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VI-ELBO.html
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<meta name="author" content="Chris Orm" />
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<meta name="keywords" content="Variational Inference, VI, ELBO, posts, " />
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<meta property="og:description" content="Deriving the Evidence Lower Bound¶Recap¶In the previous post, I talked about different types of bounds we could have, and why we need to use them. As a quick recap, we are looking to compute the marginalization of a joint distribution in a general setting: $P(X) = \int_{Z} P(X, Z) dZ$ Previously, I showed how this can be an incredibly difficult thing to compute using standard methods, such as numerical integration, because of the curse of dimensionality." />
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<meta name="twitter:title" content="Variational Inference - Deriving the ELBO ">
<meta name="twitter:description" content="Deriving the Evidence Lower Bound¶Recap¶In the previous post, I talked about different types of bounds we could have, and why we need to use them. As a quick recap, we are looking to compute the marginalization of a joint distribution in a general setting: $P(X) = \int_{Z} P(X, Z) dZ$ Previously, I showed how this can be an incredibly difficult thing to compute using standard methods, such as numerical integration, because of the curse of dimensionality.">
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<h1><a href="https://chrisorm.github.io/VI-ELBO.html"> Variational Inference - Deriving the ELBO </a></h1>
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.cm-s-ipython span.cm-number {
color: #080;
}
.cm-s-ipython span.cm-def {
color: #00f;
}
.cm-s-ipython span.cm-variable {
color: #000;
}
.cm-s-ipython span.cm-operator {
color: #AA22FF;
font-weight: bold;
}
.cm-s-ipython span.cm-variable-2 {
color: #1a1a1a;
}
.cm-s-ipython span.cm-variable-3 {
color: #333333;
}
.cm-s-ipython span.cm-comment {
color: #408080;
font-style: italic;
}
.cm-s-ipython span.cm-string {
color: #BA2121;
}
.cm-s-ipython span.cm-string-2 {
color: #f50;
}
.cm-s-ipython span.cm-meta {
color: #AA22FF;
}
.cm-s-ipython span.cm-qualifier {
color: #555;
}
.cm-s-ipython span.cm-builtin {
color: #008000;
}
.cm-s-ipython span.cm-bracket {
color: #997;
}
.cm-s-ipython span.cm-tag {
color: #170;
}
.cm-s-ipython span.cm-attribute {
color: #00c;
}
.cm-s-ipython span.cm-header {
color: blue;
}
.cm-s-ipython span.cm-quote {
color: #090;
}
.cm-s-ipython span.cm-link {
color: #00c;
}
.cm-s-ipython span.cm-error {
color: #f00;
}
.cm-s-ipython span.cm-tab {
background: url(data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADAAAAAMCAYAAAAkuj5RAAAAAXNSR0IArs4c6QAAAGFJREFUSMft1LsRQFAQheHPowAKoACx3IgEKtaEHujDjORSgWTH/ZOdnZOcM/sgk/kFFWY0qV8foQwS4MKBCS3qR6ixBJvElOobYAtivseIE120FaowJPN75GMu8j/LfMwNjh4HUpwg4LUAAAAASUVORK5CYII=);
background-position: right;
background-repeat: no-repeat;
}
div.output_wrapper {
/* this position must be relative to enable descendents to be absolute within it */
position: relative;
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: vertical;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: vertical;
-moz-box-align: stretch;
display: box;
box-orient: vertical;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: column;
align-items: stretch;
z-index: 1;
}
/* class for the output area when it should be height-limited */
div.output_scroll {
/* ideally, this would be max-height, but FF barfs all over that */
height: 24em;
/* FF needs this *and the wrapper* to specify full width, or it will shrinkwrap */
width: 100%;
overflow: auto;
border-radius: 2px;
-webkit-box-shadow: inset 0 2px 8px rgba(0, 0, 0, 0.8);
box-shadow: inset 0 2px 8px rgba(0, 0, 0, 0.8);
display: block;
}
/* output div while it is collapsed */
div.output_collapsed {
margin: 0px;
padding: 0px;
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: vertical;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: vertical;
-moz-box-align: stretch;
display: box;
box-orient: vertical;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: column;
align-items: stretch;
}
div.out_prompt_overlay {
height: 100%;
padding: 0px 0.4em;
position: absolute;
border-radius: 2px;
}
div.out_prompt_overlay:hover {
/* use inner shadow to get border that is computed the same on WebKit/FF */
-webkit-box-shadow: inset 0 0 1px #000;
box-shadow: inset 0 0 1px #000;
background: rgba(240, 240, 240, 0.5);
}
div.output_prompt {
color: #D84315;
}
/* This class is the outer container of all output sections. */
