:\n", "    draw(pm.Normal.dist(mu=np.array([0, 1, 2]), sigma=1, shape=(3, 4)))\n", "except ValueError as error:\n", "    print(error)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["如果您需要正态变量的形状为 `shape=(3, 4)`，可以在定义后对其进行转置。\n"]}, {"cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[-0.73397401, -0.18717845, -0.78548049,  1.64478883],\n", "       [ 3.54543846,  1.22954216,  2.13674063,  1.94194106],\n", "       [ 0.85294471,  3.52041332,  2.94428975,  3.25944187]])"]}, "execution_count": 16, "metadata": {}, "output_type": "execute_result"}], "source": ["transposed_normals = pm.Normal.dist(mu=np.array([0, 1, 2]), sigma=1, shape=(4, 3)).T\n", "draw(transposed_normals)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": [":::{tip} 重要的是不要混淆在分布定义中设定的维度与在后续操作（如转置、索引或广播）中设定的维度。在使用PyMC进行采样时（无论是通过前向采样还是MCMC），随机抽样始终会来自于分布形状。请注意，在以下示例中，尽管两个变量具有相同的最终形状，但实际抽取的“随机”样本数量是不同的。\n", ":::\n"]}, {"cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": ["vector_normals = pm.Normal.dist(shape=(3,))\n", "broadcasted_normal = pt.broadcast_to(pm.Normal.dist(), (3,))\n"]}, {"cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [{"data": {"text/plain": ["(array([-0.45755879,  1.59975702,  0.20546749]),\n", " array([0.29866199, 0.29866199, 0.29866199]))"]}, "execution_count": 18, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(vector_normals), draw(broadcasted_normal)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["## 多元分布示例\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["某些分布根据定义在评估时返回多个值。这可能是一个值的向量，或一个矩阵，或一个任意的多维张量。一个例子是多元正态分布，它总是返回一个向量（具有一个维度的数组）。\n"]}, {"cell_type": "code", "execution_count": 19, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["(array([0.55390975, 2.17440418, 1.83014764]), 1)"]}, "execution_count": 19, "metadata": {}, "output_type": "execute_result"}], "source": ["mvnormal_dist = pm.MvNormal.dist(mu=np.ones(3), cov=np.eye(3))\n", "mvnormal_draw = draw(mvnormal_dist)\n", "mvnormal_draw, mvnormal_draw.ndim\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["与任何分布一样，支持的维度被指定为一个固定的属性。\n"]}, {"cell_type": "code", "execution_count": 20, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["1"]}, "execution_count": 20, "metadata": {}, "output_type": "execute_result"}], "source": ["mvnormal_dist.owner.op.ndim_supp\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["即使您指定一个单维的MvNormal，您也会得到一个向量！\n"]}, {"cell_type": "code", "execution_count": 21, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["(array([-0.68893796]), 1)"]}, "execution_count": 21, "metadata": {}, "output_type": "execute_result"}], "source": ["smallest_mvnormal_dist = pm.MvNormal.dist(mu=[1], cov=[[1]])\n", "smallest_mvnormal_draw = draw(smallest_mvnormal_dist)\n", "smallest_mvnormal_draw, smallest_mvnormal_draw.ndim\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 隐式支持维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["在我们刚刚看到的MvNormal示例中，支持维度实际上是隐含的。我们并没有具体指定我们想要的抽样向量的维度是3还是1。这是从`mu`和`cov`的形状中推断出来的。因此，我们称其为*隐含支持维度*。我们可以通过使用形状来使其更加明确。\n"]}, {"cell_type": "code", "execution_count": 22, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([0.57262853, 0.34230354, 1.96818163])"]}, "execution_count": 22, "metadata": {}, "output_type": "execute_result"}], "source": ["explicit_mvnormal = pm.MvNormal.dist(mu=np.ones(3), cov=np.eye(3), shape=(3,))\n", "draw(explicit_mvnormal)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": [":::{warning} 然而，请注意，在撰写时，形状仅仅被忽略用于支持维度。它仅作为一种“类型提示”来标记预期的维度。