From 7574a3bf30462d08dc5433184ac7bef6badea10b Mon Sep 17 00:00:00 2001 From: pglpm Date: Wed, 30 Oct 2024 19:40:50 +0100 Subject: [PATCH] update beyond-ML --- beyond-ML.qmd | 2 +- docs/basic_decisions.html | 2 +- docs/beyond-ML.html | 6 +++--- docs/code_design.html | 2 +- docs/conditional_probability.html | 2 +- docs/conditional_summary.html | 2 +- docs/connection-1-ML.html | 2 +- docs/connection-2-ML.html | 2 +- docs/connection-3-ML.html | 2 +- docs/decision_trees.html | 2 +- docs/derived_rules.html | 2 +- docs/dirichlet-mixture.html | 2 +- docs/example_opm1.html | 2 +- docs/example_opm2.html | 2 +- docs/exchangeable_probabilities.html | 24 +++++++++++------------- docs/framework.html | 2 +- docs/index.html | 2 +- docs/inference.html | 2 +- docs/inference_about_freqs.html | 2 +- docs/inference_from_freqs.html | 2 +- docs/information.html | 2 +- docs/intro.html | 2 +- docs/joint_probability.html | 2 +- docs/limitations_ML.html | 2 +- docs/machine_learning_overview.html | 2 +- docs/making_decisions.html | 2 +- docs/marginal_probability.html | 2 +- docs/monty.html | 2 +- docs/neural_networks.html | 8 ++++---- docs/populations_variates.html | 2 +- docs/preface.html | 2 +- docs/probability_distributions.html | 2 +- docs/probability_inference.html | 2 +- docs/prototype_code.html | 2 +- docs/quantities_types.html | 2 +- docs/quantities_types_multi.html | 2 +- docs/references.html | 2 +- docs/samples.html | 2 +- docs/search.json | 14 +++++++------- docs/sentences.html | 2 +- docs/statistics.html | 2 +- docs/subpopulations.html | 2 +- docs/summary_formulae.html | 2 +- docs/thanks.html | 2 +- docs/truth_inference.html | 2 +- docs/utilities.html | 2 +- docs/utilities_evaluation.html | 2 +- docs/whither.html | 2 +- exchangeable_probabilities.qmd | 16 +++++++--------- 49 files changed, 76 insertions(+), 80 deletions(-) diff --git a/beyond-ML.qmd b/beyond-ML.qmd index a594f69..2c0c422 100644 --- a/beyond-ML.qmd +++ b/beyond-ML.qmd @@ -1,4 +1,4 @@ -# [Beyond machine learning]{.midgrey} {#sec-beyond-ML} +# [Beyond machine learning]{.green} {#sec-beyond-ML} {{< include macros.qmd >}} {{< include macros_prob_inference.qmd >}} {{< include macros_connection-3.qmd >}} diff --git a/docs/basic_decisions.html b/docs/basic_decisions.html index 2c539a9..9223f86 100644 --- a/docs/basic_decisions.html +++ b/docs/basic_decisions.html @@ -402,7 +402,7 @@

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    Table of contents

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    27  Beyond machine learning

    +

    27  Beyond machine learning

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    Learning and conditio
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    4&nbs
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    - 27  Beyond machine learning + 27  Beyond machine learning

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    - 27  Beyond machine learning + 27  Beyond machine learning

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    28.1 Recap

    In the chapters of part Inference I we had an overview of how an agent can draw inferences and make predictions of the most general kind, expressed by general sentences, using the four fundamental rules of inference.

    -

    Then, in part Inference II, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and machine-learning algorithms. First, inferences about measurements and observations; then, inferences about multiple instances of similar measurements and observations. The idea was that that an agent can arrive at sharper degrees of belief – that is, learn – by using information about “similar instances”.

    +

    Then, in part Inference II, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and of machine-learning algorithms. First we considered inferences about measurements and observations, then inferences about multiple instances of similar measurements and observations. The idea was that an agent can arrive at sharper degrees of belief – that is, it can learn – by using information about “similar instances”.

    For these purposes we introduced a specialized language about quantities and data types in part Data I, and about “populations” of similar “units” in part Data II.

    -

    In the Machine learning part we took an overview of current machine-learning methods, and then focused on several types of “multi-instance” tasks that popular machine-learning algorithm, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent – one that operates according to Probability Theory – can in principle perform any and all of those tasks by using the joint probability distribution

    +

    In the Machine learning part we took an overview of current machine-learning methods, and then focused on several types of tasks that popular machine-learning algorithms, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent – one that operates according to Probability Theory – can in principle perform any and all of those tasks by using the joint probability distribution

    \[ \mathrm{P}(\color[RGB]{68,119,170} Y_{N+1}\mathclose{}\mathord{\nonscript\mkern 0mu\textrm{\small=}\nonscript\mkern 0mu}\mathopen{}y_{N+1} @@ -620,8 +620,8 @@

    \(\color[RGB]{68,119,170}x_{1}, \dotsc, x_{N+1}\), \(\color[RGB]{68,119,170}y_{1}, \dotsc, y_{N+1}\), and \(N\).

    -

    For example, the task (“supervised learning”) of predicting some variates (predictands) for a new unit, from knowledge of the other variates (predictors) for the same units, and of all variates for \(N\) other units, is given by

    +

    The agent only needs to do some calculations with this joint distribution, involving sums and divisions. This distribution must be specified for all possible values of \(\color[RGB]{68,119,170}x_{1}, \dotsc, x_{N+1}\), \(\color[RGB]{68,119,170}y_{1}, \dotsc, y_{N+1}\), and \(N\).

    +

    Take “supervised learning” for example, that is, the task of predicting some variates (predictands) for a new unit, from knowledge of other variates (predictors) for the same unit and of all variates for \(N\) other units. Solving this task corresponds to calculating

    \[ \begin{aligned} @@ -660,15 +660,14 @@

    28.2 States of knowledge with symmetries

    -

    An agent’s degrees of belief about a particular population may satisfy a special symmetry called exchangeability. This symmetry can be understood from different points of view. We start from one of these viewpoints, and then make connections with alternative ones.