div.output_area {
padding: 0px;
page-break-inside: avoid;
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: horizontal;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: horizontal;
-moz-box-align: stretch;
display: box;
box-orient: horizontal;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: row;
align-items: stretch;
}
div.output_area .MathJax_Display {
text-align: left !important;
}
div.output_area
div.output_area
div.output_area img,
div.output_area svg {
max-width: 100%;
height: auto;
}
div.output_area img.unconfined,
div.output_area svg.unconfined {
max-width: none;
}
/* This is needed to protect the pre formating from global settings such
as that of bootstrap */
.output {
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: vertical;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: vertical;
-moz-box-align: stretch;
display: box;
box-orient: vertical;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: column;
align-items: stretch;
}
@media (max-width: 540px) {
div.output_area {
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: vertical;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: vertical;
-moz-box-align: stretch;
display: box;
box-orient: vertical;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: column;
align-items: stretch;
}
}
div.output_area pre {
margin: 0;
padding: 0;
border: 0;
vertical-align: baseline;
color: black;
background-color: transparent;
border-radius: 0;
}
/* This class is for the output subarea inside the output_area and after
the prompt div. */
div.output_subarea {
overflow-x: auto;
padding: 0.4em;
/* Old browsers */
-webkit-box-flex: 1;
-moz-box-flex: 1;
box-flex: 1;
/* Modern browsers */
flex: 1;
max-width: calc(100% - 14ex);
}
div.output_scroll div.output_subarea {
overflow-x: visible;
}
/* The rest of the output_* classes are for special styling of the different
output types */
/* all text output has this class: */
div.output_text {
text-align: left;
color: #000;
/* This has to match that of the the CodeMirror class line-height below */
line-height: 1.21429em;
}
/* stdout/stderr are 'text' as well as 'stream', but execute_result/error are *not* streams */
div.output_stderr {
background: #fdd;
/* very light red background for stderr */
}
div.output_latex {
text-align: left;
}
/* Empty output_javascript divs should have no height */
div.output_javascript:empty {
padding: 0;
}
.js-error {
color: darkred;
}
/* raw_input styles */
div.raw_input_container {
line-height: 1.21429em;
padding-top: 5px;
}
pre.raw_input_prompt {
/* nothing needed here. */
}
input.raw_input {
font-family: monospace;
font-size: inherit;
color: inherit;
width: auto;
/* make sure input baseline aligns with prompt */
vertical-align: baseline;
/* padding + margin = 0.5em between prompt and cursor */
padding: 0em 0.25em;
margin: 0em 0.25em;
}
input.raw_input:focus {
box-shadow: none;
}
p.p-space {
margin-bottom: 10px;
}
div.output_unrecognized {
padding: 5px;
font-weight: bold;
color: red;
}
div.output_unrecognized a {
color: inherit;
text-decoration: none;
}
div.output_unrecognized a:hover {
color: inherit;
text-decoration: none;
}
.rendered_html {
color: #000;
/* any extras will just be numbers: */
}
.rendered_html :link {
text-decoration: underline;
}
.rendered_html :visited {
text-decoration: underline;
}
.rendered_html h1:first-child {
margin-top: 0.538em;
}
.rendered_html h2:first-child {
margin-top: 0.636em;
}
.rendered_html h3:first-child {
margin-top: 0.777em;
}
.rendered_html h4:first-child {
margin-top: 1em;
}
.rendered_html h5:first-child {
margin-top: 1em;
}
.rendered_html h6:first-child {
margin-top: 1em;
}
.rendered_html * + ul {
margin-top: 1em;
}
.rendered_html * + ol {
margin-top: 1em;
}
.rendered_html pre,
.rendered_html tr,
.rendered_html th,
.rendered_html td,
.rendered_html * + table {
margin-top: 1em;
}
.rendered_html * + p {
margin-top: 1em;
}
.rendered_html * + img {
margin-top: 1em;
}
.rendered_html img,
.rendered_html img.unconfined,
div.text_cell {
/* Old browsers */
display: -webkit-box;
-webkit-box-orient: horizontal;
-webkit-box-align: stretch;
display: -moz-box;
-moz-box-orient: horizontal;
-moz-box-align: stretch;
display: box;
box-orient: horizontal;
box-align: stretch;
/* Modern browsers */
display: flex;
flex-direction: row;
align-items: stretch;
}
@media (max-width: 540px) {
div.text_cell > div.prompt {
display: none;
}
}
div.text_cell_render {
/*font-family: "Helvetica Neue", Arial, Helvetica, Geneva, sans-serif;*/
outline: none;
resize: none;
width: inherit;
border-style: none;
padding: 0.5em 0.5em 0.5em 0.4em;