\n", ":::\n"]}, {"cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([1.0623799 , 0.84622693, 0.34046237])"]}, "execution_count": 23, "metadata": {}, "output_type": "execute_result"}], "source": ["ignored_shape_mvnormal = pm.MvNormal.dist(mu=np.ones(3), cov=np.eye(3), shape=(4,))\n", "draw(ignored_shape_mvnormal)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 显式批次维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["与单变量分布一样，我们可以添加显式的批量维度。我们将使用另一个向量分布来说明这一点：多项分布。下面的代码片段定义了一个包含五个独立多项分布的矩阵，每个分布都是大小为3的向量。\n"]}, {"cell_type": "code", "execution_count": 24, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([[2, 0, 3],\n", "       [1, 1, 3],\n", "       [0, 2, 3],\n", "       [0, 1, 4],\n", "       [1, 0, 4]])"]}, "execution_count": 24, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Multinomial.dist(n=5, p=[0.1, 0.3, 0.6], shape=(5, 3)))\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": [":::{warning} 再次注意，形状对支持的维度没有影响\n", ":::\n"]}, {"cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[0, 1, 4],\n", "       [0, 0, 5],\n", "       [3, 1, 1],\n", "       [0, 1, 4],\n", "       [0, 2, 3]])"]}, "execution_count": 25, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Multinomial.dist(n=5, p=[0.1, 0.3, 0.6], shape=(5, 4)))\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["出于同样的原因，您必须始终在支持维度的“左侧”明确定义显式的批量维度。以下内容将不会按预期行为。\n"]}, {"cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[2, 0, 3],\n", "       [1, 3, 1],\n", "       [1, 1, 3]])"]}, "execution_count": 26, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Multinomial.dist(n=5, p=[0.1, 0.3, 0.6], shape=(3, 5)))\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["如果你需要多项式变量的形状为 `shape=(3, 5)`，可以在定义后进行转置。\n"]}, {"cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [{"data": {"text/plain": ["array([[0, 0, 0, 0, 0],\n", "       [2, 2, 1, 0, 3],\n", "       [3, 3, 4, 5, 2]])"]}, "execution_count": 27, "metadata": {}, "output_type": "execute_result"}], "source": ["transposed_multinomials = pm.Multinomial.dist(n=5, p=[0.1, 0.3, 0.6], shape=(5, 3)).T\n", "draw(transposed_multinomials)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 隐式批量维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["与单变量分布一样，我们可以为每个批处理维度使用不同的参数。\n"]}, {"cell_type": "code", "execution_count": 28, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([[1, 2, 2],\n", "       [0, 3, 7]])"]}, "execution_count": 28, "metadata": {}, "output_type": "execute_result"}], "source": ["multinomial_dist = pm.Multinomial.dist(n=[5, 10], p=[0.1, 0.3, 0.6])\n", "draw(multinomial_dist)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["哪一个等同于更冗长的内容\n"]}, {"cell_type": "code", "execution_count": 29, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([[2, 2, 1],\n", "       [0, 3, 7]])"]}, "execution_count": 29, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Multinomial.dist(n=[5, 10], p=[[0.1, 0.3, 0.6], [0.1, 0.3, 0.6]]))\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["如果您熟悉NumPy的广播规则，您可能会好奇PyMC是如何使其工作的。简单的广播在这里是行不通的。\n"]}, {"cell_type": "code", "execution_count": 30, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["shape mismatch: objects cannot be broadcast to a single shape.  Mismatch is between arg 0 with shape (2,) and arg 1 with shape (3,).\n"]}], "source": ["try:\n", "    np.broadcast_arrays([5, 10], [0.1, 0.3, 0.6])\n", "except ValueError as exc:\n", "    print(exc)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["为了理解发生了什么，我们需要引入参数核心维度的概念。分布参数的核心维度是定义分布所需的参数最小维数。在多项式分布中，`n` 至少必须是一个标量整数，而 `p` 至少必须是一个表示每个类别结果概率的向量。因此，对于多项式分布，`n` 的核心维度是0，而 `p` 的核心维度是1。\n", "\n", "因此，如果我们有一个包含两个 `n` 的向量，我们实际上应该将 `p` 的向量广播成一个包含两个这样的向量的矩阵，并将每个 `n` 与每个广播的 `p` 的行配对。这的工作方式与 `np.vectorize` 完全相同。