    +

    An agent’s degrees of belief about a particular population may satisfy a special symmetry called exchangeability. This symmetry can be understood from different points of view. Let’s start from one of these viewpoints, and then make connections with alternative ones.

    Take again two populations briefly mentioned in § 20.1:

    • @@ -693,8 +692,7 @@

      <


      -

      -

      Suppose that, in each of these populations, you (the agent) don’t know the variate value for unit #735, and for some reason would like to infer it. You are given the variate values for 100 other units, which you can use to improve your inference. Now consider this question:

      +Suppose that, in each of these populations, you (the agent) don’t know the variate value for unit #735, and for some reason would like to infer it. You are given the variate values for 100 other units, which you can use to improve your inference. Now consider this question:

      How much does the relative order of the 100 known units and the unknown unit matter to you, for drawing your inference?

      @@ -1396,8 +1394,8 @@

      Further constraints

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      ADA511 0.2
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      26&nb
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      26 

    Neural networks are performing extremely well on complex tasks such as language modelling and realistic image generation, although the principle behind how they work, is quite simple.

    The basic building block is a node, which receives some values as input, and computes a single output:

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    +
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    27 Preventing ov

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    Preface

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    References
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  • diff --git a/docs/search.json b/docs/search.json index 9abe3ab..76602b2 100644 --- a/docs/search.json +++ b/docs/search.json @@ -1541,7 +1541,7 @@ "text": "27.1 Machine learning from a bird’s-eye view\nThe last few chapters gave a brief introduction to and overview of popular machine-learning methods, their terminology, and the points of view that they typically adopt. Now let’s try to look at them keeping in mind our main goal in this course: exploring new inference methods, understanding their foundations, and thinking out of the box.\nIn this and the next few chapters we shall focus on the following question: to what purpose do we use machine-learning algorithms?. After answering this question and clarifying what the purpose is, we shall try to achieve it in an optimal way, according to the methods and concepts we studied in the initial part of the course. Remember that they are guaranteed to give the optimal solution (chapter  2). But we shall keep an eye open to see where our optimal methods seem to be similar or dissimilar to machine-learning methods.\nThereafetr, in the last chapters, we shall examine where the optimal solution and machine-learning methods converge and diverge, try to understand what machine-learning methods do from the point of view of our optimal solution, and think of ways to improve them.", "crumbs": [ "[**Inference III**]{.green}", - "27  [Beyond machine learning]{.midgrey}" + "27  [Beyond machine learning]{.green}" ] }, { @@ -1552,7 +1552,7 @@ "text": "27.2 A task-oriented categorization of some machine-learning problems\nFor our goal, the common machine-learning categorization and terminology discussed in chapter  24 are somewhat inadequate. Distinctions such as “supervised learning” vs “unsupervised learning” are of secondary importance to a data engineer (as opposed to a “data mechanic”) for several reasons:\n\n  They group together some types of tasks that are actually quite different from an inferential or decision-making viewpoint; and conversely they separate types of tasks that are quite similar.\n  They focus on procedures rather than on purposes.\n\nThe important questions for us, in fact, are: What do we wish to infer or choose? and From which kind of information? These questions define the problem we want to solve. \nLet’s introduce a different categorization of the kind of tasks that we want to accomplish; a categorization that tries to focus on the purpose or task, and on the types of desired information and of available information, rather than on the procedure.\nThe categorization below is informal. It only provides a starting point from which to examine a new kind of task we may face. Many tasks will fall in between categories: every data-engineering or data-science problem is unique.\n\nWe exclude from the start all tasks that require an agent to continuously and actively interact with its environment for acquiring information, making choices, getting feedback, and so on. Clearly these tasks are the domain of Decision Theory in its most complex form, with ramified decisions, strategies, and possibly the interaction with other decision-making agents. To explore and analyse this kind of tasks is beyond the purpose of this course.\n\n\n\n\n\n\n\n\n For the extra curious\n\n\n\n\nDecision Analysis\nChapters 16–18 in Artificial Intelligence\nGames and Decisions\n\n\n\n\nWe focus on tasks where multiple “instances” with similar characteristics are involved, and the agent has some question related to a “new instance”. According to the conceptual framework developed in part Data II, we can view these “instances” as units of a practically infinite population. The “characteristics” that the agent has observed or must guess are variates common to all these units.\n\n\nRemember that you can adopt any terminology you like. If you prefer “instance” and “characteristics” or some other words to “unit” and “variate”, then use them. What’s important is that you understand the ideas and methods behind these words\n\nNew unit: given vs generated\nA first distinction can be made between\n\n   Tasks where an agent must itself generate a new unit\n   Tasks where a new unit is given to an agent, who must guess some of its variates\n\nAn example of the first type of task is image generation: an algorithm is given a collection of images and is asked to generate a new image based on them.\nWe shall see that these two types of task are actually quite close to each other, from the point of view of Decision Theory and Probability Theory.\n\nThe terms “discriminative” and “generative” are sometimes associated in machine learning with the two types of task. This association, however, is quite loose, because some tasks typically called “generative” actually belong to the first type. We shall therefore avoid these terms. It’s enough to keep in mind the distinction between the two types of task above.\n\n\n\nGuessing variates: all or some\nFocusing on the second type of task (a new unit is given to the agent), we can further divide it into two subtypes:\n\n  The agent must guess all variates of the new unit\n  The agent must guess some variates of the new unit, but can observe other variates of the new unit\n\nAn example of the first subtype of task is the “urgent vs non-urgent” problem of § 17.4: having observed incoming patients, some of which where urgent and some non-urgent, the agent must guess whether the next incoming patient will be urgent or not. No other kinds of information (transport, patient characteristics, or others) are available.