color: #000;
box-sizing: border-box;
-moz-box-sizing: border-box;
-webkit-box-sizing: border-box;
}
a.anchor-link:link {
text-decoration: none;
padding: 0px 20px;
visibility: hidden;
}
h1:hover .anchor-link,
h2:hover .anchor-link,
h3:hover .anchor-link,
h4:hover .anchor-link,
h5:hover .anchor-link,
h6:hover .anchor-link {
visibility: visible;
}
.text_cell.rendered .input_area {
display: none;
}
.text_cell.rendered
.text_cell.unrendered .text_cell_render {
display: none;
}
.cm-header-1,
.cm-header-2,
.cm-header-3,
.cm-header-4,
.cm-header-5,
.cm-header-6 {
font-weight: bold;
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
.cm-header-1 {
font-size: 185.7%;
}
.cm-header-2 {
font-size: 157.1%;
}
.cm-header-3 {
font-size: 128.6%;
}
.cm-header-4 {
font-size: 110%;
}
.cm-header-5 {
font-size: 100%;
font-style: italic;
}
.cm-header-6 {
font-size: 100%;
font-style: italic;
}
</style>
<style type="text/css">.highlight .hll { background-color: #ffffcc }
.highlight { background: #f8f8f8; }
.highlight .c { color: #408080; font-style: italic } /* Comment */
.highlight .err { border: 1px solid #FF0000 } /* Error */
.highlight .k { color: #008000; font-weight: bold } /* Keyword */
.highlight .o { color: #666666 } /* Operator */
.highlight .ch { color: #408080; font-style: italic } /* Comment.Hashbang */
.highlight .cm { color: #408080; font-style: italic } /* Comment.Multiline */
.highlight .cp { color: #BC7A00 } /* Comment.Preproc */
.highlight .cpf { color: #408080; font-style: italic } /* Comment.PreprocFile */
.highlight .c1 { color: #408080; font-style: italic } /* Comment.Single */
.highlight .cs { color: #408080; font-style: italic } /* Comment.Special */
.highlight .gd { color: #A00000 } /* Generic.Deleted */
.highlight .ge { font-style: italic } /* Generic.Emph */
.highlight .gr { color: #FF0000 } /* Generic.Error */
.highlight .gh { color: #000080; font-weight: bold } /* Generic.Heading */
.highlight .gi { color: #00A000 } /* Generic.Inserted */
.highlight .go { color: #888888 } /* Generic.Output */
.highlight .gp { color: #000080; font-weight: bold } /* Generic.Prompt */
.highlight .gs { font-weight: bold } /* Generic.Strong */
.highlight .gu { color: #800080; font-weight: bold } /* Generic.Subheading */
.highlight .gt { color: #0044DD } /* Generic.Traceback */
.highlight .kc { color: #008000; font-weight: bold } /* Keyword.Constant */
.highlight .kd { color: #008000; font-weight: bold } /* Keyword.Declaration */
.highlight .kn { color: #008000; font-weight: bold } /* Keyword.Namespace */
.highlight .kp { color: #008000 } /* Keyword.Pseudo */
.highlight .kr { color: #008000; font-weight: bold } /* Keyword.Reserved */
.highlight .kt { color: #B00040 } /* Keyword.Type */
.highlight .m { color: #666666 } /* Literal.Number */
.highlight .s { color: #BA2121 } /* Literal.String */
.highlight .na { color: #7D9029 } /* Name.Attribute */
.highlight .nb { color: #008000 } /* Name.Builtin */
.highlight .nc { color: #0000FF; font-weight: bold } /* Name.Class */
.highlight .no { color: #880000 } /* Name.Constant */
.highlight .nd { color: #AA22FF } /* Name.Decorator */
.highlight .ni { color: #999999; font-weight: bold } /* Name.Entity */
.highlight .ne { color: #D2413A; font-weight: bold } /* Name.Exception */
.highlight .nf { color: #0000FF } /* Name.Function */
.highlight .nl { color: #A0A000 } /* Name.Label */
.highlight .nn { color: #0000FF; font-weight: bold } /* Name.Namespace */
.highlight .nt { color: #008000; font-weight: bold } /* Name.Tag */
.highlight .nv { color: #19177C } /* Name.Variable */
.highlight .ow { color: #AA22FF; font-weight: bold } /* Operator.Word */
.highlight .w { color: #bbbbbb } /* Text.Whitespace */
.highlight .mb { color: #666666 } /* Literal.Number.Bin */
.highlight .mf { color: #666666 } /* Literal.Number.Float */
.highlight .mh { color: #666666 } /* Literal.Number.Hex */
.highlight .mi { color: #666666 } /* Literal.Number.Integer */
.highlight .mo { color: #666666 } /* Literal.Number.Oct */
.highlight .sb { color: #BA2121 } /* Literal.String.Backtick */
.highlight .sc { color: #BA2121 } /* Literal.String.Char */
.highlight .sd { color: #BA2121; font-style: italic } /* Literal.String.Doc */
.highlight .s2 { color: #BA2121 } /* Literal.String.Double */
.highlight .se { color: #BB6622; font-weight: bold } /* Literal.String.Escape */
.highlight .sh { color: #BA2121 } /* Literal.String.Heredoc */
.highlight .si { color: #BB6688; font-weight: bold } /* Literal.String.Interpol */
.highlight .sx { color: #008000 } /* Literal.String.Other */
.highlight .sr { color: #BB6688 } /* Literal.String.Regex */