\n"]}, {"cell_type": "code", "execution_count": 31, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"name": "stdout", "output_type": "stream", "text": [">> 5 [0.1 0.3 0.6]\n", ">> 10 [0.1 0.3 0.6]\n"]}, {"data": {"text/plain": ["array([[1, 0, 4],\n", "       [1, 2, 7]])"]}, "execution_count": 31, "metadata": {}, "output_type": "execute_result"}], "source": ["def core_multinomial(n, p):\n", "    print(\">>\", n, p)\n", "    return draw(pm.Multinomial.dist(n, p))\n", "\n", "\n", "vectorized_multinomial = np.vectorize(core_multinomial, signature=\"(),(p)->(p)\")\n", "vectorized_multinomial([5, 10], [0.1, 0.3, 0.6])\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["每个分布参数的核心维度也是硬编码为每个分布的一个属性。\n"]}, {"cell_type": "code", "execution_count": 32, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["(0, 1)"]}, "execution_count": 32, "metadata": {}, "output_type": "execute_result"}], "source": ["multinomial_dist.owner.op.ndims_params\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["隐式批处理维度仍然必须遵循广播规则。以下示例无效，因为 `n` 的批处理维度为 `shape=(2,)` ，而 `p` 的批处理维度为 `shape=(3,)` ，这两个维度无法一起广播。\n"]}, {"cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["operands could not be broadcast together with remapped shapes [original->remapped]: (2,)  and requested shape (3,)\n", "Apply node that caused the error: multinomial_rv{1, (0, 1), int64, True}(RandomGeneratorSharedVariable(<Generator(PCG64) at 0x7F6425B8B060>), [], 4, [ 5 10], [[0.1 0.3 ... 0.3 0.6]])\n", "Toposort index: 0\n", "Inputs types: [RandomGeneratorType, TensorType(int64, shape=(0,)), TensorType(int64, shape=()), TensorType(int64, shape=(2,)), TensorType(float64, shape=(3, 3))]\n", "Inputs shapes: ['No shapes', (0,), (), (2,), (3, 3)]\n", "Inputs strides: ['No strides', (0,), (), (8,), (24, 8)]\n", "Inputs values: [Generator(PCG64) at 0x7F6425B8B060, array([], dtype=int64), array(4), array([ 5, 10]), 'not shown']\n", "Outputs clients: [['output'], ['output']]\n", "\n", "HINT: Re-running with most PyTensor optimizations disabled could provide a back-trace showing when this node was created. This can be done by setting the PyTensor flag 'optimizer=fast_compile'. If that does not work, PyTensor optimizations can be disabled with 'optimizer=None'.\n", "HINT: Use the PyTensor flag `exception_verbosity=high` for a debug print-out and storage map footprint of this Apply node.\n"]}], "source": ["try:\n", "    draw(pm.Multinomial.dist(n=[5, 10], p=[[0.1, 0.3, 0.6], [0.1, 0.3, 0.6], [0.1, 0.3, 0.6]]))\n", "except ValueError as error:\n", "    print(error)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["### 结合隐式和显式批量维度\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["您可以并且应该将多维参数的隐式维度与显式形状信息结合，这样更易于推理。\n"]}, {"cell_type": "code", "execution_count": 34, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"text/plain": ["array([[0, 1, 4],\n", "       [4, 1, 5]])"]}, "execution_count": 34, "metadata": {}, "output_type": "execute_result"}], "source": ["draw(pm.Multinomial.dist(n=[5, 10], p=[0.1, 0.3, 0.6], shape=(2, 3)))\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["显式的批量维度仍然可以超出任何隐式的批量维度。同样，由于广播的工作原理，显式批量维度必须始终\"在左边\"。以下情况是无效的，因为`n`具有形状为`(2,)`的批量维度，这无法广播到形状为`(2, 4)`的显式批量维度。\n"]}, {"cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["operands could not be broadcast together with remapped shapes [original->remapped]: (2,)  and requested shape (2,4)\n", "Apply node that caused the error: multinomial_rv{1, (0, 1), int64, True}(RandomGeneratorSharedVariable(<Generator(PCG64) at 0x7F6425AC8120>), [2 4], 4, [ 5 10], [0.1 0.3 0.6])\n", "Toposort index: 0\n", "Inputs types: [RandomGeneratorType, TensorType(int64, shape=(2,)), TensorType(int64, shape=()), TensorType(int64, shape=(2,)), TensorType(float64, shape=(3,))]\n", "Inputs shapes: ['No shapes', (2,), (), (2,), (3,)]\n", "Inputs strides: ['No strides', (8,), (), (8,), (8,)]\n", "Inputs values: [Generator(PCG64) at 0x7F6425AC8120, array([2, 4]), array(4), array([ 5, 10]), array([0.1, 0.3, 0.6])]\n", "Outputs clients: [['output'], ['output']]\n", "\n", "HINT: Re-running with most PyTensor optimizations disabled could provide a back-trace showing when this node was created. This can be done by setting the PyTensor flag 'optimizer=fast_compile'. If that does not work, PyTensor optimizations can be disabled with 'optimizer=None'.