\nWe shall call predictands1 the variates that the agent must guess in a new unit, and predictors those that the agent can observe.2 The first subtype above can be viewed as a special case of the second where all variates are predictands, and there are no predictors.\n1 literally “what has to be predicted”2 In machine learning and other fields, the terms “dependent variable”, “class” or “label” (for nominal variates) are often used for “predictand”; and the terms “independent variable” or “features” are often used for “predictor”.\nThe terms “unsupervised learning” and “supervised learning” are sometimes associated in machine learning with these two subtypes of task. But the association is loose and can be misleading. “Clustering” tasks, for example, are usually called “unsupervised” but they are examples of the second subtype above, where the agent has some predictors.\n\n\n\nInformation available in previous units\nFinally we can further divide the second subtype above into two or three subsubtypes, depending on the information available to the agent about previous units:\n\n   All predictors and predictands of previous units are known to the agent\n   All predictors of previous units, but not the predictands, are known to the agent\n   All predictands of previous units, but not the predictors, are known to the agent\n\n\n\nAn example of the first subsubtype of task is image classification. The agent is for example given the following 128 × 128-pixel images and character-labels from the One Punch Man series:\n\nand is then given one new 128 × 128-pixel image:\n\n\n\n\n\nof which it must guess the character-label.\nIn the example just given, the image is the predictor, the character-label is the predictand.\n\n\nA slight modification of the example above gives us a task of the second subsubtype. A different agent is given the images above, but without labels:\n\nand must then guess some kind of “label” or “group” for the new image above; and possibly also for the images already given. The kind of “group” requested depends on the specific problem.\nIn this example the image is the predictor, and the label or group is the predictand. The difference from the previous example is that the agent doesn’t have the predictand values of previous units.\n\nThe term “supervised learning” typically refer to the first subsubtype above.\nThe term “unsupervised learning” can refer to the second subsubtype, for instance in “clustering” tasks. In a clustering task, the agent tries to guess which group or “cluster” a unit belong to, given a collection of similar units, whose groups are not known either. The cluster effectively is the predictand variate. In some cases the agent may want to guess the cluster not only of a new unit, but also of all previous units.\nThe third subsubtype is very rarely considered in machine learning, yet it is not an unrealistic task.\n\nThe types, subtypes, subsubtypes above are obviously not mutually exclusive or comprehensive. We can easily imagine scenarios where an agent has some predictors & predictands available about some previous units, but only predictors or only predictands available for other previous units. This scenario falls in between the three subsubtypes above. In machine learning, hybrid situations like these are categorized as “missing data” or “imputation”.", "crumbs": [ "[**Inference III**]{.green}", - "27  [Beyond machine learning]{.midgrey}" + "27  [Beyond machine learning]{.green}" ] }, { @@ -1563,7 +1563,7 @@ "text": "27.3 Flexible categorization using probability theory\nWe have been speaking about the agent’s guessing the values of some variates. “Guessing” means that there’s a state of uncertainty: the agent can’t simply say something like “the value of the label is Saitama”, because that could be false. Uncertainty means that the most honest thing that the agent can do is to express degrees of belief about each of the possible values. Probability theory enters the scene.\nIn fact it turns out that the categorization above into subtypes and subsubtypes of tasks can be presented in a more straightforward and flexible way using probability-theory notation.\n\nNotation\nFirst let’s introduce some symbol conventions to be used in the next chapters.\n\nWe shall denote with \\({\\color[RGB]{68,119,170}Z}\\) all variates that are of interest to the agent: those to be guessed as well as those that may be already known.\nThe variates to be guessed in a new unit (the predictands) will be collectively denoted with \\({\\color[RGB]{68,119,170}Y}\\).\nThe variates that can be observed in a new unit (the predictors) will be collectively denoted with \\({\\color[RGB]{68,119,170}X}\\). In cases where there are no predictors, \\({\\color[RGB]{68,119,170}X}\\) is empty.\n\nTherefore we have \\({\\color[RGB]{68,119,170}Z}= ({\\color[RGB]{68,119,170}Y}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}{\\color[RGB]{68,119,170}X})\\). In cases where there are no predictors we have \\({\\color[RGB]{68,119,170}Z}= {\\color[RGB]{68,119,170}Y}\\).\n\n\\({\\color[RGB]{68,119,170}Z}_i\\)  denote all variates for unit #\\(i\\).\n\\({\\color[RGB]{68,119,170}Y}_i\\)  denote all predictands for unit #\\(i\\).\n\\({\\color[RGB]{68,119,170}X}_i\\)  denote all predictors for unit #\\(i\\).\n\nAs usual we number from  \\(i=1\\)  to  \\(i=N\\)  the units that serve for learning, and  \\(i=N+1\\)  is the new unit of interest to the agent.\nRecall (§ 5.3) that in probability notation\n\\[\\mathrm{P}(\\text{\\color[RGB]{238,102,119}\\small[proposal]}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\\text{\\color[RGB]{34,136,51}\\small[conditional]} \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\\]\nthe proposal contains what the agent’s belief is about, and the conditional contains what’s supposed to be known to the agent, together with the background information \\(\\mathsfit{I}\\).\n\nFinally let’s see how to express different typologies of tasks in probability notation.\n\n\n\n\n  The agent must guess all variates of the new unit\nThis kinds of guess is represented by the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nfor all possible values \\(\\color[RGB]{238,102,119}z\\) in the domain of \\({\\color[RGB]{68,119,170}Z}\\). The specific values \\(\\color[RGB]{34,136,51}z_N, \\dotsc, z_1\\) of the variate \\({\\color[RGB]{68,119,170}Z}\\) for the previous units are known to the agent.\n\n\n\n\n  The agent must guess some variates of the new unit, having observed other variates of the new unit\nThis kind of guess is represented by the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\n\\dotsb \\,\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nfor all possible values \\(\\color[RGB]{238,102,119}y\\) in the domain of the predictands \\({\\color[RGB]{68,119,170}Y}\\). The value \\(\\color[RGB]{34,136,51}x\\) of the predictors \\({\\color[RGB]{68,119,170}X}\\) for the new unit is known to the agent.