The Weighted Perplexity Benchmark: Tokenizer-Normalized Evaluation for Language Model Comparison

Wait 5 sec.

Published on July 7, 2025 9:43 PM GMTAbstractPerplexity remains the primary intrinsic metric for evaluating language models, yet direct comparison across models with different tokenization schemes presents methodological challenges. We introduce a tokenizer-normalized perplexity metric that enables consistent comparison of language models regardless of their tokenization approaches. Through empirical analysis of 19 language models across five families on WikiText-2, we demonstrate how tokenization differences can affect traditional perplexity measurements by up to 21.6%. Our normalization method reveals the magnitude of tokenization effects on model comparisons and enables analysis of architectural efficiency patterns. We identify Llama Scout as performing substantially worse than models with far fewer active parameters. The Weighted Perplexity Benchmark provides researchers with a principled approach to evaluate language models while controlling for tokenization strategies.Authors: Jessica Taylor, Vie McCoy1. IntroductionAs language model architectures, training approaches, and tokenization strategies diversify, comparing their fundamental text prediction capabilities has become increasingly difficult. Perplexity, defined as the exponentiated average negative log-likelihood of a sequence, remains the standard intrinsic evaluation metric. However, different tokenization schemes create systematic differences in perplexity comparisons that complicate cross-model evaluation.This post addresses the need for tokenizer-independent evaluation metrics through the Weighted Perplexity Benchmark (WPB). Our contributions include:A mathematical framework for normalizing perplexity scores across different tokenization schemesEmpirical evaluation of 19 language models demonstrating significant tokenization effectsAnalysis of how tokenization differences affect model comparisonsIdentification of architectural efficiency patterns when tokenization effects are controlled2. Background and Related Work2.1 Perplexity and Tokenization DependenciesFor a tokenized sequence X=(x1,x2,…,xt).mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}.mjx-math * {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}.mjx-numerator {display: block; text-align: center}.mjx-denominator {display: block; text-align: center}.MJXc-stacked {height: 0; position: relative}.MJXc-stacked > * {position: absolute}.MJXc-bevelled > * {display: inline-block}.mjx-stack {display: inline-block}.mjx-op {display: block}.mjx-under {display: table-cell}.mjx-over {display: block}.mjx-over > * {padding-left: 0px!important; padding-right: 0px!important}.mjx-under > * {padding-left: 0px!important; padding-right: 0px!important}.mjx-stack > .mjx-sup {display: block}.mjx-stack > .mjx-sub {display: block}.mjx-prestack > .mjx-presup {display: block}.mjx-prestack > .mjx-presub {display: block}.mjx-delim-h > .mjx-char {display: inline-block}.mjx-surd {vertical-align: top}.mjx-surd + .mjx-box {display: inline-flex}.mjx-mphantom * {visibility: hidden}.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}.mjx-annotation-xml {line-height: normal}.mjx-menclose > svg {fill: none; stroke: currentColor; overflow: visible}.mjx-mtr {display: table-row}.mjx-mlabeledtr {display: table-row}.mjx-mtd {display: table-cell; text-align: center}.mjx-label {display: table-row}.mjx-box {display: inline-block}.mjx-block {display: block}.mjx-span {display: inline}.mjx-char {display: block; white-space: pre}.mjx-itable {display: inline-table; width: auto}.mjx-row {display: table-row}.mjx-cell {display: table-cell}.mjx-table {display: table; width: 100%}.mjx-line {display: block; height: 0}.mjx-strut {width: 0; padding-top: 1em}.mjx-vsize {width: 0}.MJXc-space1 {margin-left: .167em}.MJXc-space2 {margin-left: .222em}.MJXc-space3 {margin-left: .278em}.mjx-test.mjx-test-display {display: table!important}.mjx-test.mjx-test-inline {display: inline!important; margin-right: -1px}.mjx-test.mjx-test-default {display: block!important; clear: both}.mjx-ex-box {display: inline-block!important; position: absolute; overflow: hidden; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0; width: 1px; height: 60ex}.mjx-test-inline .mjx-left-box {display: inline-block; width: 0; float: left}.mjx-test-inline .mjx-right-box {display: inline-block; width: 0; float: right}.mjx-test-display .mjx-right-box {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}.MJXc-TeX-unknown-R {font-family: monospace; font-style: normal; font-weight: normal}.MJXc-TeX-unknown-I {font-family: monospace; font-style: italic; font-weight: normal}.MJXc-TeX-unknown-B {font-family: monospace; font-style: normal; font-weight: bold}.MJXc-TeX-unknown-BI {font-family: monospace; font-style: italic; font-weight: bold}.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax_AMS'), local('MathJax_AMS-Regular')}@font-face {font-family: MJXc-TeX-ams-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_AMS-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_AMS-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax_Caligraphic Bold'), local('MathJax_Caligraphic-Bold')}@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax_Caligraphic'); font-weight: bold}@font-face {font-family: MJXc-TeX-cal-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Caligraphic-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Caligraphic-Bold.otf') format('opentype')}@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax_Fraktur'), local('MathJax_Fraktur-Regular')}@font-face {font-family: MJXc-TeX-frak-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Fraktur-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Fraktur-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-frak-B; src: local('MathJax_Fraktur Bold'), local('MathJax_Fraktur-Bold')}@font-face {font-family: MJXc-TeX-frak-Bx; src: local('MathJax_Fraktur'); font-weight: bold}@font-face {font-family: MJXc-TeX-frak-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Fraktur-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Fraktur-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Fraktur-Bold.otf') format('opentype')}@font-face {font-family: MJXc-TeX-math-BI; src: local('MathJax_Math BoldItalic'), local('MathJax_Math-BoldItalic')}@font-face {font-family: MJXc-TeX-math-BIx; src: local('MathJax_Math'); font-weight: bold; font-style: italic}@font-face {font-family: MJXc-TeX-math-BIw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Math-BoldItalic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Math-BoldItalic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Math-BoldItalic.otf') format('opentype')}@font-face {font-family: MJXc-TeX-sans-R; src: local('MathJax_SansSerif'), local('MathJax_SansSerif-Regular')}@font-face {font-family: MJXc-TeX-sans-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-sans-B; src: local('MathJax_SansSerif Bold'), local('MathJax_SansSerif-Bold')}@font-face {font-family: MJXc-TeX-sans-Bx; src: local('MathJax_SansSerif'); font-weight: bold}@font-face {font-family: MJXc-TeX-sans-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Bold.otf') format('opentype')}@font-face {font-family: MJXc-TeX-sans-I; src: local('MathJax_SansSerif Italic'), local('MathJax_SansSerif-Italic')}@font-face {font-family: MJXc-TeX-sans-Ix; src: local('MathJax_SansSerif'); font-style: italic}@font-face {font-family: MJXc-TeX-sans-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_SansSerif-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_SansSerif-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_SansSerif-Italic.otf') format('opentype')}@font-face {font-family: MJXc-TeX-script-R; src: local('MathJax_Script'), local('MathJax_Script-Regular')}@font-face {font-family: MJXc-TeX-script-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Script-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Script-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Script-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-type-R; src: local('MathJax_Typewriter'), local('MathJax_Typewriter-Regular')}@font-face {font-family: MJXc-TeX-type-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Typewriter-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Typewriter-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Typewriter-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-cal-R; src: local('MathJax_Caligraphic'), local('MathJax_Caligraphic-Regular')}@font-face {font-family: MJXc-TeX-cal-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Caligraphic-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Caligraphic-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Caligraphic-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-main-B; src: local('MathJax_Main Bold'), local('MathJax_Main-Bold')}@font-face {font-family: MJXc-TeX-main-Bx; src: local('MathJax_Main'); font-weight: bold}@font-face {font-family: MJXc-TeX-main-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Bold.otf') format('opentype')}@font-face {font-family: MJXc-TeX-main-I; src: local('MathJax_Main Italic'), local('MathJax_Main-Italic')}@font-face {font-family: MJXc-TeX-main-Ix; src: local('MathJax_Main'); font-style: italic}@font-face {font-family: MJXc-TeX-main-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Italic.otf') format('opentype')}@font-face {font-family: MJXc-TeX-main-R; src: local('MathJax_Main'), local('MathJax_Main-Regular')}@font-face {font-family: MJXc-TeX-main-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Main-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Main-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Main-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-math-I; src: local('MathJax_Math Italic'), local('MathJax_Math-Italic')}@font-face {font-family: MJXc-TeX-math-Ix; src: local('MathJax_Math'); font-style: italic}@font-face {font-family: MJXc-TeX-math-Iw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Math-Italic.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Math-Italic.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Math-Italic.otf') format('opentype')}@font-face {font-family: MJXc-TeX-size1-R; src: local('MathJax_Size1'), local('MathJax_Size1-Regular')}@font-face {font-family: MJXc-TeX-size1-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size1-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size1-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size1-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-size2-R; src: local('MathJax_Size2'), local('MathJax_Size2-Regular')}@font-face {font-family: MJXc-TeX-size2-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size2-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size2-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size2-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-size3-R; src: local('MathJax_Size3'), local('MathJax_Size3-Regular')}@font-face {font-family: MJXc-TeX-size3-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size3-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size3-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size3-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-size4-R; src: local('MathJax_Size4'), local('MathJax_Size4-Regular')}@font-face {font-family: MJXc-TeX-size4-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Size4-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Size4-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Size4-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-vec-R; src: local('MathJax_Vector'), local('MathJax_Vector-Regular')}@font-face {font-family: MJXc-TeX-vec-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Regular.otf') format('opentype')}@font-face {font-family: MJXc-TeX-vec-B; src: local('MathJax_Vector Bold'), local('MathJax_Vector-Bold')}@font-face {font-family: MJXc-TeX-vec-Bx; src: local('MathJax_Vector'); font-weight: bold}@font-face {font-family: MJXc-TeX-vec-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/eot/MathJax_Vector-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/woff/MathJax_Vector-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/fonts/HTML-CSS/TeX/otf/MathJax_Vector-Bold.otf') format('opentype')}and model M, perplexity is defined as:PPLM(X)=exp(−1tt∑i=1lnpM(xi|x