\n", "HINT: Use the PyTensor flag `exception_verbosity=high` for a debug print-out and storage map footprint of this Apply node.\n"]}], "source": ["try:\n", "    draw(pm.Multinomial.dist(n=[5, 10], p=[0.1, 0.3, 0.6], shape=(2, 4, 3)))\n", "except ValueError as error:\n", "    print(error)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["## 使用模型图检查维度\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["在PyMC模型中，分布通常会被频繁使用，因此存在一些工具可以在该上下文中帮助推理分布的形状。\n"]}, {"cell_type": "code", "execution_count": 36, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["          x: shape=(3,)\n", "sigma_log__: shape=()\n", "      sigma: shape=()\n", "          y: shape=(3,)\n"]}], "source": ["with pm.Model() as pmodel:\n", "    mu = pm.Normal(\"x\", mu=0, shape=(3))\n", "    sigma = pm.HalfNormal(\"sigma\")\n", "    y = pm.Normal(\"y\", mu=mu, sigma=sigma)\n", "\n", "for rv, shape in pmodel.eval_rv_shapes().items():\n", "    print(f\"{rv:>11}: shape={shape}\")\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["一个更强大的工具来理解和调试PyMC中的维度是{func}`~pymc.model_to_graphviz`函数。我们可以通过读取Graphviz输出而不是检查数组输出，以理解变量的维度。\n"]}, {"cell_type": "code", "execution_count": 37, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"image/svg+xml": ["<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", "<!-- Generated by graphviz version 7.1.0 (20230122.1345)\n", " -->\n", "<!-- Pages: 1 -->\n", "<svg width=\"282pt\" height=\"249pt\"\n", " viewBox=\"0.00 0.00 281.71 248.91\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 244.91)\">\n", "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-244.91 277.71,-244.91 277.71,4 -4,4\"/>\n", "<g id=\"clust1\" class=\"cluster\">\n", "<title>cluster3</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M20,-8C20,-8 110,-8 110,-8 116,-8 122,-14 122,-20 122,-20 122,-220.91 122,-220.91 122,-226.91 116,-232.91 110,-232.91 110,-232.91 20,-232.91 20,-232.91 14,-232.91 8,-226.91 8,-220.91 8,-220.91 8,-20 8,-20 8,-14 14,-8 20,-8\"/>\n", "<text text-anchor=\"middle\" x=\"109\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n", "</g>\n", "<!-- y -->\n", "<g id=\"node1\" class=\"node\">\n", "<title>y</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"65\" cy=\"-76.48\" rx=\"49.49\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">y</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "<!-- x -->\n", "<g id=\"node2\" class=\"node\">\n", "<title>x</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"65\" cy=\"-187.43\" rx=\"49.49\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-198.73\" font-family=\"Times,serif\" font-size=\"14.00\">x</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-183.73\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-168.73\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "<!-- x&#45;&gt;y -->\n", "<g id=\"edge2\" class=\"edge\">\n", "<title>x&#45;&gt;y</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M65,-149.54C65,-141.91 65,-133.76 65,-125.79\"/>\n", "<polygon fill=\"black\" stroke=\"black\" points=\"68.5,-125.84 65,-115.84 61.5,-125.84 68.5,-125.84\"/>\n", "</g>\n", "<!