\nThe remaining information “\\(\\dotsb\\)” contained in the conditional depends on the subsubtype of task:\n\n\n\n   All predictors and predictands of previous units are known to the agent\nThis corresponds to the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nfor all possible \\(\\color[RGB]{238,102,119}y\\). All information about predictands \\({\\color[RGB]{68,119,170}Y}\\) and predictors \\({\\color[RGB]{68,119,170}X}\\) for previous units appears in the conditional.\nIn the example with image classification, a pictorial representation of this probability would be\n\nwhere \\({\\color[RGB]{238,102,119}y} \\in \\set{\\color[RGB]{238,102,119}{\\small\\verb;Saitama;}, {\\small\\verb;Fubuki;}, {\\small\\verb;Genos;}, {\\small\\verb;MetalBat;}, \\dotsc \\color[RGB]{0,0,0}}\\).\n\n\n\n\n   All predictors of previous units, but not their predictands, are known to the agent\nThis corresponds to the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nfor all possible \\(\\color[RGB]{238,102,119}y\\). All information about predictors \\({\\color[RGB]{68,119,170}X}\\) for the previous units, but not that about their predictands \\({\\color[RGB]{68,119,170}Y}\\), appears in the conditional.\n\n\n\n\n\nMore general and hybrid tasks\nConsider a task that doesn’t fit into any of the types discussed above: The agent wants to guess the predictands for a new unit, say #3, after observing that its predictors have value \\(\\color[RGB]{34,136,51}x\\). Of two previous units, the agent knows the predictor value \\(\\color[RGB]{34,136,51}x_1\\) of the first, and the predictand value \\(\\color[RGB]{34,136,51}y_2\\) of the second. This task is expressed by the probability\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{3}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{3}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{2}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{2}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\n\n\n\n\n\n\n\n\n\n Exercises\n\n\n\n\nWrite down the general probability expression for the task of subsubtype “all predictands of previous units, but not their predictors, are known to the agent”.\nWhat kind of task does the following probability express?:\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}Y_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{2}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{2}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}X_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{2}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{2}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nWhat kind of task could it represent in machine-learning terminology?\n\n\n\n\n\n\n   Tasks where an agent must itself generate a new unit\nOur very first categorization included the task of generating a new unit, given previous examples. In this kind of task there are possible alternatives that the agent could generate. How should one alternative be chosen? A moment’s thought shows that the probabilities for the possible alternatives should enter the choice.\nSuppose, as a very simple example, that a generative agent has been shown, in an unsystematic order, 30 copies of the symbol and 10 copies of the symbol , and is asked to generate a new symbol out of these examples. Intuitively we expect that it will generate , but we cannot and don’t want to exclude the possibility that it will generate . These two generation possibilities should simply have different probabilities and, in the long run, appear with different frequencies.\nAlso in this kind of task, therefore, we have the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nthe difference from before is that the sentence \\(\\color[RGB]{238,102,119}Z_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\\) represents not the hypothesis that a given new unit has value \\(\\color[RGB]{238,102,119}z\\), but the possibility of generating a new unit with that value. In other words, the symbol “\\(\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}\\)” here means “must be set to…” rather than “would be observed to be…”. Remember the discussion and warnings in § 6.3?\n\n\nOur general conclusion is this:\n\n\n\n\n\n\n \n\n\n\n\nProbability distribution such as those discussed above should intrinsically enter all types of machine-learning algorithms.\n\n\n\nThis is the condition for machine-learning algorithms to be optimal and self-consistent. The less an algorithm satisfies that condition, the less optimal and less consistent it is.", "crumbs": [ "[**Inference III**]{.green}", - "27  [Beyond machine learning]{.midgrey}" + "27  [Beyond machine learning]{.green}" ] }, { @@ -1574,7 +1574,7 @@ "text": "27.4 The underlying distribution\nA remarkable feature of all the probabilities discussed in the above task categorization is that they can all be calculated from one and the same probability distribution. We briefly discussed and used this feature in chapter  17.\nA conditional probability such as \\(\\mathrm{P}(\\mathsfit{\\color[RGB]{238,102,119}A}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\\mathsfit{\\color[RGB]{34,136,51}B} \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\\) can always be written, by the and-rule, as the ratio of two probabilities:\n\\[\n\\mathrm{P}(\\mathsfit{\\color[RGB]{238,102,119}A}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\\mathsfit{\\color[RGB]{34,136,51}B} \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n=\n\\frac{\n\\mathrm{P}(\\mathsfit{\\color[RGB]{238,102,119}A}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{\\color[RGB]{34,136,51}B} \\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\mathrm{P}(\\mathsfit{\\color[RGB]{34,136,51}B} \\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\]\nTherefore we have, for the probabilities of some of the tasks above,\n\n\\[\n\\begin{aligned}\n&\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n=\n\\frac{\n\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\mathrm{P}(\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\\\[2em]\n&\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\\\[2ex]\n&\\qquad{}=\n\\frac{\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\mathrm{P}(\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\\\[2em]\n&\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\\\[2ex]\n&\\qquad{}=\n\\frac{\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\mathrm{P}(\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\end{aligned}\n\\]\n\n\n\nWe also know the marginalization rule (chapter  16.1): any quantity \\(\\color[RGB]{204,187,68}C\\) with values \\(\\color[RGB]{204,187,68}c\\) can be introduced into the proposal of a probability via the or-rule:\n\\[\n\\mathrm{P}( {\\color[RGB]{34,136,51}\\boldsymbol{\\dotsb}} \\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I}) =\n\\sum_{\\color[RGB]{204,187,68}c}\\mathrm{P}({\\color[RGB]{204,187,68}C\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}c} \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}{\\color[RGB]{34,136,51}\\boldsymbol{\\dotsb}} \\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n\\]\nUsing the marginalization rule we find these final expressions for the probabilities of some machine-learning tasks discussed so far:\n\n\n\n\n\n\n\n \n\n\n\n\n Guess all