-- sigma -->\n", "<g id=\"node3\" class=\"node\">\n", "<title>sigma</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"203\" cy=\"-187.43\" rx=\"70.92\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"203\" y=\"-198.73\" font-family=\"Times,serif\" font-size=\"14.00\">sigma</text>\n", "<text text-anchor=\"middle\" x=\"203\" y=\"-183.73\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"203\" y=\"-168.73\" font-family=\"Times,serif\" font-size=\"14.00\">HalfNormal</text>\n", "</g>\n", "<!-- sigma&#45;&gt;y -->\n", "<g id=\"edge1\" class=\"edge\">\n", "<title>sigma&#45;&gt;y</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M164.19,-155.79C146.52,-141.84 125.55,-125.28 107.45,-110.99\"/>\n", "<polygon fill=\"black\" stroke=\"black\" points=\"110.03,-108.57 100.02,-105.12 105.7,-114.07 110.03,-108.57\"/>\n", "</g>\n", "</g>\n", "</svg>\n"], "text/plain": ["<graphviz.graphs.Digraph at 0x7f64258f4a00>"]}, "execution_count": 37, "metadata": {}, "output_type": "execute_result"}], "source": ["pm.model_to_graphviz(pmodel)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["在上面的示例中，每个框（或板）左下角的数字表示其中分布的维度。如果某个分布在任何带数字的框之外，它具有标量形状。\n", "\n", "让我们利用这个工具来回顾隐式和显式维度：\n"]}, {"cell_type": "code", "execution_count": 38, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"image/svg+xml": ["<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", "<!-- Generated by graphviz version 7.1.0 (20230122.1345)\n", " -->\n", "<!-- Pages: 1 -->\n", "<svg width=\"618pt\" height=\"138pt\"\n", " viewBox=\"0.00 0.00 617.92 137.95\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 133.95)\">\n", "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-133.95 613.92,-133.95 613.92,4 -4,4\"/>\n", "<g id=\"clust1\" class=\"cluster\">\n", "<title>cluster3</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M205.92,-8C205.92,-8 383.92,-8 383.92,-8 389.92,-8 395.92,-14 395.92,-20 395.92,-20 395.92,-109.95 395.92,-109.95 395.92,-115.95 389.92,-121.95 383.92,-121.95 383.92,-121.95 205.92,-121.95 205.92,-121.95 199.92,-121.95 193.92,-115.95 193.92,-109.95 193.92,-109.95 193.92,-20 193.92,-20 193.92,-14 199.92,-8 205.92,-8\"/>\n", "<text text-anchor=\"middle\" x=\"382.92\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">3</text>\n", "</g>\n", "<g id=\"clust2\" class=\"cluster\">\n", "<title>cluster4</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M415.92,-8C415.92,-8 589.92,-8 589.92,-8 595.92,-8 601.92,-14 601.92,-20 601.92,-20 601.92,-109.95 601.92,-109.95 601.92,-115.95 595.92,-121.95 589.92,-121.95 589.92,-121.95 415.92,-121.95 415.92,-121.95 409.92,-121.95 403.92,-115.95 403.92,-109.95 403.92,-109.95 403.92,-20 403.92,-20 403.92,-14 409.92,-8 415.92,-8\"/>\n", "<text text-anchor=\"middle\" x=\"588.92\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">4</text>\n", "</g>\n", "<!-- scalar (support) -->\n", "<g id=\"node1\" class=\"node\">\n", "<title>scalar (support)</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"91.92\" cy=\"-76.48\" rx=\"91.85\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"91.92\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">scalar (support)</text>\n", "<text text-anchor=\"middle\" x=\"91.92\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"91.92\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "<!-- vector (implicit) -->\n", "<g id=\"node2\" class=\"node\">\n", "<title>vector (implicit)</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"294.92\" cy=\"-76.48\" rx=\"92.76\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"294.92\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">vector (implicit)</text>\n", "<text text-anchor=\"middle\" x=\"294.92\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"294.92\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "<!