variates:\n\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n=\n\\frac{\n\\mathrm{P}(\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\sum_{\\color[RGB]{170,51,119}z}\n\\mathrm{P}(\n\\color[RGB]{238,102,119}\nZ_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}{\\color[RGB]{170,51,119}z}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\color[RGB]{34,136,51}\nZ_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nZ_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}z_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\]\n\n\n\n All previous predictors and predictands known:\n\n\\[\n\\begin{aligned}\n&\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\\\[2ex]\n&\\qquad{}=\n\\frac{\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\sum_{\\color[RGB]{170,51,119}y}\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}{\\color[RGB]{170,51,119}y}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\end{aligned}\n\\]\n\n\n\n Previous predictors known, previous predictands unknown:\n\n\\[\n\\begin{aligned}\n&\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\\\[2ex]\n&\\quad{}=\n\\frac{\n\\sum_{\\color[RGB]{204,187,68}y_{N}, \\dotsc, y_{1}}\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\color[RGB]{0,0,0}\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\n\\color[RGB]{204,187,68}\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\color[RGB]{34,136,51}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{204,187,68}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\color[RGB]{34,136,51}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}{\n\\sum_{{\\color[RGB]{170,51,119}y}, \\color[RGB]{204,187,68}y_{N}, \\dotsc, y_{1}}\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}{\\color[RGB]{170,51,119}y}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{34,136,51}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\color[RGB]{0,0,0}\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\n\\color[RGB]{204,187,68}\nY_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\color[RGB]{34,136,51}\nX_{N}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\n\\color[RGB]{204,187,68}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\color[RGB]{34,136,51}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\end{aligned}\n\\]\n\n\n\nAll these formulae, even for hybrid tasks, involve sums and ratios of only one distribution:\n\\[\\boldsymbol{\n\\mathrm{P}(\\color[RGB]{68,119,170}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n}\n\\]\nStop for a moment and contemplate some of the consequences of this remarkable fact:\n\n  An agent that can perform one of the tasks above can, in principle, also perform all other tasks.\nThis is why a perfect agent, working with probability, in principle does not have to worry about “supervised”, “unsupervised”, “missing data”, “imputation”, and similar situations. This also shows what was briefly mentioned before: all these task typologies are much closer to one another than it might look like from the perspective of current machine-learning methods.\n\n\n\nThe acronym OPM can stand for Optimal Predictor Machine or Omni-Predictor Machine\n\n  The probability distribution above encodes the agent’s background knowledge and assumptions; different agents differ only in the values of that distribution.\nIf two agents yield different probability values in the same task, with the same variates and same training data, the difference must come from the joint probability distribution above. And, since the data given to the two agents are exactly the same, the difference must lie in the agents’ background information \\(\\mathsfit{I}\\).\n  Data cannot “speak for themselves”\nGiven some data, we can choose two different joint distributions for these data, and therefore get different results in our inferences and tasks. This means that the data alone cannot determine the result: specific background information and assumptions, whether acknowledged or not, always affect the result.\n\nThe qualification “in principle” in the first consequence is important. Some of the sums that enter the formulae above are computationally extremely expensive and, with current technologies and maths techniques, cannot be performed within a reasonable time. But new technologies and new maths discoveries could make these calculations possible. This is why a data engineer cannot simply brush them aside and forget them.\nAs regards the third consequence, we shall see that there are different states of knowledge which can converge to the same results, as the number of training data increases.\n\n\n\n\n\n\n Exercise\n\n\n\nIn a previous example of “hybrid” task we had the probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{238,102,119}\nY_{3}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{}\n\\color[RGB]{34,136,51}\nX_{3}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x\n\\, \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\,\nY_{2}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{2}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\mathsfit{I})\n\\]\nRewrite it in terms of the underlying joint distribution.", "crumbs": [ "[**Inference III**]{.green}", - "27  [Beyond machine learning]{.midgrey}" + "27  [Beyond machine learning]{.green}" ] }, { @@ -1585,7 +1585,7 @@ "text": "27.5 Plan for the next few chapters\nOur goal in building an “Optimal Predictor Machine” is now clear: we must find a way to\n\n\n\n\nassign the joint probability distribution above, in such a way that it reflects some reasonable background information\nencode the distribution in a computationally useful way\n\nThe “encode” goal sounds quite challenging, because the number \\(N\\) of units can in principle be infinite; we have an infinite probability distribution.\nIn the next Inference III part we shall see that partially solving the “assign” goal actually makes the “encode” goal feasible.\n\nOne question arises if we now look at machine-learning methods from our Probability Theory perspective. Some machine-learning methods, including many popular ones, don’t give us probabilities about values. They return one definite value. How do we reconcile this with the probabilistic point of view above? We shall answer this question in full in the last chapters; but a short, intuitive answer can already be given now.\nIf there are several possible correct answers to a given guess, but a machine-learning algorithm gives us only one answer, then the algorithm must have internally chosen one of them. In other words, the machine-learning algorithm is internally doing decision-making. We know from chapters Chapter 2 and Chapter 3 that this process should obey Decision Theory and therefore must involve:\n\n  the probabilities of the possible correct answers\n  the utilities of the possible answer choices\n\nNon-probabilistic machine-learning algorithms must therefore be approximations of an Optimal Predictor Machine that, after computing probabilities, selects one particular answer by using utilities.", "crumbs": [ "[**Inference III**]{.green}", - "27  [Beyond machine learning]{.midgrey}" + "27  [Beyond machine learning]{.green}" ] }, { @@ -1593,7 +1593,7 @@ "href": "exchangeable_probabilities.html", "title": "28  Exchangeable beliefs", "section": "", - "text": "28.1 Recap\nIn the chapters of part Inference I we had an overview of how an agent can draw inferences and make predictions of the most general kind, expressed by general sentences, using the four fundamental rules of inference.\nThen, in part Inference II, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and machine-learning algorithms. First, inferences about measurements and observations; then, inferences about multiple instances of similar measurements and observations. The idea was that that an agent can arrive at sharper degrees of belief – that is, learn – by using information about “similar instances”.\nFor these purposes we introduced a specialized language about quantities and data types in part Data I, and about “populations” of similar “units” in part Data II.\nIn the Machine learning part we took an overview of current machine-learning methods, and then focused on several types of “multi-instance” tasks that popular machine-learning algorithm, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent – one that operates according to Probability Theory – can in principle perform any and all of those tasks by using the joint probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{68,119,170}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n\\]\nin some calculations involving sums and divisions. This distribution must be specified for all possible values of \\(\\color[RGB]{68,119,170}x_{1}, \\dotsc, x_{N+1}\\), \\(\\color[RGB]{68,119,170}y_{1}, \\dotsc, y_{N+1}\\), and \\(N\\).\nFor example, the task (“supervised learning”) of predicting some variates (predictands) for a new unit, from knowledge of the other variates (predictors) for the same units, and of all variates for \\(N\\) other units, is given by\nOur problem now is to (1) choose such a distribution according to reasonable assumptions and background information, (2) encode it in a computationally feasible way.\nFor these two purposes we shall now narrow our focus further, upon inferences satisfying a property that greatly simplifies the calculations, and that is reasonable in some contexts. Luckily such contexts are typical of many “supervised” and “unsupervised” machine-learning applications.", + "text": "28.1 Recap\nIn the chapters of part Inference I we had an overview of how an agent can draw inferences and make predictions of the most general kind, expressed by general sentences, using the four fundamental rules of inference.\nThen, in part Inference II, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and of machine-learning algorithms. First we considered inferences about measurements and observations, then inferences about multiple instances of similar measurements and observations. The idea was that an agent can arrive at sharper degrees of belief – that is, it can learn – by using information about “similar instances”.\nFor these purposes we introduced a specialized language about quantities and data types in part Data I, and about “populations” of similar “units” in part Data II.\nIn the Machine learning part we took an overview of current machine-learning methods, and then focused on several types of tasks that popular machine-learning algorithms, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent – one that operates according to Probability Theory – can in principle perform any and all of those tasks by using the joint probability distribution\n\\[\n\\mathrm{P}(\\color[RGB]{68,119,170}\nY_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{N+1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{N+1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\\dotsb \\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nY_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}y_{1}\n\\mathbin{\\mkern-0.5mu,\\mkern-0.5mu}\nX_{1}\\mathclose{}\\mathord{\\nonscript\\mkern 0mu\\textrm{\\small=}\\nonscript\\mkern 0mu}\\mathopen{}x_{1}\n\\color[RGB]{0,0,0}\\nonscript\\:\\vert\\nonscript\\:\\mathopen{} \\mathsfit{I})\n\\]\nThe agent only needs to do some calculations with this joint distribution, involving sums and divisions. This distribution must be specified for all possible values of \\(\\color[RGB]{68,119,170}x_{1}, \\dotsc, x_{N+1}\\), \\(\\color[RGB]{68,119,170}y_{1}, \\dotsc, y_{N+1}\\), and \\(N\\).\nTake “supervised learning” for example, that is, the task of predicting some variates (predictands) for a new unit, from knowledge of other variates (predictors) for the same unit and of all variates for \\(N\\) other units. Solving this task corresponds to calculating\nTo build an AI agent that deals with these kinds of task we must: (1) choose a joint distribution according to reasonable assumptions and background information, (2) encode it in a computationally feasible way.\nIn order to reach these two goals we shall now narrow our focus further, upon inferences satisfying a condition that greatly simplifies the calculations, and that is also reasonable in many real inference problems – and it is moreover typical of many “supervised” and “unsupervised” machine-learning applications.", "crumbs": [ "[**Inference III**]{.green}", "28  [Exchangeable beliefs]{.green}" @@ -1615,7 +1615,7 @@ "href": "exchangeable_probabilities.html#sec-excheable-beliefs", "title": "28  Exchangeable beliefs", "section": "28.2 States of knowledge with symmetries", - "text": "28.2 States of knowledge with symmetries\nAn agent’s degrees of belief about a particular population may satisfy a special symmetry called exchangeability. This symmetry can be understood from different points of view. We start from one of these viewpoints, and then make connections with alternative ones.\nTake again two populations briefly mentioned in § 20.1:\n\n\nStock exchange\n\nThe daily change in closing price of a stock during 1000 consecutive days. Each day the change can be positive or zero: \\({\\color[RGB]{34,136,51}{\\small\\verb;+;}}\\), or negative: \\({\\color[RGB]{238,102,119}{\\small\\verb;-;}}\\).\n\n\n\n\n\n\n\n\nMars prospecting\n\nA collection of 1000 similar-sized rocks gathered from a specific, large crater on Mars. Each rock either contains haematite: \\({\\color[RGB]{102,204,238}{\\small\\verb;Y;}}\\), or it doesn’t: \\({\\color[RGB]{204,187,68}{\\small\\verb;N;}}\\).\n\n\n\n\n\n\n\n\nSuppose that, in each of these populations, you (the agent) don’t know the variate value for unit #735, and for some reason would like to infer it. You are given the variate values for 100 other units, which you can use to improve your inference. Now consider this question:\n\nHow much does the relative order of the 100 known units and the unknown unit matter to you, for drawing your inference?\n\nWe know, from information theory, that it never hurts having extra information, such as the units’ order. But you probably judge the units’ order to be much more important for your inference in the stock-exchange case than in the Mars-prospecting one. In the stock-exchange case it would be more informative to have data from units temporally close to unit #735; for example units #635–#734, or #736–#835, or #685–#734 & #736–#785, or similar ranges. But in the Mars-prospecting case you might find it acceptable if the 100 known units were picked up in some unsystematic way from the catalogue of remaining 999 units. There are reasons, boiling down to physics, behind this kind of judgement.