-- vector (explicit) -->\n", "<g id=\"node3\" class=\"node\">\n", "<title>vector (explicit)</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"502.92\" cy=\"-76.48\" rx=\"91.43\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"502.92\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">vector (explicit)</text>\n", "<text text-anchor=\"middle\" x=\"502.92\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"502.92\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "</g>\n", "</svg>\n"], "text/plain": ["<graphviz.graphs.Digraph at 0x7f642591f250>"]}, "execution_count": 38, "metadata": {}, "output_type": "execute_result"}], "source": ["with pm.Model() as pmodel:\n", "    pm.Normal(\"scalar (support)\")\n", "    pm.Normal(\"vector (implicit)\", mu=[1, 2, 3])\n", "    pm.Normal(\"vector (explicit)\", shape=(4,))\n", "\n", "pm.model_to_graphviz(pmodel)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["## 维度\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["PyMC 支持 `dims` 的概念。对于许多随机变量来说，哪个维度对应于哪个“现实世界”的概念，例如观察数量、处理单元数量等，可能会变得令人困惑。`dims` 参数是一个额外的易读标签，可以传达这种含义。当单独使用时，`dims` 必须与显式的 `shape` 信息结合使用。\n"]}, {"cell_type": "code", "execution_count": 39, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"image/svg+xml": ["<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", "<!-- Generated by graphviz version 7.1.0 (20230122.1345)\n", " -->\n", "<!-- Pages: 1 -->\n", "<svg width=\"138pt\" height=\"138pt\"\n", " viewBox=\"0.00 0.00 138.00 137.95\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 133.95)\">\n", "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-133.95 134,-133.95 134,4 -4,4\"/>\n", "<g id=\"clust1\" class=\"cluster\">\n", "<title>clusteryear (3)</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M20,-8C20,-8 110,-8 110,-8 116,-8 122,-14 122,-20 122,-20 122,-109.95 122,-109.95 122,-115.95 116,-121.95 110,-121.95 110,-121.95 20,-121.95 20,-121.95 14,-121.95 8,-115.95 8,-109.95 8,-109.95 8,-20 8,-20 8,-14 14,-8 20,-8\"/>\n", "<text text-anchor=\"middle\" x=\"86\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">year (3)</text>\n", "</g>\n", "<!-- profit -->\n", "<g id=\"node1\" class=\"node\">\n", "<title>profit</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"65\" cy=\"-76.48\" rx=\"49.49\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">profit</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "</g>\n", "</svg>\n"], "text/plain": ["<graphviz.graphs.Digraph at 0x7f6425ac2b60>"]}, "execution_count": 39, "metadata": {}, "output_type": "execute_result"}], "source": ["with pm.Model() as model:\n", "    pm.Normal(\"profit\", shape=(3,), dims=\"year\")\n", "\n", "pm.model_to_graphviz(model)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["在 `dims` 的使用中，最强大的地方在于模型中指定的 `coords`。这为每个 `dim` 条目提供了一个独特的标签，使其更具意义。\n"]}, {"cell_type": "code", "execution_count": 40, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"image/svg+xml": ["<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", "<!-- Generated by graphviz version 7.1.0 (20230122.1345)\n", " -->\n", "<!-- Pages: 1 -->\n", "<svg width=\"138pt\" height=\"138pt\"\n", " viewBox=\"0.00 0.00 138.00 137.95\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 133.95)\">\n", "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-133.95 134,-133.95 134,4 -4,4\"/>\n", "<g id=\"clust1\" class=\"cluster\">\n", "<title>clusteryear (3)</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M20,-8C20,-8 110,-8 110,-8 116,-8 122,-14 122,-20 122,-20 122,-109.95 122,-109.95 122,-115.95 116,-121.95 110,-121.95 110,-121.95 20,-121.95 20,-121.95 14,-121.95 8,-115.95 8,-109.95 8,-109.95 8,-20 8,-20 8,-14 14,-8 20,-8\"/>\n", "<text text-anchor=\"middle\" x=\"86\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">year (3)</text>\n", "</g>\n", "<!