\nThe question above could also be replaced by others, slightly different but still connected to the same issue. For example:\n\nHow strongly would you like to be able to choose which 100 units you can have data from, in order to draw your inference?\n\nor\n\nHow much would you be upset if the original order of the population units were destroyed by accidental shuffling?\n\nor\n\nWould it be acceptable to you if only the frequencies of the values (\\(\\set{{\\color[RGB]{34,136,51}{\\small\\verb;+;}},{\\color[RGB]{238,102,119}{\\small\\verb;-;}}}\\) in one case, \\(\\set{{\\color[RGB]{102,204,238}{\\small\\verb;Y;}},{\\color[RGB]{204,187,68}{\\small\\verb;N;}}}\\) in the other) for the 100 known units were given to you?\n\n\n\n\n\n\n\n Exercise\n\n\n\n\nFind examples of populations where the units have some kind of ordering that you think would be very important for drawing inferences about some units, given other units. Examine why you judge such ordering to be important. (The ordering doesn’t need to be one-dimensional. For instance, the pixel intensities of an image also have a two-dimensional relative order or position: is that important if you want to draw inferences about the intensities of some pixels from those of other pixels?)\nFind examples of populations where any potential ordering of the units would not be very important for drawing inferences about some units, given other units. Or, put it otherwise, you wouldn’t be excessively upset or worried if such order were lost owing to accidental shuffling of the units.\n\n\n\nMany kinds of inference considered in data science and engineering, and all inferences done with “supervised” or “unsupervised” machine-learning algorithms, are examples where any ordering of the data used for learning is deemed irrelevant and is, in fact, often lost. This irrelevance is clear from the data-shuffling involved in many procedures that accompany these algorithms.\nWe shall thus restrict our attention to situations and kinds of background information where this judgement of irrelevance is considered appropriate. In reality this is not a black-or-white situation: it is possible that some kind of ordering information would improve our inferences; what we are assuming here is that this improvement is so small that it can be neglected altogether.", + "text": "28.2 States of knowledge with symmetries\nAn agent’s degrees of belief about a particular population may satisfy a special symmetry called exchangeability. This symmetry can be understood from different points of view. Let’s start from one of these viewpoints, and then make connections with alternative ones.\nTake again two populations briefly mentioned in § 20.1:\n\n\nStock exchange\n\nThe daily change in closing price of a stock during 1000 consecutive days. Each day the change can be positive or zero: \\({\\color[RGB]{34,136,51}{\\small\\verb;+;}}\\), or negative: \\({\\color[RGB]{238,102,119}{\\small\\verb;-;}}\\).\n\n\n\n\n\n\n\n\nMars prospecting\n\nA collection of 1000 similar-sized rocks gathered from a specific, large crater on Mars. Each rock either contains haematite: \\({\\color[RGB]{102,204,238}{\\small\\verb;Y;}}\\), or it doesn’t: \\({\\color[RGB]{204,187,68}{\\small\\verb;N;}}\\).\n\n\n\n\n\n\n\nSuppose that, in each of these populations, you (the agent) don’t know the variate value for unit #735, and for some reason would like to infer it. You are given the variate values for 100 other units, which you can use to improve your inference. Now consider this question:\n\nHow much does the relative order of the 100 known units and the unknown unit matter to you, for drawing your inference?\n\nWe know, from information theory, that it never hurts having extra information, such as the units’ order. But you probably judge the units’ order to be much more important for your inference in the stock-exchange case than in the Mars-prospecting one. In the stock-exchange case it would be more informative to have data from units temporally close to unit #735; for example units #635–#734, or #736–#835, or #685–#734 & #736–#785, or similar ranges. But in the Mars-prospecting case you might find it acceptable if the 100 known units were picked up in some unsystematic way from the catalogue of remaining 999 units. There are reasons, boiling down to physics, behind this kind of judgement.\nThe question above could also be replaced by others, slightly different but still connected to the same issue. For example:\n\nHow strongly would you like to be able to choose which 100 units you can have data from, in order to draw your inference?\n\nor\n\nHow much would you be upset if the original order of the population units were destroyed by accidental shuffling?\n\nor\n\nWould it be acceptable to you if only the frequencies of the values (\\(\\set{{\\color[RGB]{34,136,51}{\\small\\verb;+;}},{\\color[RGB]{238,102,119}{\\small\\verb;-;}}}\\) in one case, \\(\\set{{\\color[RGB]{102,204,238}{\\small\\verb;Y;}},{\\color[RGB]{204,187,68}{\\small\\verb;N;}}}\\) in the other) for the 100 known units were given to you?\n\n\n\n\n\n\n\n Exercise\n\n\n\n\nFind examples of populations where the units have some kind of ordering that you think would be very important for drawing inferences about some units, given other units. Examine why you judge such ordering to be important. (The ordering doesn’t need to be one-dimensional. For instance, the pixel intensities of an image also have a two-dimensional relative order or position: is that important if you want to draw inferences about the intensities of some pixels from those of other pixels?)\nFind examples of populations where any potential ordering of the units would not be very important for drawing inferences about some units, given other units. Or, put it otherwise, you wouldn’t be excessively upset or worried if such order were lost owing to accidental shuffling of the units.\n\n\n\nMany kinds of inference considered in data science and engineering, and all inferences done with “supervised” or “unsupervised” machine-learning algorithms, are examples where any ordering of the data used for learning is deemed irrelevant and is, in fact, often lost. This irrelevance is clear from the data-shuffling involved in many procedures that accompany these algorithms.\nWe shall thus restrict our attention to situations and kinds of background information where this judgement of irrelevance is considered appropriate. In reality this is not a black-or-white situation: it is possible that some kind of ordering information would improve our inferences; what we are assuming here is that this improvement is so small that it can be neglected altogether.", "crumbs": [ "[**Inference III**]{.green}", "28  [Exchangeable beliefs]{.green}" diff --git a/docs/sentences.html b/docs/sentences.html index c90ae81..e3b7321 100644 --- a/docs/sentences.html +++ b/docs/sentences.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/statistics.html b/docs/statistics.html index d7aaf31..22fda24 100644 --- a/docs/statistics.html +++ b/docs/statistics.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/subpopulations.html b/docs/subpopulations.html index 38cfb3d..ae91c95 100644 --- a/docs/subpopulations.html +++ b/docs/subpopulations.html @@ -402,7 +402,7 @@