-- profit -->\n", "<g id=\"node1\" class=\"node\">\n", "<title>profit</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"65\" cy=\"-76.48\" rx=\"49.49\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">profit</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"65\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">Normal</text>\n", "</g>\n", "</g>\n", "</svg>\n"], "text/plain": ["<graphviz.graphs.Digraph at 0x7f6425ac3c40>"]}, "execution_count": 40, "metadata": {}, "output_type": "execute_result"}], "source": ["coords = {\"year\": [2020, 2021, 2022]}\n", "with pm.Model(coords=coords) as model:\n", "    pm.Normal(\"profit\", dims=\"year\")\n", "\n", "pm.model_to_graphviz(model)\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["在这种情况下，分布的维度实际上可以通过使用的 `dims` 来定义。我们不需要传递形状或定义隐式批处理维度。\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["让我们通过一个多元正态分布的例子来回顾不同维度的特点。\n"]}, {"cell_type": "code", "execution_count": 41, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"data": {"image/svg+xml": ["<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n", "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n", " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n", "<!-- Generated by graphviz version 7.1.0 (20230122.1345)\n", " -->\n", "<!-- Pages: 1 -->\n", "<svg width=\"586pt\" height=\"138pt\"\n", " viewBox=\"0.00 0.00 586.00 137.95\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n", "<g id=\"graph0\" class=\"graph\" transform=\"scale(1 1) rotate(0) translate(4 133.95)\">\n", "<polygon fill=\"white\" stroke=\"none\" points=\"-4,4 -4,-133.95 582,-133.95 582,4 -4,4\"/>\n", "<g id=\"clust1\" class=\"cluster\">\n", "<title>clustersupport (3)</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M20,-8C20,-8 142,-8 142,-8 148,-8 154,-14 154,-20 154,-20 154,-109.95 154,-109.95 154,-115.95 148,-121.95 142,-121.95 142,-121.95 20,-121.95 20,-121.95 14,-121.95 8,-115.95 8,-109.95 8,-109.95 8,-20 8,-20 8,-14 14,-8 20,-8\"/>\n", "<text text-anchor=\"middle\" x=\"106\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">support (3)</text>\n", "</g>\n", "<g id=\"clust2\" class=\"cluster\">\n", "<title>clusterbatch (4) x support (3)</title>\n", "<path fill=\"none\" stroke=\"black\" d=\"M174,-8C174,-8 558,-8 558,-8 564,-8 570,-14 570,-20 570,-20 570,-109.95 570,-109.95 570,-115.95 564,-121.95 558,-121.95 558,-121.95 174,-121.95 174,-121.95 168,-121.95 162,-115.95 162,-109.95 162,-109.95 162,-20 162,-20 162,-14 168,-8 174,-8\"/>\n", "<text text-anchor=\"middle\" x=\"481.5\" y=\"-15.8\" font-family=\"Times,serif\" font-size=\"14.00\">batch (4) x support (3)</text>\n", "</g>\n", "<!-- vector -->\n", "<g id=\"node1\" class=\"node\">\n", "<title>vector</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"81\" cy=\"-76.48\" rx=\"65.11\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"81\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">vector</text>\n", "<text text-anchor=\"middle\" x=\"81\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"81\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">MvNormal</text>\n", "</g>\n", "<!-- matrix (explicit) -->\n", "<g id=\"node2\" class=\"node\">\n", "<title>matrix (explicit)</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"469\" cy=\"-76.48\" rx=\"92.76\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"469\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">matrix (explicit)</text>\n", "<text text-anchor=\"middle\" x=\"469\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"469\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">MvNormal</text>\n", "</g>\n", "<!