    22&nb
  • - 27  Beyond machine learning + 27  Beyond machine learning
  • diff --git a/docs/summary_formulae.html b/docs/summary_formulae.html index ca06feb..35d8053 100644 --- a/docs/summary_formulae.html +++ b/docs/summary_formulae.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/thanks.html b/docs/thanks.html index 68e072a..1e7e699 100644 --- a/docs/thanks.html +++ b/docs/thanks.html @@ -401,7 +401,7 @@

    Thanks

    - 27  Beyond machine learning + 27  Beyond machine learning
  • diff --git a/docs/truth_inference.html b/docs/truth_inference.html index de416fd..4e30551 100644 --- a/docs/truth_inference.html +++ b/docs/truth_inference.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/utilities.html b/docs/utilities.html index 6901b96..236c583 100644 --- a/docs/utilities.html +++ b/docs/utilities.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/utilities_evaluation.html b/docs/utilities_evaluation.html index 9861c9c..76e433c 100644 --- a/docs/utilities_evaluation.html +++ b/docs/utilities_evaluation.html @@ -402,7 +402,7 @@

    - 27  Beyond machine learning + 27  Beyond machine learning

  • diff --git a/docs/whither.html b/docs/whither.html index 3112041..767d534 100644 --- a/docs/whither.html +++ b/docs/whither.html @@ -402,7 +402,7 @@

    What next?
  • - 27  Beyond machine learning + 27  Beyond machine learning
  • diff --git a/exchangeable_probabilities.qmd b/exchangeable_probabilities.qmd index f3f13e8..77e95d5 100644 --- a/exchangeable_probabilities.qmd +++ b/exchangeable_probabilities.qmd @@ -6,11 +6,11 @@ In the chapters of part [Inference I]{.green} we had an overview of how an agent can draw inferences and make predictions of the most general kind, expressed by general sentences, using the four fundamental rules of inference. -Then, in part [Inference II]{.green}, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and machine-learning algorithms. First, inferences about measurements and observations; then, inferences about multiple instances of similar measurements and observations. The idea was that that an agent can arrive at sharper degrees of belief -- that is, learn -- by using information about "similar instances". +Then, in part [Inference II]{.green}, we successively narrowed our focus on more and more specialized kinds of inference, typical of engineering and data-science problems and of machine-learning algorithms. First we considered inferences about measurements and observations, then inferences about multiple instances of similar measurements and observations. The idea was that an agent can arrive at sharper degrees of belief -- that is, it can *learn* -- by using information about "similar instances". For these purposes we introduced a specialized language about quantities and data types in part [Data I]{.yellow}, and about "populations" of similar "units" in part [Data II]{.yellow}. -In the [Machine learning]{.red} part we took an overview of current machine-learning methods, and then focused on several types of "multi-instance" tasks that popular machine-learning algorithm, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent -- one that operates according to Probability Theory -- can in principle perform *any and all* of those tasks by using the joint probability distribution +In the [Machine learning]{.red} part we took an overview of current machine-learning methods, and then focused on several types of tasks that popular machine-learning algorithms, such as deep networks and random forests, purport to solve. We found a remarkable result: a perfect agent -- one that operates according to Probability Theory -- can in principle perform *any and all* of those tasks by using the joint probability distribution $$ \P(\blue @@ -24,9 +24,9 @@ X_{1}\mo x_{1} \black \| \yI) $$ -in some calculations involving sums and divisions. This distribution must be specified for all possible values of $\blue x_{1}, \dotsc, x_{N+1}$, $\blue y_{1}, \dotsc, y_{N+1}$, and $N$. +The agent only needs to do some calculations with this joint distribution, involving sums and divisions. This distribution must be specified for all possible values of $\blue x_{1}, \dotsc, x_{N+1}$, $\blue y_{1}, \dotsc, y_{N+1}$, and $N$. -For example, the task ("supervised learning") of predicting some variates (predictands) for a new unit, from knowledge of the other variates (predictors) for the same units, and of all variates for $N$ other units, is given by +Take "supervised learning" for example, that is, the task of predicting some variates (predictands) for a new unit, from knowledge of other variates (predictors) for the same unit and of all variates for $N$ other units. Solving this task corresponds to calculating :::{.column-page-inset-right} $$ @@ -67,16 +67,15 @@ $$ ::: \ +To build an AI agent that deals with these kinds of task we must: (1) choose a joint distribution according to reasonable assumptions and background information, (2) encode it in a computationally feasible way. -Our problem now is to (1) choose such a distribution according to reasonable assumptions and background information, (2) encode it in a computationally feasible way. - -For these two purposes we shall now narrow our focus further, upon inferences satisfying a property that greatly simplifies the calculations, and that is reasonable in some contexts. Luckily such contexts are typical of many "supervised" and "unsupervised" machine-learning applications. +In order to reach these two goals we shall now narrow our focus further, upon inferences satisfying a condition that greatly simplifies the calculations, and that is also reasonable in many real inference problems -- and it is moreover typical of many "supervised" and "unsupervised" machine-learning applications. \ ## States of knowledge with symmetries {#sec-excheable-beliefs} -An agent's degrees of belief about a particular population may satisfy a special symmetry called [**exchangeability**]{.blue}. This symmetry can be understood from different points of view. We start from one of these viewpoints, and then make connections with alternative ones. +An agent's degrees of belief about a particular population may satisfy a special symmetry called [**exchangeability**]{.blue}. This symmetry can be understood from different points of view. Let's start from one of these viewpoints, and then make connections with alternative ones. Take again two populations briefly mentioned in [§@sec-collections]: @@ -96,7 +95,6 @@ Take again two populations briefly mentioned in [§@sec-collections]: ::: \ - Suppose that, in each of these populations, you (the agent) don't know the variate value for unit [#735]{.midgrey}, and for some reason would like to infer it. You are given the variate values for 100 other units, which you can use to improve your inference. Now consider this question: > [*How much does the relative order of the 100 known units and the unknown unit matter to you, for drawing your inference?*]{.blue}