-- matrix (implicit) -->\n", "<g id=\"node3\" class=\"node\">\n", "<title>matrix (implicit)</title>\n", "<ellipse fill=\"none\" stroke=\"black\" cx=\"264\" cy=\"-76.48\" rx=\"94.09\" ry=\"37.45\"/>\n", "<text text-anchor=\"middle\" x=\"264\" y=\"-87.78\" font-family=\"Times,serif\" font-size=\"14.00\">matrix (implicit)</text>\n", "<text text-anchor=\"middle\" x=\"264\" y=\"-72.78\" font-family=\"Times,serif\" font-size=\"14.00\">~</text>\n", "<text text-anchor=\"middle\" x=\"264\" y=\"-57.78\" font-family=\"Times,serif\" font-size=\"14.00\">MvNormal</text>\n", "</g>\n", "</g>\n", "</svg>\n"], "text/plain": ["<graphviz.graphs.Digraph at 0x7f6425711510>"]}, "execution_count": 41, "metadata": {}, "output_type": "execute_result"}], "source": ["coords = {\n", "    \"batch\": [0, 1, 2, 3],\n", "    \"support\": [\"A\", \"B\", \"C\"],\n", "}\n", "with pm.Model(coords=coords) as model:\n", "    pm.MvNormal(\"vector\", mu=[0, 0, 0], cov=np.eye(3), dims=(\"support\",))\n", "    pm.MvNormal(\"matrix (implicit)\", mu=np.zeros((4, 3)), cov=np.eye(3), dims=(\"batch\", \"support\"))\n", "    pm.MvNormal(\n", "        \"matrix (explicit)\", mu=[0, 0, 0], cov=np.eye(3), shape=(4, 3), dims=(\"batch\", \"support\")\n", "    )\n", "\n", "pm.model_to_graphviz(model)\n"]}, {"cell_type": "markdown", "metadata": {}, "source": [":::{tip} 对于最终模型的发布，我们建议将尺寸和坐标作为标签，因为这些标签将传递给 {class}`arviz.InferenceData`。这既是最佳实践，确保透明性和可读性，方便他人理解。同时，在单开发者的工作流程中，这也是非常有用的，例如，当存在三维或更高维度的分布时，它将有助于指明哪个维度对应于哪个模型概念。\n", ":::\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["## 调试形状问题的提示\n"]}, {"cell_type": "markdown", "metadata": {"pycharm": {"name": "#%% md\n"}}, "source": ["尽管我们提供了所有这些工具以方便用户，并且 PyMC 尽力理解用户的意图，但混合维度工具的结果可能并不总是符合最终的预期维度。有时模型可能在采样时不会指示错误，或者根本不会指示问题。在处理维度，特别是更复杂的维度时，我们建议：\n", "\n", "* 在采样之前使用 `model_to_graphviz` 可视化模型\n", "* 使用 `draw` 或 `sample_prior predictive` 早期捕捉错误\n", "* 检查返回的 `az.InferenceData` 对象，以确保所有数组大小符合预期\n", "* 在追踪错误时使用质数定义形状。\n"]}, {"cell_type": "code", "execution_count": 42, "metadata": {"pycharm": {"name": "#%%\n"}}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["Last updated: Mon Jul 17 2023\n", "\n", "Python implementation: CPython\n", "Python version       : 3.10.9\n", "IPython version      : 8.11.0\n", "\n", "pytensor: 2.12.3\n", "\n", "pymc    : 5.2.0\n", "numpy   : None\n", "pytensor: 2.12.3\n", "\n", "Watermark: 2.3.1\n", "\n"]}], "source": ["%load_ext watermark\n", "\n", "%watermark -n -u -v -iv -w -p pytensor\n"]}, {"cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": ["\n"]}], "metadata": {"hide_input": false, "interpreter": {"hash": "f574fac5b7e4a41f7640949d1e1759089329dd116ff7b389caa9cf21f93d872d"}, "kernelspec": {"display_name": "pymc", "language": "python", "name": "pymc"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 3}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.9"}, "toc": {"base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true}, "varInspector": {"cols": {"lenName": 16, "lenType": 16, "lenVar": 40}, "kernels_config": {"python": {"delete_cmd_postfix": "", "delete_cmd_prefix": "del ", "library": "var_list.py", "varRefreshCmd": "print(var_dic_list())"}, "r": {"delete_cmd_postfix": ") ", "delete_cmd_prefix": "rm(", "library": "var_list.r", "varRefreshCmd": "cat(var_dic_list()) "}}, "types_to_exclude": ["module", "function", "builtin_function_or_method", "instance", "_Feature"], "window_display": false}}, "nbformat": 4, "nbformat_minor": 4}